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23.74,"ASIN":"0195061373","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":20.25,"ASIN":"0195061365","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":22.36,"ASIN":"0195061357","isPreorder":0}],"shippingId":"0195061373::0dKpVwQI7S8Of%2B9fy94Pktt8CpltXVyfjpP9wp1Mmqz6Cc%2FM9RmtZ9qkwQrKMUvUTnvqTRtuBldO2RORpfNZbuMzmrlH%2BZtvinqJk%2B03xkE%3D,0195061365::IUhYsWRpnZNJ1jwaA7mmUiO6JvupOoz9fOYs%2BPavdGIfwFyqdq4wXoJ9LVNJjgvSQauYb9uWRfs6bkdKoW7yZFqavtdRzjdtpgAhyDzae68%3D,0195061357::debHDu3xgvJUibW1pLqKM6wfDlDNGyO0Vl6ExcX3fpZKTQdvQlG%2Fs7GqYkHr1jVKKrqw2FAliiaQfa%2FUwlWk9zJftMk2CX7BKGU2ufcw2EU thorough exposition of the history of mathematics for mathematicians. For non-mathematicians, I recommend Mathematics for the Nonmathematician by the same author. This will be more accessible to the layperson like this one, just put Morris Kline in the search bar at the top of any Amazon page!
This book starts with "modern" topics like projective geometry after leaving off with calculus in the second volume. Like his intuitive calculus book, it alsoThe cool thing is that, unlike his books written in the 60's, this volume was 1990, just at the "reincarnation" of projective geometry and other "historical" math phenomena like quaternions, in new trends like video games and 3D computer animation. Still very relevant, and one of the best ways to learn more advanced math, due to Kline's wonderful teaching style, intuitive explanations, and comparisons with "everyday" physical happenings that FINALLY (at least in my case) helps you get what that equation really "means!"
If you see reviews trashing this or any other math book due to Kindle, don't fault the book!Read more › |
Topics include the algebra of whole numbers, integers, and rational numbers; binary operations involving polynomials; introduction of the laws of exponents; equation-solving techniques for first-degree equations; solving simultaneous linear equations by graphing, substitution and addition methods; word problems. Assistance is available in the Center for Academic Success. No previous knowledge of algebra is assumed. Three class hours weekly. |
• Written by a trusted maths teacher and author, Collins New GCSE Maths Edexcel Linear Grade C Booster Workbook is perfect for students on the D / C borderline who want to make sure they get their Grade C. • Lots of questions at Grade C (with some Grade D and some Grade B) help students focus on the key topics required to get their grade. • Assess how well students are doing with Grade progression maps that clearly show how to move from a Grade D to C. • More in-depth questions assess students' understanding, and test problem-solving and functional maths skills • Perfect for revision, booster classes, retakes, intervention groups. |
MATH 505 Discrete Mathematics
(Either semester/3 credits)
Introduction to the basic mathematical structures and methods used to solve problems that are inherently finite in nature. Topics include logic, Boolean algebra, sets, relations, functions, matrices, induction and elementary recursion, and introductory treatments of combinatorics and graph theory |
The Pearson Guide To Quantitative Aptitude For Competitive Examination
(Paperback)
The Pearson Guide To Quantitative Aptitude For Competitive Examination Book Description
The quantitative aptitude section occupies a very important place in any competitive examination today. The QA questions assess your basic computation skills and the ability to reason mathematically. The Pearson Guide to Quantitative Aptitude for Competitive Examinations is a unique self-help manual that familiarizes you with basic mathematical concepts and enables you to apply them to a range of calculation-based problems. This book is divided into 31 chapters and covers a wide variety of topics. Fundamental principles are explained with the help of easy-to-understand examples. Practice exercises at the end of the chapters further refine problem-solving skills. Quantitative Aptitude, thus, is a complete preparation tool you can t afford to miss.
Features
More than 5,000 solved problems to help you develop problem-solving skills
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The book The Pearson Guide To Quantitative Aptitude For Competitive Examination by Dinesh Khattar
(author) is published or distributed by PEARSON EDUCATION LIMITED [, 9788131719565].
This particular edition was published on or around 2008 date.
The Pearson Guide To Quantitative Aptitude For Competitive Examination is available for use in Paperback binding.
This book by Dinesh Khatt |
9780201726343
ISBN:
0201726343
Edition: 5 Pub Date: 2003 Publisher: Pearson
Summary: This text is organised into 4 main parts - discrete mathematics, graph theory, modern algebra and combinatorics (flexible modular structuring). It includes a large variety of elementary problems allowing students to establish skills as they practice |
Every president, from Johnson to Obama, has made big promises when it comes to "fixing" education in America. And almost every parent, from then until now has asked themselves an essential question—"Is my child getting a good education?" Regardless of neighborhood or income, it's a concern that keeps parents up at night, and the answer rests at the heart of a national firestorm brewing over education.
"Waiting for Superman": What it Means for You and Your Child
ted
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The Messenger Series
Hi. This is the first lecture in MIT's course 18.06, linear algebra, and I'm Gilbert Strang. The text for the course is this book, Introduction to Linear Algebra. And the course web page, which has got a lot of exercises from the past, MatLab codes, the syllabus for the course, is web.mit.edu/18.06. And this is the first lecture, lecture one.
Mathematics | 18.06 Linear Algebra, Spring 2010 | Video Lectures | Lecture 1: The geometry of linear equations
fun Smaller cells are easily visible under a light microscope.
Luminosity: Fλ is the radiative flux at the stellar surface. Energy may be lost due to neutrinos or direct mass loss. Flux: At the Earth's surface, observed flux is Stellar flux
Astrophysics I « Physics Made Easy |
Mathematics
Calculus
Class Level: Freshman
Credits: 4
Department: Mathematics and Computing
Term:
Description: The concept of differentiation is developed using limits and focusing on algebraic, exponential, and logarithmic functions. Applications of derivatives in the sciences and economics are presented, and an introduction to integration concludes the course. T
Overheard
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Franklin College has prepared me by teaching me how to adapt to situations quickly." |
GRE Subject Mathematics
The test consists of approximately 66 multiple-choice questions drawn from courses commonly offered at the undergraduate level.
Approximately 50 percent of the questions involve calculus and its applications — subject matter that can be assumed to be common to the backgrounds of almost all mathematics majors.
About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra and number theory. The remaining questions deal with other areas of mathematics currently studied by undergraduates in many institutions.
The following content descriptions may assist students in preparing for the test. The percents given are estimates; actual percents will vary somewhat from one edition of the test to another.
CALCULUS — 50%
Material learned in the usual sequence of elementary calculus courses — differential and integral calculus of one and of several variables — includes calculus-based applications and connections with coordinate geometry, trigonometry, differential equations and other branches of mathematics.
ALGEBRA — 25%
Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics
Other topics: general topology, geometry, complex variables, probability and statistics, and numerical analysis
The above descriptions of topics covered in the test should not be considered exhaustive; it is necessary to understand many other related concepts. Prospective test takers should be aware that questions requiring no more than a good precalculus background may be quite challenging; such questions can be among the most difficult questions on the test. In general, the questions are intended not only to test recall of information but also to assess test takers' understanding of fundamental concepts and the ability to apply those concepts in various situations |
Geometry Seeing, Doing, Understanding
9780716743613
ISBN:
0716743612
Edition: 3 Pub Date: 2003 Publisher: W H Freeman & Co
Summary: Jacobs innovative discussions, anecdotes, examples, and exercises to capture and hold students' interest. Although predominantly proof-based, more discovery based and informal material has been added to the text to help develop geometric intuition.
Jacobs, Harold R. is the author of Geometry Seeing, Doing, Understanding, published 2003 under ISBN 9780716743613 and 0716743612. One hundred fifty five Geometry ...Seeing, Doing, Understanding textbooks are available for sale on ValoreBooks.com, eighteen used from the cheapest price of $71.65, or buy new starting at $168 |
This book is intended for a graduate course in complex analysis, where the main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two- and three-manifolds,... more...
Part of a series surveying the theory of theta functions which play a central role in the fields of complex analysis, algebraic geometry, number theory and most recently particle physics. This volume constitutes an exposition of theta functions, beginning with their historical roots as analytic functions in one variable. more... |
MCA - I Year - I SEMMATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCEObjectives:
To develop problem-solving techniques and explore topics in a variety of areas of discrete mathematics,including but not limited to logic, graph theory, set theory, recursions, combinatorics, and algorithms.Students will learn to express statements in the language of formal logic and draw conclusions, modelsituations in terms of graph and set theory, find and interpret recursive definitions for mathematicalsequences, use combinatorial methods to approach counting problems.
Outcomes:
• Ability to Illustrate by examples the basic terminology of functions, relations, andsets and demonstrate knowledge of their associated operations.• Ability to Demonstrate in practical applications the use of basic countingprinciples of permutations, combinations, inclusion/exclusion principleand the pigeonhole methodology.• Ability to represent and Apply Graph theory in solving computer science problems |
Discrete Mathematics for Teachers - 05 edition
Summary: Part of a new generation of textbooks for in-service and pre-service teachers at the junior-senior level, this text teaches in three main ways: it extends students' breadth of knowledge beyond, but related to, the topics covered in elementary and middle-grade curriculums; it increases prospective teachers' depth of mathematical understanding by providing problems rich in exploration and mathematical communication; and it models the most current ways of teaching mathe...show morematics. ...show less
0618433929235 |
for sophomore-level or junior/senior-level first courses in linear algebra and assumes calculus as a prerequisite.
This thorough and accessible text, from one of the leading figures in the use of technology in linear algebra, gives students a challenging and broad understanding of the subject. The author infuses key concepts with their modern practical applications to offer students examples of how mathematics is used in the real world. Each chapter contains integrated worked examples and chapter tests. The book stresses the important roles geometry and visualization play in understanding linear algebra.
Features
Extensive applications of linear algebra concepts to a variety of real world situations. These applications introduce new material and show relevance of the material covered. Students learn how theories and concepts of linear algebra can help solve modern day problems.
Interesting and current examples include the application of linear transformations to an airplane, eigenvectors determining the orientation of a space shuttle, and how Google Inc. makes use of linear algebra to rank and order search results.
Abundant computer exercises, more extensive than any other linear algebra book on the market, help students to visualize and discover linear algebra and allow them to explore more realistic applications that are too computationally intensive to work out by hand.
These exercises also provide students with experience in performing matrix computations.
Worked out examples illustrate new concepts, making the material less abstract and helping students quickly build their understanding.
Two chapter testsfor every chapter help students study for exams and get the practice they need to master the material.
A comprehensive MATLAB® appendix gives users all the information they need to use the latest version of MATLAB. A series of MATLAB and Maple™ guides are also available at no additional charge with this text.
New To This Edition
1. New Section on Matrix Arithmetic
One of the longer sections in the previous edition was the section on matrix algebra in Chapter 1. The material in that section has been expanded further for the current edition. Rather than include an overly long revised section, we have divided the material into sections titled Matrix Arithmetic and Matrix Algebra.
2. New Exercises
After seven editions it was quite a challenge to come up with additional original exercises. This eighth edition has more than 130 new exercises. The new exercises are not evenly distributed throughout the book. Some sections have many new exercises and others have few or none.
3. New Subsections and Applications
A new subsection on cross products has been included in Section 3 of Chapter 2. A new application to Newtonian Mechanics has also been added to that section. In Section 4 of Chapter 6 (Hermitian Matrices), a new subsection on the Real Schur Decomposition has been added.
4. New and Improved Notation
The standard notation for the jth column vector of a matrix A is aj , however, there seems to be no universally accepted notation for row vectors. In the MATLAB package, the ith row of A is denoted by A(i, :). In previous editions of this book we used a similar notation a(i, :), however, this notation seems somewhat artificial. For this edition we use the same notations as for a column vector except we put a horizontal arrow above the letter to indicate that the vector is a row vector (an horizontal array) rather than a column vector (a vertical array).
We have also introduced improved notation for the standard Euclidean vector spaces and their complex counterparts.
5. Other Revisions
Various other revisions have been made throughout the text. Many of these revisions were suggested by reviewers.
6. Special Web Site and Supplemental Web Materials
Pearson has developed a special Web site to accompany the 8th edition. This site includes a host of materials for both students and instructors.
Table of Contents
Preface
What's New in the Eighth Edition?
Computer Exercises
Overview of Text
Suggested Course Outlines
Supplementary Materials
Acknowledgments
1. Matrices and Systems of Equations
1.1 Systems of Linear Equations
1.2 Row Echelon Form
1.3 Matrix Arithmetic
1.4 Matrix Algebra
1.5 Elementary Matrices
1.6 Partitioned Matrices
Matlab Exercises
Chapter Test A
Chapter Test B
2. Determinants
2.1 The Determinant of a Matrix
2.2 Properties of Determinants
2.3 Additional Topics and Applications
Matlab Exercises
Chapter Test A
Chapter Test B
3. Vector Spaces
3.1 Definition and Examples
3.2 Subspaces
3.3 Linear Independence
3.4 Basis and Dimension
3.5 Change of Basis
3.6 Row Space and Column Space
Matlab Exercises
Chapter Test A
Chapter Test B
4. Linear Transformations
4.1 Definition and Examples
4.2 Matrix Representations of Linear Transformations
4.3 Similarity
Matlab Exercises
Chapter Test A
Chapter Test B
5. Orthogonality
5.1 The Scalar Product in Rn
5.2 Orthogonal Subspaces
5.3 Least Squares Problems
5.4 Inner Product Spaces
5.5 Orthonormal Sets
5.6 The Gram—Schmidt Orthogonalization Process
5.7 Orthogonal Polynomials
Matlab Exercises
Chapter Test A
Chapter Test B
6. Eigenvalues
6.1 Eigenvalues and Eigenvectors
6.2 Systems of Linear Differential Equations
6.3 Diagonalization
6.4 Hermitian Matrices
6.5 The Singular Value Decomposition
6.6 Quadratic Forms
6.7 Positive Definite Matrices
6.8 Nonnegative Matrices
Matlab Exercises
Chapter Test A
Chapter Test B
7. Numerical Linear Algebra
7.1 Floating-Point Numbers
7.2 Gaussian Elimination
7.3 Pivoting Strategies
7.4 Matrix Norms and Condition Numbers
7.5 Orthogonal Transformations
7.6 The Eigenvalue Problem
7.7 Least Squares Problems
Matlab Exercises
Chapter Test A
Chapter Test B
Appendix: MATLAB
The MATLAB Desktop Display
Basic Data Elements
Submatrices
Generating Matrices
Matrix Arithmetic
MATLAB Functions
Programming Features
M-files
Relational and Logical Operators
Columnwise Array Operators
Graphics
Symbolic Toolbox
Help Facility
Conclusions
Bibliography
A. Linear Algebra and Matrix Theory
B. Applied and Numerical Linear Algebra
C. Books of Related Interest
Answers to Selected Exercises
About the Author(s)
Steven J. Leon is a Chancellor Professor of Mathematics at the University of Massachusetts Dartmouth. He has been a Visiting Professor at Stanford University, ETH Zurich (the Swiss Federal Institute of Technology), KTH (the Royal Institute of Technology in Stockholm), UC San Diego, and Brown University. His areas of specialty are linear algebra and numerical analysis.
Leon is currently serving as Chair of the Education Committee of the International Linear Algebra Society and as Contributing Editor to Image, the Bulletin of the International Linear Algebra Society. He had previously served as Editor-in-Chief of Image from 1989 to 1997. In the 1990's he also served as Director of the NSF sponsored ATLAST Project (Augmenting the Teaching of Linear Algebra using Software Tools). The project conducted 18 regional faculty workshops during the period from 1992–1997.
PearsonChoices
Give your students choices! PearsonChoices products are designed to give your |
This module is included inLens:Community College Open Textbook Collaborative By: CC Open Textbook CollaborativeAs a part of collection: "Elementary Algebra"
Comments:
"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"
Click the
"College Open Textbooks"
link to see all content they endorseSystems of Linear Equations: Objectives
Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
Beginning with the graphical solution of systems, this chapter includes an interpretation of independent, inconsistent, and dependent systems and examples to illustrate the applications for these systems. The substitution method and the addition method of solving a system by elimination are explained, noting when to use each method. The five-step method is again used to illustrate the solutions of value and rate problems (coin and mixture problems), using drawings that correspond to the actual situation.
This module contains the objectives for the chapter "Systems of Linear Equations". |
Books
Geometry & Topology
From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The second part, a very short one, consists of hints... The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem....
This is not an introduction to Hilbert space theory. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book."
Mathematics education in schools has seen a revolution in recent years. Students everywhere expect the subject to be well-motivated, relevant and practical. When such students reach higher education, the traditional development of analysis, often divorced from the calculus they learned at school, seems highly inappropriate. Shouldn't every step in a first course in analysis arise naturally from the student's experience of functions and calculus in school? And shouldn't such a course take every opportunity to endorse and extend the student's basic knowledge of functions? In Yet Another Introduction to Analysis, the author steers a simple and well-motivated path through the central ideas of real analysis. Each concept is introduced only after its need has become clear and after it has already been used informally. Wherever appropriate, new ideas are related to common topics in math curricula and are used to extend the reader's understanding of those topics. In this book the readers are led carefully through every step in such a way that they will soon be predicting the next step for themselves. In this way students will not only understand analysis, but also enjoy it.
The fundamental concepts of general topology are covered in this text whic can be used by students with only an elementary background in calculus. Chapters cover: sets; functions; topological spaces; subspaces; and homeomorphisms.
The object of this book is two-fold -- on the one hand it conveys to mathematical readers a rigorous presentation and exploration of the important applications of analysis leading to numerical calculations. On the other hand, it presents physics readers with a body of theory in which the well-known formulae find their justification. The basic study of fundamental notions, such as Lebesgue integration and theory of distribution, allow the establishment of the following areas: Fourier analysis and convolution Filters and signal analysis time-frequency analysis (gabor transforms and wavelets). The whole is rounded off with a large number of exercises as well as selected worked-out solutions.
Complex Analysis with Mathematica offers a new way of learning and teaching a subject that lies at the heart of many areas of pure and applied mathematics, physics, engineering and even art. This book offers teachers and students an opportunity to learn about complex numbers in a state-of-the-art computational environment. The innovative approach also offers insights into many areas too often neglected in a student treatment, including complex chaos and mathematical art. Thus readers can also use the book for self-study and for enrichment. The use of Mathematica enables the author to cover several topics that are often absent from a traditional treatment. Students are also led, optionally, into cubic or quartic equations, investigations of symmetric chaos, and advanced conformal mapping. A CD is included which contains a live version of the book, and the Mathematica code enables the user to run computer experiments.
This book develops methods which explore some new interconnections and interrelations between analysis and topology and their applications. Emphasis is given to several recent results which have been obtained mainly during the last years and which cannot be found in other books in nonlinear analysis. Interest in this subject area has rapidly increased over the last decade, yet the presentation of research has been confined mainly to journal articles.
A substantially revised edition of the UTM volume, with a view to making the book far more accessible to undergraduates. It contains a larger number of detailed explanations and exercises, together with fully worked solutions to the essential problems and a new chapter on the historical aspects.
This advanced textbook on topology has three unusual features. First, the introduction is from the locale viewpoint, motivated by the logic of finite observations: this provides a more direct approach than the traditional one based on abstracting properties of open sets in the real line. Second, the author freely exploits the methods of locale theory. Third, there is substantial discussion of some computer science applications. As computer scientists become more aware of the mathematical foundations of their discipline, it is appropriate that such topics are presented in a form of direct relevance and applicability. This book goes some way towards bridging the gap for computer scientists.
The purpose of this book is to explore the rich and elegant interplay that exists between the two main currents of mathematics, the continuous and the discrete. Such fundamental notions in discrete mathematics as induction, recursion, combinatorics, number theory, discrete probability, and the algorithmic point of view as a unifying principle are continually explored as they interact with traditional calculus. The book is addressed primarily to well-trained calculus students and those who teach them, but it can also serve as a supplement in a traditional calculus course for anyone who wants to see more. The problems, taken for the most part from probability, analysis, and number theory, are an integral part of the text. There are over 400 problems presented in this book. |
Amparo Gil, Javier Segura, and Nico M. Temme
"This book is inventive and original—an exceedingly valuable collection of methods to compute special functions, bringing together material usually found only in disparate sources." —Van Snyder, Jet Propulsion Laboratory, California Institute of Technology.
Special functions arise in many problems of pure and applied mathematics, mathematical statistics, physics, and engineering. This book provides an up-to-date overview of numerical methods for computing special functions and discusses when to use these methods depending on the function and the range of parameters. Not only are standard and simple parameter domains considered, but methods valid for large and complex parameters are described as well.
The first part of the book (basic methods) covers convergent and divergent series, Chebyshev expansions, numerical quadrature, and recurrence relations. Its focus is on the computation of special functions; however, it is suitable for general numerical courses. Pseudoalgorithms are given to help students write their own algorithms. In addition to these basic tools, the authors discuss other useful and efficient methods, such as methods for computing zeros of special functions, uniform asymptotic expansions, Padé approximations, and sequence transformations. The book also provides specific algorithms for computing several special functions (like Airy functions and parabolic cylinder functions, among others).
Audience This book is intended for researchers in applied mathematics, scientific computing, physics, engineering, statistics, and other scientific disciplines in which special functions are used as computational tools. Some chapters can be used in general numerical analysis courses.
About the Authors Amparo Gil is an Associate Professor of Applied Mathematics at Universidad de Cantabria (Spain). Her main research interests are in numerical analysis, with special emphasis on numerical methods and software for the computation of special functions.
Javier Segura is an Associate Professor of Mathematical Analysis at Universidad de Cantabria (Spain). His research focuses on numerical analysis and, in particular, the computation of special functions and numerical software.
Since his retirement from CWI in Amsterdam, Nico M. Temme has been a guest researcher at CWI. He has served as editor for a number of scientific journals, is the author of Special Functions: An Introduction to the Classical Functions of Mathematical Physics (Wiley, 1996), and wrote four chapters for the Digital Library of Mathematical Functions, the replacement for the Handbook of Mathematical Functions (Abramowitz and Stegun, 1964). He also serves on the editorial board for this project. His research interest is asymptotic analysis and the numerical evaluation of special functions. |
Laplace transformsDocument Transcript
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Fourier and Laplace Transforms
This book presents in a unified manner the fundamentals of both continuous and
discrete versions of the Fourier and Laplace transforms. These transforms play an
important role in the analysis of all kinds of physical phenomena. As a link between
the various applications of these transforms the authors use the theory of signals and
systems, as well as the theory of ordinary and partial differential equations.
The book is divided into four major parts: periodic functions and Fourier series,
non-periodic functions and the Fourier integral, switched-on signals and the Laplace
transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform.
Each part closes with a separate chapter on the applications of the specific transform
to signals, systems, and differential equations. The book includes a preliminary part
which develops the relevant concepts in signal and systems theory and also contains
a review of mathematical prerequisites.
This textbook is designed for self-study. It includes many worked examples, together with more than 450 exercises, and will be of great value to undergraduates
and graduate students in applied mathematics, electrical engineering, physics and
computer science.
Preface
This book arose from the development of a course on Fourier and Laplace
transforms for the Open University of the Netherlands. Originally it was the
intention to get a suitable course by revising part of the book Analysis and
numerical analysis, part 3 in the series Mathematics for higher education
by R. van Asselt et al. (in Dutch). However, the revision turned out to be so
thorough that in fact a completely new book was created. We are grateful
that Educaboek was willing to publish the original Dutch edition of the book
besides the existing series.
In writing this book, the authors were led by a twofold objective:
- the 'didactical structure' should be such that the book is suitable for those
who want to learn this material through self-study or distance teaching,
without damaging its usefulness for classroom use;
- the material should be of interest to those who want to apply the Fourier
and Laplace transforms as well as to those who appreciate a mathematically
sound treatment of the theory.
We assume that the reader has a mathematical background comparable
to an undergraduate student in one of the technical sciences. In particular
we assume a basic understanding and skill in differential and integral calculus. Some familiarity with complex numbers and series is also presumed,
although chapter 2 provides an opportunity to refresh this subject.
The material in this book is subdivided into parts. Each part consists of a
number of coherent chapters covering a specific part of the field of Fourier
and Laplace transforms. In each chapter we accurately state all the learning
objectives, so that the reader will know what we expect from him or her when
studying that particular chapter. Besides this, we start each chapter with an
introduction and we close each chapter with a summary and a selftest. The
selftest consists of a series of exercises that readers can use to test their own
knowledge and skills. For selected exercises, answers and extensive hints
will be available on the CUP website.
Sections contain such items as definitions, theorems, examples, and so on.
These are clearly marked in the left margin, often with a number attached to
them. In the remainder of the text we then refer to these numbered items.
ix
x
Preface
For almost all theorems proofs are given following the heading Proof. The
end of a proof is indicated by a right-aligned black square: . In some cases
it may be wise to skip the proof of a theorem in a first reading, in order not
to lose the main line of argument. The proof can be studied later on.
Examples are sometimes included in the running text, but often they are
presented separately. In the latter case they are again clearly marked in the
left margin (with possibly a number, if this is needed as a reference later on).
The end of an example is indicated by a right-aligned black triangle: .
Mathematical formulas that are displayed on a separate line may or may
not be numbered. Only formulas referred to later on in the text have a number
(right-aligned and in brackets).
Some parts of the book have been marked with an asterisk: ∗ . This concerns elements such as sections, parts of sections, or exercises which are
considerably more difficult than the rest of the text. In those parts we go
deeper into the material or we present more detailed background material.
The book is written in such a way that these parts can be omitted.
The major part of this book has been written by Dr R.J. Beerends and
Dr H.G. ter Morsche. Smaller parts have been written by Drs J.C. van
den Berg and Ir E.M. van de Vrie. In writing this book we gratefully used
the comments made by Prof. Dr J. Boersma and the valuable remarks of
Ir G. Verkroost, Ir R. de Roo and Ir F.J. Oosterhof.
Finally we would like to thank Drs A.H.D.M. van Gijsel, E.D.S. van den
Heuvel, H.M. Welte and P.N. Truijen for their unremitting efforts to get this
book to the highest editorial level possible.
Introduction
Fourier and Laplace transforms are examples of mathematical operations which can
play an important role in the analysis of mathematical models for problems originating from a broad spectrum of fields. These transforms are certainly not new,
but the strong development of digital computers has given a new impulse to both
the applications and the theory. The first applications actually appeared in astronomy, prior to the publication in 1822 of the famous book Th´ orie analytique de la
e
chaleur by Joseph Fourier (1768 – 1830). In astronomy, sums of sine and cosine
functions were already used as a tool to describe periodic phenomena. However, in
Fourier's time one came to the surprising conclusion that the Fourier theory could
also be applied to non-periodic phenomena, such as the important physical problem
of heat conduction. Fundamental for this was the discovery that an arbitrary function could be represented as a superposition of sine and cosine functions, hence, of
simple periodic functions. This also reflects the essential starting point of the various Fourier and Laplace transforms: to represent functions or signals as a sum or an
integral of simple functions or signals. The information thus obtained turns out to
be of great importance for several applications. In electrical networks, for example,
the sinusoidal voltages or currents are important, since these can be used to describe
the operation of such a network in a convenient way. If one now knows how to
express the voltage of a voltage source in terms of these sinusoidal signals, then this
information often enables one to calculate the resulting currents and voltages in the
network.
Applications of Fourier and Laplace transforms occur, for example, in physical
problems, such as heat conduction, and when analyzing the transfer of signals in various systems. Some examples are electrical networks, communication systems, and
analogue and digital filters. Mechanical networks consisting of springs, masses and
dampers, for the production of shock absorbers for example, processes to analyze
chemical components, optical systems, and computer programs to process digitized
sounds or images, can all be considered as systems for which one can use Fourier
and Laplace transforms as well. The specific Fourier and Laplace transform being
used may differ from application to application. For electrical networks the Fourier
and Laplace transforms are applied to functions describing a current or voltage as
function of time. In heat conduction problems, transforms occur that are applied
to, for example, a temperature distribution as a function of position. In the modern theory of digital signal processing, discrete versions of the Fourier and Laplace
transforms are used to analyze and process a sequence of measurements or data,
originating for example from an audio signal or a digitized photo.
In this book the various transforms are all treated in detail. They are introduced
in a mathematically sound way, and many mutually related properties are derived,
so that the reader may experience not only the differences, but above all the great
coherence between the various transforms.
As a link between the various applications of the Fourier and Laplace transforms,
we use the theory of signals and systems as well as the theory of ordinary and partial
1
2
Introduction
FIGURE 0.1
When digitizing a photo, information is lost. Conditions under which a good reconstruction can be obtained will be discussed in part 5. Copyright: Archives de
l'Acad´ mie des Sciences de Paris, Paris
e
differential equations. We do not assume that the reader is familiar with systems
theory. It is, however, an advantage to have some prior knowledge of some of the
elementary properties of linear differential equations.
Considering the importance of the applications, our first chapter deals with signals
and systems. It is also meant to incite interest in the theory of Fourier and Laplace
transforms. Besides this, part 1 also contains a chapter with mathematical preparations for the parts to follow. Readers with a limited mathematical background are
offered an opportunity here to supplement their knowledge.
In part 2 we meet our first transform, specifically meant for periodic functions
or signals. This is the theory of Fourier series. The central issue in this part is to
investigate the information on a periodic function that is contained in the so-called
Fourier coefficients, and especially if and how a periodic function can be described
by these Fourier coefficients. The final chapter of this part examines some of the applications of Fourier series in continuous-time systems and in solving ordinary and
partial differential equations. Differential equations often originate from a physical
problem, such as heat conduction, or from electrical networks.
Part 3 treats the Fourier integral as a transform that is applied to functions which
are no longer periodic. In order to construct a sound theory for the Fourier integral – keeping the applications in mind – we can no longer content ourselves with
the classical notion of a function. In this part we therefore pay special attention in
chapters 8 and 9 to distributions, among which is the well-known delta function.
Usually, a consistent treatment of the theory of distributions is only found in advanced textbooks on mathematics. This book shows that a satisfactory treatment is
also feasible for readers without a background in theoretical mathematics. In the
final chapter of this part, the use of the Fourier integral in systems theory and in
solving partial differential equations is explained in detail.
The Laplace transform is the subject of part 4. This transform is particularly relevant when we are dealing with phenomena that are switched on. In the first chapter
an introduction is given to the theory of complex functions. It is then easier for the
reader to conceive of a Laplace transform as a function defined on the complex numbers. The treatment in part 4 proceeds more or less along the same lines as in parts 2
and 3, with a focus on the applications in systems theory and in solving differential
equations in the closing chapter.
In parts 2, 3 and 4, transforms were considered for functions defined on the
real numbers or on a part of these real numbers. Part 5 is dedicated to the discrete transforms, which are intended for functions or signals defined on the integers.
Introduction
3
These functions or signals may arise by sampling a continuous-time signal, as in the
digitization of an audiosignal (or a photo, as in figure 0.1). In the first chapter of this
part, we discuss how this can be achieved without loss of information. This results
in the important sampling theorem. The second chapter in this part starts with the
treatment of the first discrete transform in this book, which is the so-called discrete
Fourier transform, abbreviated as DFT. The Fast Fourier Transform, abbreviated as
FFT, is the general term for several fast algorithms to calculate the DFT numerically.
In the third chapter of part 5 an FFT, based on the popular situation where the 'length
of the DFT' is a power of two, is treated extensively. In part 5 we also consider the
z-transform, which plays an important role in the analysis of discrete systems. The
final chapter is again dedicated to the applications. This time, the use of discrete
transforms in the study of discrete systems is explained.
CHAPTER 1
Signals and systems
INTRODUCTION
Fourier and Laplace transforms provide a technique to solve differential equations
which frequently occur when translating a physical problem into a mathematical
model. Examples are the vibrating string and the problem of heat conduction. These
will be discussed in chapters 5, 10 and 14.
Besides solving differential equations, Fourier and Laplace transforms are important tools in analyzing signals and the transfer of signals by systems. Hence,
the Fourier and Laplace transforms play a predominant role in the theory of signals
and systems. In the present chapter we will introduce those parts of the theory of
signals and systems that are crucial to the application of the Fourier and Laplace
transforms. In chapters 5, 10, 14 and 19 we will then show how the Fourier and
Laplace transforms are utilized.
Signals and systems are introduced in section 1.1 and then classified in sections
1.2 and 1.3, which means that on the basis of a number of properties they will
be divided into certain classes that are relevant to applications. The fundamental
signals are the sinusoidal signals (i.e. sine-shaped signals) and the time-harmonic
signals. Time-harmonic signals are complex-valued functions (the values of these
functions are complex numbers) which contain only one frequency. These constitute
the fundamental building blocks of the Fourier and Laplace transforms.
The most important properties of systems, treated in section 1.3, are linearity
and time-invariance. It is these two properties that turn Fourier and Laplace transforms into an attractive tool. When a linear time-invariant system receives a timeharmonic signal as input, the resulting signal is again a time-harmonic signal with
the same frequency. The way in which a linear time-invariant system transforms a
time-harmonic signal is expressed by the so-called frequency response, which will
also be considered in section 1.3.
The presentation of the theory of signals and systems, and of the Fourier and
Laplace transforms as well, turns out to be much more convenient and much simpler
if we allow the signals to have complex numbers as values, even though in practice
the values of signals will usually be real numbers. This chapter will therefore assume that the reader has some familiarity with the complex numbers; if necessary
one can first consult part of chapter 2, where the complex numbers are treated in
more detail.
7
8
1 Signals and systems
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know what is meant by a signal and a system
- can distinguish between continuous-time, discrete-time, real, complex, periodic,
power, energy and causal signals
- know what a sinusoidal and a time-harmonic signal are
- are familiar with the terms amplitude, frequency and initial phase of a sinusoidal
and a time-harmonic signal
- know what is meant by the power- and energy-content of a signal and in particular
know what the power of a periodic signal is
- can distinguish between continuous-time, discrete-time, time-invariant, linear,
real, stable and causal systems
- know what is meant by the frequency response, amplitude response and phase
response for a linear time-invariant system
- know the significance of a sinusoidal signal for a real linear time-invariant system
- know the significance of causal signals for linear time-invariant causal systems.
1.1
Signals and systems
To clarify what will be meant by signals and systems in this book, we will first
consider an example.
In figure 1.1 a simple electric network is shown in which we have a series connection of a resistor R, a coil L and a voltage generator. The generator in the network
i (t )
R
+
E (t )
L
_
FIGURE 1.1
Electric network with resistor, coil and voltage generator.
supplies a voltage E(t) and as a consequence a current i(t) will flow in the network.
From the theory of electrical networks it follows that the current i(t) is determined
unambiguously by the voltage E(t), assuming that before we switch on the voltage
generator, the network is at rest and hence there is no current flowing through the coil
and resistor. We say that the current i(t) is uniquely determined by the voltage E(t).
Using the Kirchhoff voltage-law and the current–voltage relationship for the resistor R and coil L, one can derive an equation from which the current i(t) can be
calculated explicitly as a function of time. Here we shall not be concerned with this
derivation and merely state the result:
i(t) =
1 t
e−(t−τ )R/L E(τ ) dτ.
L −∞
(1.1)
This is an integral relationship of a type that we shall encounter quite frequently
in this book. The causal relation between E(t) and i(t) can be represented by the
diagram of figure 1.2. The way in which i(t) follows from E(t) is thus given by the
1.1 Signals and systems
9
E (t )
i (t )
network
FIGURE 1.2
The relation between E(t) and i(t).
relation (1.1). Mathematically, this can be viewed as a mapping which assigns to a
function E(t) the function i(t). In systems theory this mapping is called a system.
The functions E(t) and i(t) are called the input and output respectively.
So a system is determined once the relationship is known between input and
corresponding output. It is of no importance how this relationship can be realized
physically (in our example by the electrical network). Often a system can even
be realized in several ways. To this end we consider the mechanical system in
figure 1.3, where a point-mass P with mass m is connected by an arm to a damper D.
The point-mass P is acted upon by a force F(t). As a result of the force the point P
D
F (t )
m
v (t )
P
FIGURE 1.3
Mechanical system.
moves with velocity v(t). The movement causes a frictional force K in the damper
which is proportional to the velocity v(t), but in direction opposite to the direction
of v(t). Let k be the proportionality constant (the damping constant of the damper),
then K = −kv(t). Using Newton's law one can derive an equation of motion for the
velocity v(t). Given F(t) one can then obtain from this equation a unique solution
for the velocity v(t), assuming that when the force F(t) starts acting, the mechanical
system is at rest. Again we shall not be concerned with the derivation and only state
the result:
v(t) =
1 t
e−(t−τ )k/m F(τ ) dτ.
m −∞
(1.2)
Relation (1.2) defines, in the same way as relation (1.1), a system which assigns to
an input F(t) the output v(t). But when R = k and L = m then, apart from the
dimensions of the physical quantities involved, relations (1.1) and (1.2) are identical
and hence the systems are equal as well. The realizations however, are different!
This way of looking at systems has the advantage that the properties which can
be deduced from a system apply to all realizations. This will in particular be the
case for the applications of the Fourier and Laplace transforms.
It is now the right moment to introduce the concept of a signal. The previous
examples give rise to the following description of the notion of a signal.
Signal
A signal is a function.
Thus, in the example of the electrical network, the voltage E(t) is a signal, which is
defined as a function of time. The preceding description of the concept of a signal
10
1 Signals and systems
is very general and has thus a broad application. It merely states that it is a function.
Even the domain, the set on which the function is defined, and the range, the set of
function-values, are not prescribed. For instance, the yearly energy consumption in
the Netherlands can be considered as a signal. See figure 1.4.
energy consumption
(in 1015 joule)
3000
2500
2000
1500
'81
'82
'83
'84
'85
'86
'87
'88
'89
years
FIGURE 1.4
Energy consumption in the Netherlands.
Now that we have introduced the notion of a signal, it will also be clear from the
foregoing what the concept of a system will mean in this book.
System
A system is a mapping L assigning to an input u a unique output y.
Response
It is customary to represent a system as a 'black box' with an input and an output
(see figure 1.5). The output y corresponding to the input u is uniquely determined
by u and is called the response of the system to the input u.
u
y
L
FIGURE 1.5
System.
When y is the response of a system L to the input u, then, depending on the
context, we use either of the two notations
y = Lu,
u → y.
Our description of the concept of a system allows only one input and one output. In
general more inputs and outputs are possible. In this book we only consider systems
with one input and one output.
In the next section, signals will be classified on the basis of a number of
properties.
1.2 Classification of signals
1.2
Real signal
11
Classification of signals
The values that a signal can attain will in general be real numbers. This has been the
case in all previous examples. Such signals are called real or real-valued signals.
However, in the treatment of Fourier and Laplace transforms it is a great advantage
to work with signals that have complex numbers as values. This means that we will
suppose that a signal f has the form
f = f1 + i f2 ,
Complex signal
Null-signal
where i is the imaginary unit for which i 2 = −1, and f 1 and f 2 are two real-valued
signals. The signal f 1 is called the real part of the complex signal f (notation
Re f ) and f 2 the imaginary part (notation Im f ). If necessary, one can first consult chapter 2, where a review of the theory of complex numbers can be found.
In section 1.2.2 we will encounter an important example of a complex signal, the
so-called time-harmonic signal.
Note that two complex signals are equal if the real parts and the imaginary parts
of the complex signals agree. When for a signal f one has that f 2 = Im f = 0,
then the signal is real. When f 1 = Re f = 0 and f 2 = Im f = 0, then the signal f
is equal to zero. This signal is called the null-signal.
Usually, the signals occurring in practice are real. Hence, when dealing with
results obtained from the application of Fourier and Laplace transforms, it will be
important to consider specifically the consequences for real signals.
1.2.1
Continuous-time signal
Discrete-time signal
Continuous-time and discrete-time signals
In electrical networks and mechanical systems, the signals are a function of the timevariable t, a real variable which may assume all real values. Such signals are called
continuous-time signals. However, it is not necessary that the adjective continuoustime has any relation with time as a variable. It only expresses the fact that the
function is defined on R or a subinterval of R. Hence, a continuous-time signal is a
function defined on R or a subinterval of R. One should not confuse the concept of
a continuous-time signal with the concept of a continuous function as it is used in
mathematics.
In the example of the yearly energy consumption in the Netherlands, the signal
is not defined on R, but only defined for discrete moments of time. Such a signal
can be considered as a function defined on a part of Z, which is the set of integers.
In our example the value at n ∈ Z is the energy consumption in year n. A signal
defined on Z, or on a part of Z, will be called a discrete-time signal.
As a matter of fact we assume in this book, unless explicitly stated otherwise,
that continuous-time signals are defined on the whole of R and discrete-time signals
on the whole of Z. In theory, a signal can always be extended to, respectively, the
whole of R or the whole of Z.
We denote continuous-time signals by f (t), g(t), etc. and discrete-time signals
by f [n], g[n], etc., hence using square brackets surrounding the argument n.
The introduction of continuous-time and discrete-time signals that we have given
above excludes functions of more than one variable. In this book we thus confine
ourselves to signals depending on one variable only. As a consequence we also
confine ourselves to systems where the occurring signals depend on one variable
only.
12
1 Signals and systems
1.2.2
Periodic signals
An important class of signals consists of the periodic signals.
Periodic continuoustime signal
A continuous-time signal f (t) is called periodic with period T > 0 if
Periodic discretetime signal
A discrete-time signal f [n] is called periodic with period N ∈ N if
Sinusoidal signal
In the class of periodic signals the so-called sinusoidal signals play an important
role. These are real signals which, in the continuous-time case, can be written as:
f (t + T ) = f (t)
for t ∈ R.
for n ∈ Z.
f [n + N ] = f [n]
f (t) = A cos(ωt + φ0 )
Amplitude
Frequency
Initial phase
Here A is the amplitude, ω the (radial)frequency and φ0 the initial phase of the
signal. The period T equals T = 2π/ω.
In the discrete-time case the sinusoidal signals have the form:
f [n] = A cos(ωn + φ0 )
Modulus
Argument
for t ∈ R.
for n ∈ N.
Again A is the amplitude and φ0 the initial phase. The period N equals N = 2π/ω.
From this it follows that ω cannot be arbitrary since N is a natural number!
We now introduce an important complex periodic signal which we will repeatedly
come across in Fourier transforms. In order to do so, we will use Euler's formula
for complex numbers, which is treated extensively in chapter 2, but will also be
introduced here in a nutshell.
A complex number z = x +i y with real part x and imaginary part y is represented
in the complex plane by a point with coordinates (x, y). Then the distance r =
x 2 + y 2 to the origin is called the modulus |z| of z, while the angle φ of the radius
vector with the positive real axis is called the argument of z; notation φ = arg z. The
argument is thus determined up to an integral multiple of 2π . See figure 1.6. Using
z
r
y
ϕ
x
FIGURE 1.6
Representation of a complex number in the complex plane.
Euler's formula
polar coordinates in the complex plane, the complex number z can also be written as
z = r (cos φ + i sin φ). For the complex number z = cos φ + i sin φ Euler's formula
gives the following representation as a complex exponential:
eiφ = cos φ + i sin φ.
1.2 Classification of signals
13
Hence, a complex number z with modulus r and argument φ can be written as
z = r eiφ .
This formula has major advantages. One can compute with this complex exponential
as if it were a real exponential. The most important rule is the product formula.
When z 1 = r1 eiφ1 and z 2 = r2 eiφ2 , then one has, as expected,
z 1 z 2 = r1 r2 ei(φ1 +φ2 ) .
This means that |z 1 z 2 | = |z 1 ||z 2 | and arg(z 1 z 2 ) = arg z 1 + arg z 2 . The latter
relationship obviously only holds up to an integral multiple of 2π.
There are, however, also differences with the real exponential. For the real exponential it follows from e x = e y that x = y. This does not hold for complex
exponentials because ei(φ+2π) = eiφ for all real φ, since e2πi = 1.
In this chapter it is not our intention to go into the theory of complex numbers
any further. We will return to this in chapter 2. The preceding part of the theory
of complex numbers is only intended to allow the introduction of the following
complex periodic signal.
Let ω ∈ R and c be a complex constant. The complex signal f (t) is called a timeharmonic continuous-time signal when it is given by
Time-harmonic
continuous-time signal
f (t) = ceiωt
for t ∈ R.
If we write the complex number c as c = Aeiφ0 , where A is the modulus of c and
φ0 the argument, then the time-harmonic signal can also be written as follows:
f (t) = Aeiφ0 eiωt = Aei(ωt+φ0 ) .
Amplitude, frequency and
initial phase of a
time-harmonic signal
For a given value of t, one can represent f (t) in the complex plane by a point on
the circle having the origin as centre and A as radius. At time t = 0 the argument
is equal to φ0 , the initial phase. In the complex plane the signal f (t) corresponds
to a circular movement with constant angular velocity |ω|. See figure 1.7. The
movement is in the clockwise direction if ω < 0 and counter-clockwise if ω > 0.
Note that the time-harmonic signal f (t) is periodic with period 2π/ | ω |. The real
number ω is called the frequency of the time-harmonic signal, A the amplitude and
φ0 the initial phase. Hence, the frequency can be negative and it then loses its
physical meaning. In the complex plane the sign of ω does indicate the direction of
the circular movement and then | ω | is the frequency.
Above we introduced a time-harmonic signal in the continuous-time case. Similarly we define in the discrete-time case:
Let ω ∈ R and c be a complex constant. The discrete-time signal f [n] is called a
time-harmonic discrete-time signal when it is given by
Time-harmonic discrete-time
signal
f [n] = ceiωn
for n ∈ Z.
In contrast to time-harmonic continuous-time signals, a time-harmonic discrete-time
signal will in general not be periodic. Only when | ω | = 2π/N for some positive
integer N will the time-harmonic discrete-time signal be periodic with period N .
A final remark we wish to make concerns the relationship between a timeharmonic signal and a sinusoidal signal. We only consider the continuous-time
case; the discrete-time case follows from this by replacing t with the variable n.
From Euler's formula it follows that
f (t) = Aei(ωt+φ0 ) = A(cos(ωt + φ0 ) + i sin(ωt + φ0 )).
14
1 Signals and systems
ωt
(t = 0)
ϕ0
A
FIGURE 1.7
Time-harmonic signal.
We conclude that a sinusoidal signal is the real part of a time-harmonic signal. A
sinusoidal signal can also be written as a combination of time-harmonic signals. For
this we use the complex conjugate z = x − i y of a complex number z = x + i y.
Then the complex conjugate of eiφ equals e−iφ . Since cos φ = (eiφ + e−iφ )/2,
one has (verify this):
A cos(ωt + φ0 ) =
1.2.3
ceiωt + ce−iωt
2
with c = Aeiφ0 .
Power and energy signals
In electrical engineering it is customary to define the power of an element in an
electric network, through which a current i(t) flows and which has a drop in voltage
v(t), as the product i(t)v(t). The average power over the time-interval [t0 , t1 ] then
equals
t1
1
i(t)v(t) dt.
t1 − t0 t0
For a resistor of 1 ohm one has, ignoring the dimensions of the quantities involved,
that v(t) = i(t), so that in this case the average power equals
t1
1
i 2 (t) dt.
t1 − t0 t0
In signal theory this expression is called the average power of the signal i(t) over the
time-interval [t0 , t1 ]. The limiting-case, in which the average power is taken over
ever increasing time-intervals, leads to the definition of the power of a signal.
1.2 Classification of signals
Power (continuous-time)
The power P of a continuous-time signal f (t) is defined by
A
1
| f (t) |2 dt.
A→∞ 2A −A
P = lim
Power-signal
(continuous-time)
15
(1.3)
If the power of a signal is finite, then the signal is called a power-signal. Notice
that in equation (1.3) we have not used f 2 (t), but | f (t) |2 . The reason is that the
power of a signal should not be negative, while for complex numbers the square
could indeed become negative, since i 2 = −1, and could even result in a complex
value.
An example of a power-signal is a periodic signal. One can calculate the limit in
(1.3) explicitly for a periodic signal f (t) with period T . We formulate the result,
without proof, as follows.
Let f (t) be a periodic continuous-time signal with period T . Then f (t) is a powersignal with power P equal to
Power of a periodic
continuous-time signal
P=
1 T /2
| f (t) |2 dt.
T −T /2
Besides the power-signals, one also has the so-called energy-signals. In the preceding example of the resistor of 1 ohm, the amount of energy absorbed by the
resistor during the time-interval [t0 , t1 ] equals
t1
t0
i 2 (t) dt.
The definition of the energy-content of a signal concerns the time-interval −∞ to
∞. Hence, the energy-content of a continuous-time signal f (t) is defined by
∞
| f (t) |2 dt.
Energy-content
(continuous-time)
E=
Energy-signal
(continuous-time)
A continuous-time signal with a finite energy-content is called an energy-signal.
−∞
For discrete-time signals one uses analogous concepts. Integrals turn into sums. We
will contend ourselves here with stating the definitions.
The power P of a discrete-time signal f [n] is defined by
M
1
| f [n] |2 .
M→∞ 2M
n=−M
Power (discrete-time)
P = lim
Power-signal (discrete-time)
If the power of a discrete-time signal is finite, then the signal is called a powersignal. For a periodic discrete-time signal f [n] with period N one has, as for the
continuous-time case, that
Power of a periodic
discrete-time signal
P=
1 N −1
| f [n] |2 .
N n=0
The energy-content E of a discrete-time signal f [n] is defined by
Energy-content
(discrete-time)
Energy-signal (discrete-time)
E=
∞
| f [n] |2 .
n=−∞
If the energy-content of a signal is finite, then the signal is called an energy-signal.
16
1 Signals and systems
1.2.4
Causal signals
A final characteristic of a signal that we want to mention in this section has to do
with causality. In the next section the concept of causality is also introduced for
systems. In that context it will be easier to understand why the following definition
of a causal signal is used.
A continuous-time signal f (t), or a discrete-time signal f [n] respectively, is called
causal if
Causal
f (t) = 0
for t < 0,
f [n] = 0
for n < 0.
Periodic signals are thus not causal, with the exception of the null-signal. If a signal
has the property that f (t) = 0 for t < t0 and some t0 , then we call t0 the switch-on
time of the signal f (t). Note that this definition does not fix the switch-on time
uniquely. If a signal has the property that f (t) = 0 for all t < 1, then besides t = 1
as switch-on time, one can also use t = 0. Similar definitions apply to discrete-time
signals. Notice that causal signals have switch-on time t = 0, or n = 0 in the
discrete case.
Switch-on time
EXERCISES
1.1
Given are the two sinusoidal signals f 1 (t) = A1 cos(ωt + φ1 ) and f 2 (t) =
A2 cos(ωt +φ2 ) with the same frequency ω. Show that the sum f 1 (t)+ f 2 (t) is also
a sinusoidal signal with frequency ω and determine its amplitude and initial phase.
1.2
Show that the sum of two time-harmonic signals f 1 (t) and f 2 (t) with the same frequency ω and with amplitudes A1 and A2 and initial phases φ1 and φ2 respectively
is again a time-harmonic signal with frequency ω and determine its amplitude and
initial phase.
1.3
Show that the sum of two discrete-time sinusoidal signals with the same frequency is
again a discrete-time sinusoidal signal and determine its amplitude and initial phase.
1.4
Show that the sum of two discrete-time time-harmonic signals with the same frequency is again a discrete-time time-harmonic signal and determine its amplitude
and initial phase.
1.5
Calculate the power of the sinusoidal signal f (t) = A cos(ωt + φ0 ).
1.6
Calculate the energy-content of the signal f (t) given by
f (t) =
e−t
0
for t > 0,
for t ≤ 0.
1.7
Calculate the power of the periodic discrete-time signal f [n] = cos(nπ/2).
1.8
Calculate the energy-content of the causal discrete-time signal f [n] given by
f [n] =
1.3
e−n
0
for n ≥ 0,
for n < 0.
Classification of systems
Besides signals one can also classify systems. It is customary to do this on the basis
of the type of signals being processed by that system.
1.3 Classification of systems
1.3.1
Continuous-time system
Discrete-time system
17
Continuous-time and discrete-time systems
A continuous-time system is a system for which the input and output signals are
continuous-time signals.
A discrete-time system is a system for which the input and output signals are discretetime signals.
Discrete-time systems are of major importance in the modern field of digital signal
processing (e.g. Van den Enden and Verhoeckx (in Dutch), 1987 – see Literature at
the back of the book).
1.3.2
Linear time-invariant systems
We will now formulate two properties of systems that are crucial for the application of Fourier and Laplace transforms. The first property concerns the linearity of
systems, while the second one concerns the time-invariance.
DEFINITION 1.1
Linear system
A system L is called linear if for each two inputs u and v and arbitrary complex a
and b one has
L(au + bv) = aLu + bLv.
(1.4)
For continuous-time systems this property can be denoted as
au(t) + bv(t) → a(Lu)(t) + b(Lv)(t)
and for discrete-time systems as
au[n] + bv[n] → a(Lu)[n] + b(Lv)[n].
Note that in the preceding linear combination au + bv of the signals u and v, the
coefficients a and b may be complex. Since in general we assume that the signals
are complex, we will also allow complex numbers a and b in (1.4).
EXAMPLE 1.1
Let L be the continuous-time system described by equation (1.1). In order to show
that the system is linear, we simply have to use the fact that integration is a linear
operation. The proof then proceeds as follows:
1 t
e−(t−τ )R/L (au(τ ) + bv(τ )) dτ
L −∞
a t
b t
=
e−(t−τ )R/L u(τ ) dτ +
e−(t−τ )R/L v(τ ) dτ
L −∞
L −∞
au(t) + bv(t) →
= a(Lu)(t) + b(Lv)(t).
EXAMPLE 1.2
For a discrete-time system L, the response y[n] to an input u[n] is given by
y[n] =
u[n] + 2u[n − 1] + u[n − 2]
4
for n ∈ Z.
18
1 Signals and systems
The output at time n is apparently a weighted average of the input u[n] at times n,
n − 1, n − 2. We verify the linearity of this system as follows:
au[n] + bv[n] →
au[n] + bv[n] + 2au[n − 1] + 2bv[n − 1] + au[n − 2] + bv[n − 2]
4
a(u[n] + 2u[n − 1] + u[n − 2]) b(v[n] + 2v[n − 1] + v[n − 2])
+
=
4
4
= a(Lu)[n] + b(Lv)[n].
The second property, the so-called time-invariance, has to do with the behaviour
of a system with respect to time-delays, or, more generally, shifts in the variable t
or n. When a system has the property that a time-shift in the input results in the
same time-shift in the output, then the system is called time-invariant. A precise
description is given in the following definition.
DEFINITION 1.2
Time-invariant system
A continuous-time system is called time-invariant if for each input u(t) and each
t0 ∈ R one has:
if
u(t) → y(t)
then
u(t − t0 ) → y(t − t0 ).
(1.5)
A discrete-time system is called time-invariant if for each input u[n] and each n 0 ∈ Z
one has:
if
EXAMPLE 1.3
u[n] → y[n]
then
u[n − n 0 ] → y[n − n 0 ].
(1.6)
Once again we consider the continuous-time system described by (1.1). This is a
time-invariant system, which can be verified as follows:
u(t − t0 ) →
1 t
e−(t−τ )R/L u(τ − t0 ) dτ
L −∞
1 t−t0 −(t−t0 −ξ )R/L
=
e
u(ξ ) dξ = y(t − t0 ).
L −∞
In this calculation a new integration variable ξ = τ − t0 was introduced.
EXAMPLE 1.4
Again consider the discrete-time system given in example 1.2. This system is timeinvariant as well. This immediately follows from condition (1.6):
u[n − n 0 ] →
u[n − n 0 ] + 2u[n − 1 − n 0 ] + u[n − 2 − n 0 ]
= y[n − n 0 ].
4
Linear time-invariant system
A system which is both linear and time-invariant is called a linear time-invariant
system. It is precisely these linear time-invariant systems for which the Fourier
and Laplace transforms form a very attractive tool. These systems have the nice
property that the response to a time-harmonic signal, whenever this response exists,
is again a time-harmonic signal with the same frequency. However, the existence
of the response to a time-harmonic signal is not ensured. This has to do with the
eigenfrequencies and the stability of the system. We will discuss the role of stability
following our next theorem. Treatment of eigenfrequencies will be postponed until
chapter 5.
THEOREM 1.1
Let L be a linear time-invariant system and u a time-harmonic input with frequency
ω ∈ R for which the response exists. Then the output y is also a time-harmonic
signal with the same frequency ω.
1.3 Classification of systems
19
Proof
We will only prove the case of a continuous-time system. The proof for a discretetime system can be given analogously. Let u(t) be a time-harmonic input with
frequency ω and let y(t) be the corresponding output. Hence, u(t) = ceiωt , where
c is a complex constant and ω ∈ R. The system is time-invariant and so one has for
each τ ∈ R:
ceiω(t−τ ) → y(t − τ ).
On the other hand one has that
u(t − τ ) = ceiω(t−τ ) = ce−iωτ eiωt = e−iωτ u(t).
Because of the linearity of the system, the response to u(t − τ ) is then also equal to
e−iωτ y(t). We conclude that for each t ∈ R and τ ∈ R one has
y(t − τ ) = e−iωτ y(t).
Substitution of t = 0, and then replacing −τ by t, leads to y(t) = y(0)eiωt , so y(t)
is again a time-harmonic signal with frequency ω.
Frequency response
System function
Transfer function
From the preceding proof it follows that the response y(t) to the input u(t) = eiωt
equals Ceiωt for some complex constant C, which can still depend on ω, meaning
that C is a function of ω. We call this function the frequency response of the system.
Often one also uses the term system function or transfer function. For continuoustime systems the frequency response will be denoted by H (ω) and for discrete-time
systems by H (eiω ). The reason for the different notation in the discrete case will
not be explained until chapter 19.
The frequency response of a linear time-invariant system is thus defined by the
following relations:
eiωt → H (ω)eiωt
(1.7)
for a continuous-time system and
eiωn → H (eiω )eiωn
(1.8)
for a discrete-time system. The frequency response H (ω) is complex and so can be
written in the form
H (ω) = | H (ω) | ei (ω) .
Amplitude response
Phase response
EXAMPLE 1.5
Here | H (ω) | and (ω) are, respectively, the modulus and the argument of H (ω).
The function | H (ω) | is called the amplitude response and (ω) the phase response.
Once more we consider the system described by (1.1), which originated from the
RL-network of figure 1.1. The response y(t) to the input u(t) = eiωt equals
1 ∞ −ξ R/L iω(t−ξ )
1 t
e−(t−τ )R/L eiωτ dτ =
e
e
dξ
L −∞
L 0
1 ∞ −ξ R/L −iωξ
e
e
dξ eiωt .
=
L 0
y(t) =
20
1 Signals and systems
This expression already shows that the response is again a time-harmonic signal
with the same frequency ω. The frequency response in this example equals
H (ω) =
1 ∞ −ξ R/L −iωξ
e
e
dξ .
L 0
In chapter 6 we will learn how to calculate this kind of integral. This is because we
are already dealing here with a Fourier transform. The result is:
H (ω) =
EXAMPLE 1.6
1
.
R + iωL
We consider the discrete-time system given in example 1.2, and calculate the response y[n] to the input u[n] = eiωn as follows:
y[n] =
eiωn (1 + 2e−iω + e−2iω )
eiωn + 2eiω(n−1) + eiω(n−2)
=
.
4
4
Again we see that the response is a time-harmonic signal. Apparently the frequency
response equals
H (eiω ) =
1.3.3
(1 + 2e−iω + e−2iω )
.
4
Stable systems
Prior to theorem 1.1 we observed that the response to a time-harmonic signal doesn't
always exist. For so-called stable systems, however, the response exists for all frequencies, and so the frequency response is defined for each ω. In order to describe
what will be meant by a stable system, we first give the definition of a bounded
signal.
Bounded signal
A continuous-time signal f (t), or a discrete-time signal f [n] respectively, is called
bounded if there exists a positive constant K such that
| f (t) | ≤ K
for t ∈ R (continuous-time),
| f [n] | ≤ K
for n ∈ N (discrete-time).
The definition of a stable system is now as follows.
DEFINITION 1.3
Stable system
A system L is called stable if the response to each bounded signal is again bounded.
A time-harmonic signal is an example of a bounded signal, since
ceiωt = | c | eiωt = | c |
for t ∈ R.
Hence, for a stable system the response to the input ceiωt exists and this response is
bounded too.
1.3.4
Real system
Real systems
In our description of a system we assumed that the inputs and outputs are complex.
In principle it is then possible that a real input leads to a complex output. A system
is called real if the response to every real input is again real. Systems occurring in
practice are mostly real.
If we apply a complex input u to a real linear system, so u = u 1 + iu 2 with u 1
and u 2 real signals, then, by the linearity property, the response y will equal y1 +i y2
1.3 Classification of systems
21
where y1 is the response to u 1 and y2 the response to u 2 . Since the system is real,
the signals y1 and y2 are also real. For real linear systems one thus has the following
property.
The response to the real part of an input u is equal to the real part of the output y
and the response to the imaginary part of u is equal to the imaginary part of y.
One can use this property of real systems to calculate the response to a sinusoidal
signal in a clever way in the case when the frequency response is known. For the
continuous-time case this can be done as follows.
Let u(t) be the given sinusoidal input u(t) = A cos(ωt + φ0 ). Using Euler's
formula we can consider the signal u(t) as the real part of the time-harmonic signal
ceiωt with c = Aeiφ0 . According to the definition of the frequency response, the
response to this signal equals cH (ω)eiωt . The system being real, this implies that
the response to the sinusoidal signal u(t) is equal to the real part of cH (ω)eiωt . In
order to calculate this real part, we write H (ω) in the form
H (ω) = | H (ω) | ei (ω) ,
where | H (ω) | is the modulus of H (ω) and
y(t) to the input u(t) we then find that
(ω) the argument. For the response
y(t) = Re(Aeiφ0 | H (ω) | ei (ω) eiωt ) = A | H (ω) | cos(ωt + φ0 +
(ω)).
For real systems and real signals one can thus benefit from working with complex
numbers. The response y(t) is again a sinusoidal signal with amplitude A | H (ω) |
and initial phase φ0 + (ω). The amplitude is multiplied by the factor | H (ω) | and
one has a phase-shift (ω). On the basis of these properties it is clear why | H (ω) |
is called the amplitude response and (ω) the phase response of the system.
1.3.5
Causal systems
A system for which the response to an input at any given time t0 only depends on
the input at times prior to t0 , hence, only on the 'past' of the input, is called a causal
system. A precise formulation is as follows.
DEFINITION 1.4
Causal system
A continuous-time system L is called causal if for each two inputs u(t) and v(t) and
for each t0 ∈ R one has:
if u(t) = v(t) for t < t0 , then (Lu)(t) = (Lv)(t) for t < t0 .
(1.9)
A discrete-time system L is called causal if for each two inputs u[n] and v[n] and
for each n 0 ∈ Z one has:
if u[n] = v[n] for n < n 0 , then (Lu)[n] = (Lv)[n] for n < n 0 .
(1.10)
Systems occurring in practice are mostly causal. The notion of causality can be
simplified for linear time-invariant systems, since the following theorem holds for
linear time-invariant systems.
THEOREM 1.2
A linear time-invariant system L is causal if and only if the response to each causal
input is again causal.
Proof
Once again we confine ourselves to the case of continuous-time systems, since the
proof for discrete-time systems is almost exactly the same and there are only some
differences in notation.
22
1 Signals and systems
Assume that the system L is causal and let u(t) be a causal input. This means that
u(t) = 0 for t < 0, so u(t) equals the null-signal for t < 0. Since for linear systems
the response to the null-signal is again the null-signal, it follows from (1.9) that the
response y(t) to u(t) has to agree with the null-signal for t < 0, which means that
y(t) is causal.
Next assume that the response to each causal input is again a causal signal. Let
u(t) and v(t) be two inputs for which u(t) = v(t) for t < t0 . Now introduce
w(t) = u(t + t0 ) − v(t + t0 ). Then the signal w(t), and so the response (Lw)(t) as
well, is causal. Since the system is linear and time-invariant, one has that (Lw)(t) =
(Lu)(t + t0 ) − (Lv)(t + t0 ). Hence, (Lu)(t + t0 ) = (Lv)(t + t0 ) for t < 0, that is,
(Lu)(t) = (Lv)(t) for t < t0 . This finishes the proof.
EXAMPLE 1.7
The linear time-invariant system described by (1.1) is causal. If we substitute a
causal input u(t) in (1.1), then for t < 0 the integrand equals 0 on the interval of
integration (−∞, t], and so the integral also equals 0 for t < 0.
EXAMPLE 1.8
The discrete-time system introduced in example 1.2 is causal. We have seen that
the system is linear and time-invariant. Substitution of a causal signal u[n] in the
relation
y[n] =
u[n] + 2u[n − 1] + u[n − 2]
4
leads for n < 0 to the value y[n] = 0. So the response is causal and hence the
system is causal.
1.3.6
Systems described by differential equations
For an important class of linear time-invariant continuous-time systems, the relation
between the input u(t) and the output y(t) is described by a differential equation of
the form+ b0 u.
= bn n + bn−1 n−1 + · · · + b1
dt
dt
dt
For example, in electrical networks one can derive these differential equations from
the so-called Kirchhoff laws. The differential equation above is called a linear differential equation with constant coefficients, and for these there exist general solution
methods. In this chapter we will not pursue these matters any further. In chapters 5,
10 and 14, systems described by differential equations will be treated in more detail.
1.3.7
Systems described by difference equations
For the linear time-invariant discrete-time case, the role of differential equations is
taken over by the so-called difference equations of the type
b0 y[n] + b1 y[n − 1] + · · · + b M y[n − M]
= a0 u[n] + a1 u[n − 1] + · · · + a N [n − N ].
This equation for the input u[n] and the output y[n] is called a linear difference
equation with constant coefficients. The systems described by difference equations
1.3 Classification of systems
23
are of major importance for the practical realization of systems. These will be discussed in detail in chapter 19.
EXERCISES
1.9
For a continuous-time system the response y(t) to an input u(t) is given by
y(t) =
a
b
c
d
e
f
g
1.10
t
t−1
u(τ ) dτ.
Show that the system is real.
Show that the system is stable.
Show that the system is linear time-invariant.
Calculate the response to the input u(t) = cos ωt.
Calculate the response to the input u(t) = sin ωt.
Calculate the amplitude response of the system.
Calculate the frequency response of the system.
For a discrete-time system the response y[n] to an input u[n] is given by
y[n] = u[n − 1] − 2u[n] + u[n + 1].
a
b
c
d
1.11
Cascade system
Show that the system is linear time-invariant.
Is the system causal? Justify your answer.
Is the system stable? Justify your answer.
Calculate the frequency response of the system.
Two linear time-invariant continuous-time systems L1 and L2 are given with, respectively, frequency response H1 (ω) and H2 (ω), amplitude response A1 (ω) and
A2 (ω) and phase response 1 (ω) and 2 (ω). The system L is a cascade connection of L1 and L2 as drawn below.
u
y
L1
L2
FIGURE 1.8
Cascade connection of L1 and L2 .
a
b
c
1.12
Determine the frequency response of L.
Determine the amplitude response of L.
Determine the phase response of L.
For a linear time-invariant discrete-time system the frequency response is given by
H (eiω ) = (1 + i)e−2iω .
a Determine the amplitude response of the system.
b Determine the response to the input u[n] = 1 for all n.
c Determine the response to the input u[n] = cos ωn.
d Determine the response to the input u[n] = cos2 2ωn.
SUMMARY
An important field for the applications of the Fourier and Laplace transforms is
signal and systems theory. In this chapter we therefore introduced a number of
important concepts relating to signals and systems.
24
1 Signals and systems
Mathematically speaking, a system can be interpreted as a mapping which assigns
in a unique way an output y to an input u. What matters here is the relation between
input and output, not the physical realization of the system.
Mathematically, a signal is a function defined on R or Z. The function values are
allowed to be complex numbers.
In practice, various types of signal occur. Hence, the signals in this book were
subdivided into continuous-time signals, which are defined on R, and discrete-time
signals, which are defined on Z. An important class of signals is the periodic signals. Another subdivision is obtained by differentiating between energy- and powersignals. Signals occurring in practice are mostly real-valued. These are called real
signals. An important real signal is the sinusoidal signal which, for a given frequency ω, initial phase φ0 and amplitude A, can be written as f (t) = A cos(ωt +φ0 )
in the continuous-time case and as f [n] = A cos(ωn +φ0 ) in the discrete-time case.
The sinusoidal signals are periodic in the continuous-time case. In general this is
not true in the discrete-time case. A sinusoidal signal can be considered as the real
part of a complex signal, the so-called time-harmonic signal ceiωt or ceiωn , with
frequency ω and complex constant c.
Time-harmonic signals play an important role, on the one hand in all of the
Fourier transforms, and on the other hand in systems that are both linear and timeinvariant. These are precisely the systems suitable for an analysis using Fourier and
Laplace transforms, because these linear time-invariant systems have the property
that time-harmonic input result in outputs which are again time-harmonic with the
same frequency. For a linear time-invariant system, the relation between a timeharmonic input u and the response y can be expressed using the so-called frequency
response H (ω) or H (eiω ) of the system:
eiωt → H (ω)eiωt
eiωn → H (eiω )eiωn
(continuous-time system),
(discrete-time system).
The modulus of the frequency response, H (ω) or H (eiω ) respectively, is called
the amplitude response, while the argument of the frequency response is called the
phase response of the system. Of practical importance are furthermore the real, the
stable and the causal systems.
Real systems have the property that the response to a real input is again real. The
response of a sinusoidal signal is then a sinusoidal signal as well, with the same
frequency.
Stable systems have the property that bounded inputs result in outputs that are
also bounded. For these systems the frequency response is well-defined for each ω.
The response of a causal system at a specific time t depends only on the input at
earlier times, hence only on the 'past' of the input. For linear time-invariant systems
causality means that the response to a causal input is causal too. Here a signal is
called causal if it is switched on at time t0 ≥ 0.
SELFTEST
Calculate the power of the signal f (t) = A cos ωt + B cos(ωt + φ0 ).
Calculate the energy-content of the signal f (t) given by
for t < 0,
0
sin(π t) for 0 ≤ t < 1,
f (t) =
0
for t ≥ 1.
1.13
a
b
1.14
Show that the power of the time-harmonic signal f (t) = ceiωt equals | c |2 .
1.3 Classification of systems
1.15
a
b
Calculate the power of the signal f [n] = A cos(π n/4) + B sin(π n/2).
Calculate the energy-content of the signal f [n] given by
f [n] =
1.16
25
0
1 n
2
for n < 0,
for n ≥ 0.
For a linear time-invariant continuous-time system the frequency response is given
by
H (ω) =
eiω
.
ω2 + 1
a Calculate the amplitude and phase response of the system.
b The time-harmonic signal u(t) = ieit is applied to the system. Calculate the
response y(t) to u(t).
1.17
For a real linear time-invariant discrete-time system the amplitude response A(eiω )
and phase response (eiω ) are given by A(eiω ) = 1/(1 + ω2 ) and (eiω ) = ω
respectively. To the system the sinusoidal signal u[n] = sin 2n is applied.
a Is the signal u[n] periodic? Justify your answer.
b Show that the output is also a sinusoidal signal and determine the amplitude and
initial phase of this signal.
1.18
For a continuous-time system the relation between the input u(t) and the corresponding output y(t) is given by
y(t) = u(t − t0 ) +
a
b
c
d
1.19
t
t−1
u(τ ) dτ.
For which values of t0 is the system causal?
Show that the system is stable.
Is the system real? Justify your answer.
Calculate the response to the sinusoidal signal u(t) = sin πt.
For a discrete-time system the relation between the input u[n] and the corresponding
output y[n] is given by
n
y[n] = u[n − n 0 ] +
u[l] .
l=n−2
a
b
c
d
For which values of n 0 ∈ Z is the system causal?
Show that the system is stable.
Is the system real? Justify your answer.
Calculate the response to the input u[n] = cos πn.
CHAPTER 2
Mathematical prerequisites
INTRODUCTION
In this chapter we present an overview of the necessary basic knowledge that will
be assumed as mathematical prerequisite in the chapters to follow. It is presupposed
that the reader already has previous knowledge of the subject matter in this chapter.
However, it is advisable to read this chapter thoroughly, and not only because one
may discover, and fill in, possible gaps in mathematical knowledge. This is because
in Fourier and Laplace transforms one uses the complex numbers quite extensively;
in general the functions that occur are complex-valued, sequences and series are sequences and series of complex numbers or of complex-valued functions, and power
series are in general complex power series. In introductory courses one usually restricts the treatment of these subjects to real numbers and real functions. This will
not be the case in the present chapter. Complex numbers will play a prominent role.
In section 2.1 the principal properties of the complex numbers are discussed, as
well as the significance of the complex numbers for the zeros of polynomials. In
section 2.2 partial fraction expansions are treated, which is a technique to convert a
rational function into a sum of simple fractions. Section 2.3 contains a short treatment of differential and integral calculus for complex-valued functions, that is, functions which are defined on the real numbers, but whose function values may indeed
be complex numbers. One will find, however, that the differential and integral calculus for complex-valued functions hardly differs from the calculus of real-valued
functions. In the same section we also introduce the class of piecewise continuous
functions, and the class of piecewise smooth functions, which are of importance
later on for the Fourier and Laplace transforms. In section 2.4 general convergence
properties, and other fundamental properties, of sequences and series of complex
numbers are considered. Again there will be some similarities with the theory of
sequences and series of real numbers. Fourier series are series of complex-valued
functions. Therefore some attention is paid to series of functions in section 2.4.3.
Finally, we treat the complex power series in section 2.5. These possess almost
identical properties to real power series.
27
28
2 Mathematical prerequisites
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- can perform calculations with complex numbers, in cartesian as well as in polar
and exponential form
- know that by using complex numbers, a polynomial can be factorized entirely into
linear factors and that you are able to perform this factorization in simple cases
- know what is meant by the nth roots of unity
- know the technique of (complex) partial fraction expansions
- can apply differential and integral calculus to complex-valued functions
- know what is meant by a piecewise continuous and a piecewise smooth function
- can apply the theory of sequences and series to sequences and series with complex
terms
- are familiar with the concept of radius of convergence for complex power series
and can calculate the radius of convergence in simple situations
- know the properties of the sum of a power series.
2.1
Complex numbers, polynomials and rational functions
2.1.1
Elementary properties of complex numbers
The complex numbers are necessary in order to determine the solutions of all
quadratic equations. The equation x 2 − 2x + 5 = 0 has no solution in the real
numbers. This is because by completing the square it follows that x 2 − 2x + 5 =
(x − 1)2 + 4 > 0 for all real x. If we now introduce the imaginary unit i, which by
definition satisfies
i 2 = −1,
and subsequently the complex number x = 1 + 2i, then (x − 1)2 + 4 = (2i)2 + 4 =
−4 + 4 = 0. Apparently the complex number x = 1 + 2i is a solution of the
given equation. Complex numbers are therefore defined as the numbers z that can
be written as
z = x + iy
Real part
Imaginary part
with x, y ∈ R.
(2.1)
The collection of all these numbers is denoted by C. The real number x is called the
real part of z and denoted by x = Re z. The real number y is called the imaginary
part of z and denoted by y = Im z. Two complex numbers are equal if the real parts
and the imaginary parts are equal. For the complex number z one has that z = 0 if
Re z = 0 and Im z = 0. For the addition and multiplication of two complex numbers
z = x + i y and w = u + iv one has by definition that
z + w = (x + i y) + (u + iv) = (x + u) + i(y + v),
z · w = (x + i y)(u + iv) = (xu − yv) + i(xv + yu).
For subtraction and division one subsequently finds:
z − w = (x − u) + i(y − v),
z
x + iy
x + i y u − iv
xu + yv
yu − xv
=
=
·
= 2
+i 2
w
u + iv
u + iv u − iv
u + v2
u + v2
for w = 0.
We see that both sum and product as well as difference and quotient of two complex
numbers have been written in the form z = Re z + iIm z again. The set C is an
2.1 Complex numbers and functions
Complex conjugate
29
extension of R, since a real number x can be written as x = x + 0 · i. The complex
conjugate of a complex number z is defined as
z = x − i y.
Note that zz = x 2 + y 2 and z = z. One can easily check that the complex conjugate
has the following properties:
z + w = z + w,
zw = zw,
z
w
=
z
w
if w = 0.
Using the complex conjugate one can express the real and imaginary part of a complex number as follows:
Re z =
Cartesian coordinates
Complex plane
Real axis
Imaginary axis
z+z
2
and
Im z =
z−z
.
2i
Since we need two real numbers x and y to describe a complex number z = x +
i y, one can assign to each complex number a point in the plane with rectangular
coordinates x and y, as shown in figure 2.1. The coordinates x and y are called
the cartesian coordinates. Figure 2.1 moreover shows the complex conjugate of z.
The plane in figure 2.1 is called the complex plane and the axes the real axis and the
imaginary axis. In this figure we see how the location of z can also be determined
Im z
z = x + iy
y
r
ϕ
0
–ϕ
x
Re z
r
–y
z = x – iy
FIGURE 2.1
The complex number z and its complex conjugate z.
Polar coordinates
by using polar coordinates r and φ. Here r is the distance from z to the origin and φ
is the angle, expressed in radians, between the positive real axis and the vector from
the origin to z. Since x = r cos φ and y = r sin φ one has that
z = r (cos φ + i sin φ).
Cartesian form
Polar form
Modulus
Unit circle
(2.2)
Expressing z as in (2.1) is called the cartesian form of z, while expressing it as
in (2.2) is called the polar form. As we shall see in a moment, multiplication of
complex numbers is much more convenient in polar form than in cartesian form.
The number r , the distance from z to the origin, is called the modulus or absolute
value of z and is denoted by | z |. The complex numbers with | z | = 1 all have
distance 1 to the origin and so they form a circle with radius 1 and the origin as
centre. This circle is called the unit circle. The modulus is a generalization of
the absolute value for real numbers. For the modulus of z = x + i y one has (see
32
2 Mathematical prerequisites
2.1.2
Zeros of polynomials
We know that quadratic equations do not always admit real solutions. A quadratic
equation is an example of an equation of the more general type
an z n + an−1 z n−1 + · · · + a1 z + a0 = 0
Root
Polynomial
Zero
(2.15)
in the unknown z, where we assume that the coefficients a0 , a1 ,. . . , an may be
complex with an = 0. In this subsection we will pay special attention to the solutions or the roots of this equation. The left-hand side will be denoted by P(z), so
P(z) = an z n + an−1 z n−1 + · · · + a1 z + a0 , and is called a polynomial of degree
n. Hence, solving equation (2.15) means determining the zeros of a polynomial.
Using algebra, one can show that if z = a is a zero of P(z) (where a may also be
complex), then the polynomial P(z) can be written as P(z) = (z − a)Q 1 (z) for
some polynomial Q 1 (z). We then say that the linear factor z − a divides P(z). If
Q 1 (a) = 0 as well, then we can write P(z) as P(z) = (z − a)2 Q 2 (z). Of course,
we can continue in this way if Q 2 (a) = 0 as well. Ultimately, this leads to the
following statement.
If z = a is a zero of a polynomial P(z), then there exists a positive integer ν such
that
P(z) = (z − a)ν Q(z)
Multiplicity
for some polynomial Q(z) with Q(a) = 0. The number ν is called the multiplicity
of the zero a.
EXAMPLE 2.1
Let P(z) be the polynomial of degree four given by
P(z) = z 4 − 2z 3 + 5z 2 − 8z + 4.
Now z = 1 is a zero of P(z). This implies that we can divide by the factor z − 1,
which results in P(z) = (z − 1)(z 3 − z 2 + 4z − 4). However, z = 1 is also a zero
of z 3 − z 2 + 4z − 4. Again dividing by z − 1 results in P(z) = (z − 1)2 (z 2 + 4).
Since z = 1 is no longer a zero of z 2 + 4, z = 1 is a zero of P(z) of multiplicity 2.
We also note that z = 2i and z = −2i are zeros of P(z). Factorizing z 2 + 4 gives
P(z) = (z − 1)2 (z − 2i)(z + 2i).
The polynomial has a zero z = 1 of multiplicity 2 and zeros z = 2i and z = −2i of
multiplicity 1.
Simple zero
Zeros of multiplicity 1 are also called simple zeros. In the preceding example we
saw that in the complex plane the polynomial P(z) is factorized entirely into linear
factors. This is a major advantage of the introduction of complex numbers. The fact
is that any arbitrary polynomial can be factorized entirely into linear factors. This
statement is based on the so-called fundamental theorem of algebra, which states
that (2.15) always has a solution in C. The treatment of the fundamental theorem of
algebra falls outside the scope of this book. We thus have the following important
property:
Let P(z) = an z n + an−1 z n−1 + · · · + a1 z + a0 be a polynomial of degree n. Then
P(z) can be written as
P(z) = an (z − z 1 )ν1 · · · (z − z k )νk .
(2.16)
Here z 1 , . . . , z k are the distinct zeros of P(z) in C with their respective multiplicities.
2.1 Complex numbers and functions
33
From (2.16) we can immediately conclude that the degree of P(z) equals the sum of
the multiplicities. If we count the number of zeros of a polynomial, with each zero
counted according to its multiplicity, then it follows that a polynomial of degree n
has precisely n zeros.
When in particular the coefficients of P(z) are real, then
P(z) = an z n + an−1 z n−1 + · · · + a1 z + a0 = P(z).
Hence, if P(a) = 0, then P(a) = 0 as well. Now if a is a non-real zero, that is if
a = a, then
P(z) = (z − a)(z − a)Q(z) = (z 2 − (a + a)z + aa)Q(z)
=
z 2 − (2Re a)z + | a |2 Q(z).
Apparently the polynomial P(z) contains a quadratic factor with real coefficients
z 2 − (2Re a)z + | a |2 , which cannot be factorized any further into real linear factors.
As a consequence we have:
A polynomial with real coefficients can always be factorized into factors which are
linear or quadratic and having real coefficients. The zeros are real or they occur in
pairs of complex conjugates.
EXAMPLE 2.2
The polynomial z 4 + 4 is a polynomial with real coefficients. The complex zeros are
(see exercise 2.6a): 1 + i, 1 − i, −(1 + i), −(1 − i). So z 4 + 4 = (z − 1 − i)(z − 1 +
i)(z + 1 + i)(z + 1 − i). Since (z − 1 − i)(z − 1 + i) = (z − 1)2 + 1 = z 2 − 2z + 2
and (z + 1 + i)(z + 1 − i) = (z + 1)2 + 1 = z 2 + 2z + 2, one has that
z 4 + 4 = (z 2 − 2z + 2)(z 2 + 2z + 2).
This factorizes the polynomial z 4 + 4 into two factors with real coefficients, which
can still be factorized into linear factors, but then these will no longer have real
coefficients.
In the theory of the discrete Fourier transform, an important role is played by the
roots of the equation
z n = 1.
nth roots of unity
These roots are called the nth roots of unity. We know that the number of zeros of
z n − 1 is equal to n. First we determine the moduli of the zeros. From (2.4) follows
that z n = | z |n and since z n = 1, we obtain that | z | = 1. From this we conclude
that all solutions lie on the unit circle. The arguments of the solutions can be found
as follows:
n arg z = arg(z n ) = arg 1 = 2kπ.
Dividing by n leads to arg z = 2kπ/n. For k = 0, 1, . . . , n − 1 this gives n distinct
solutions. The nth roots of unity are thus
z k = ei2kπ/n = cos
2kπ
n
+ i sin
2kπ
n
for k = 0, 1, . . . , n − 1.
These solutions are drawn in figure 2.2 for the case n = 5. Note that the roots
display a symmetry with respect to the real axis. This is a consequence of the fact
that the polynomial z n − 1 has real coefficients and that thus the complex conjugate
of a zero is a zero as well.
The method described for solving the equation z n = 1 can be extended to equations of the type z n = a, where a is an arbitrary complex number. We will illustrate
this using the following example.
36
2 Mathematical prerequisites
Pole
The numerator and denominator consist of polynomials in the complex variable z.
Denote the numerator by P(z) and the denominator by Q(z). We assume that bm =
0 and an = 0. The degree of the numerator is n and the degree of the denominator
is m. The zeros of Q(z) are called the poles of F(z). In section 2.1.2 we noted that,
as a consequence of the fundamental theorem of algebra, in the complex plane each
polynomial can be factorized entirely into linear factors. The denominator Q(z) can
thus be written as
Q(z) = bm (z − z 1 )ν1 (z − z 2 )ν2 · · · (z − z k )νk ,
Order of pole
where z 1 , z 2 , . . . , z k are the distinct zeros of Q(z). The point z j is then called a
pole of order ν j of F(z).
Before starting with a partial fraction expansion, one should first check whether
or not the degree of the numerator is smaller than the degree of the denominator. If
this is not the case, then we first divide by the denominator, that is, we determine
polynomials D(z) and R(z) such that P(z) = D(z)Q(z) + R(z), where the degree
of R(z) is smaller than the degree of the denominator Q(z). As a consequence, F(z)
can be written as
F(z) = D(z) +
R(z)
.
Q(z)
The rational function R(z)/Q(z) in the right-hand side now does have the property
that the degree of the numerator is smaller than the degree of the denominator. We
shall illustrate this using an example.
EXAMPLE 2.5
The rational function F(z) is given by
z4 + 1
F(z) = 2
.
z −1
The degree of the numerator is greater than the degree of the denominator. We
therefore perform the following long division:
z2 − 1 / z4
z4
− z2
+ 1 z2 + 1
z2 + 1
z2 − 1
2
From this it follows that z 4 + 1 = (z 2 + 1)(z 2 − 1) + 2 and hence
2
.
F(z) = z 2 + 1 + 2
z −1
Henceforth we will assume that we are dealing with a rational function F(z) as
given by (2.17) but with the additional condition that n < m. The purpose of a
partial fraction expansion is then to write this rational function as a sum of fractions
with the numerators being (complex) constants and the denominators z − z 1 , z − z 2 ,
. . . , z − z k . If the order ν j of a pole is greater than 1, then denominators (z − z j )2 ,
(z − z j )3 , . . . , (z − z j )ν j also occur. If the coefficients of Q(z) are real, then Q(z)
can be written as a product of linear and quadratic factors with real coefficients,
and F(z) can then be expanded into fractions with (powers of) linear and quadratic
denominators with real coefficients. For the quadratic denominators, the numerators
may be linear.
2.2 Partial fraction expansions
37
The first step in the technique of the partial fraction expansion consists of a factorization of the denominator Q(z). This means determining the zeros of Q(z) together
with their multiplicities. Next, the actual partial fraction expansion takes place. The
following two examples will show how partial fraction expansions are carried out in
the case where Q(z) is factorized entirely into linear factors.
EXAMPLE 2.6
Let the rational function be given by
F(z) =
z
(z − 1)2 (z 2 + 1)
.
The denominator Q(z) can be factorized into linear factors as follows: Q(z) =
(z − 1)2 (z − i)(z + i). There is a zero z = 1 with multiplicity 2 and there are
simple zeros for z = ±i. A partial fraction expansion is aimed at writing F(z) in
the following form:
F(z) =
A
B
D
C
+
+
.
+
2
z − 1 (z − 1)
z−i
z+i
Here A, B, C and D are constants, still to be determined. Note that the zero of
multiplicity 2 has two fractions linked to it, while the simple zeros have only one.
The constants can be calculated as follows. Multiplying the expansion above by the
denominator Q(z) gives the identity
z = A(z − 1)(z 2 + 1) + B(z 2 + 1) + C(z − 1)2 (z + i) + D(z − 1)2 (z − i).(2.18)
Formally one should exclude the zeros of Q(z) in this identity, since these are the
poles of F(z). However, the right-hand and left-hand sides contain polynomials and
by a limit transition one can prove that for these values of z the identity remains
valid. Substituting the zeros z = 1, z = i and z = −i of Q(z) in (2.18) gives the
following results:
1 = 2B,
i = 4C,
−i = 4D.
We still lack one equation. To find it, we use yet another property of polynomials.
Namely, two polynomials in z are equal if and only if the coefficients of corresponding powers are equal. Comparing the coefficients of z 3 in both sides of equation
(2.18) establishes that
0 = A + C + D.
From the preceding equations it follows easily that A = 0, B = 1/2, C = i/4,
D = −i/4. Ultimately, the partial fraction expansion is as follows:
F(z) =
EXAMPLE 2.7
1/2
i/4
1
i
i/4
i
−
=
−
.
+
+
z−i
z+i
4(z − i) 4(z + i)
(z − 1)2
2(z − 1)2
Let the function F(z) be given by
z
.
F(z) = 2
z − 6i z − 8
We factorize the denominator Q(z) by first completing the square: Q(z) = (z −
3i)2 + 1 = (z − 3i − i)(z − 3i + i) = (z − 4i)(z − 2i). There are simple zeros at
z = 4i and at z = 2i. The rational function can then be expanded as follows:
F(z) =
B
A
+
.
z − 4i
z − 2i
38
2 Mathematical prerequisites
The constants A and B can be found by first multiplying by the denominator and
then substituting its zeros. It then follows that A = 2 and B = −1. The partial
fraction expansion is as follows:
F(z) =
2
1
−
.
z − 4i
z − 2i
For some applications it is more convenient to obtain, starting from a rational
function with real coefficients in the numerator as well as in the denominator, a partial fraction expansion where only fractions having real coefficients occur as well.
When all zeros of the denominator Q(z) are real, this is no problem; the partial fraction expansion can then be performed as in the previous examples. If, however, Q(z)
has also non-real zeros, then linear factors will no longer suffice. In a real factorization of Q(z), quadratic factors will then appear as well. In the following examples
we will show how in these circumstances one can determine a real expansion.
EXAMPLE 2.8
Let the function F(z) from example 2.6 be given:
z
F(z) =
.
(z − 1)2 (z 2 + 1)
The denominator contains a quadratic factor having no factorization into linear factors with real coefficients. With this factor we associate a fraction of the form
Az + B
z2 + 1
with real coefficients A and B. The partial fraction expansion now looks like this:
A
B
Cz + D
z
=
.
+
+ 2
z − 1 (z − 1)2
(z − 1)2 (z 2 + 1)
z +1
Multiplying by the denominator of F(z) leads to the following identity:
z = A(z − 1)(z 2 + 1) + B(z 2 + 1) + (C z + D)(z − 1)2 .
Substitution of z = 1 gives B = 1/2. Next we equate coefficients of corresponding
powers of z. For the coefficient of z 3 , z 2 and z it subsequently follows that
0 = A + C,
0 = −A + B − 2C + D,
1 = A + C − 2D.
The solution to this system of equations is A = 0, B = 1/2, C = 0, D = −1/2 and
so the real partial fraction expansion looks like this:
z
(z − 1)2 (z 2 + 1)
=
1/2
1
1/2
1
=
.
− 2
−
2
2
2 + 1)
(z − 1)
z +1
2(z − 1)
2(z
We finish with an example where a quadratic factor occurs twice in the denominator of F(z).
EXAMPLE 2.9
Let the function F(z) be given by
F(z) =
z 2 + 3z + 3
.
(z 2 + 2z + 4)2
The quadratic factor in the denominator cannot be factorized into linear factors with
real coefficients. Since the quadratic factor occurs twice, the partial fraction expansion has the following form:
z 2 + 3z + 3
Cz + D
Az + B
+ 2
= 2
.
2 + 2z + 4)2
(z
z + 2z + 4 (z + 2z + 4)2
2.3 Complex-valued functions
41
From this it follows that if limt→a | f (t) − L | = 0, then limt→a u(t) = Re L.
Similarly one proves that limt→a v(t) = Im L. This completes the proof.
EXAMPLE 2.10
We show that for each real ω one has: limt→∞ eiωt /t = 0. Since |eiωt /t| =
1/t and limt→∞ 1/t = 0 we have limt→∞ |eiωt /t| = 0 and hence, according to
theorem 2.3, limt→∞ eiωt /t = 0.
Continuity
Differentiability
Concepts like continuity and differentiability of a function are defined using limits. For instance, a function is continuous at t = a if limt→a f (t) = f (a) and
differentiable at t = a if limt→a ( f (t) − f (a))/(t − a) exists. One can show (this
is not very hard, but it will be omitted here) that for a complex-valued function
f (t) = u(t) + iv(t), the continuity of f (t) is equivalent to the continuity of both
the real part u(t) and the imaginary part v(t), and also that the differentiability of
f (t) is equivalent to the differentiability of u(t) and v(t). Moreover, one has for the
derivative at a point t that
f (t) = u (t) + iv (t).
(2.19)
Consequently, for the differentiation of complex-valued functions the same rules apply as for real-valued functions. Complex numbers may be considered as constants
here.
EXAMPLE 2.11
If f (t) = eat with a ∈ C, then f (t) = aeat . We can show this as follows: put
a = x + i y and write f (t) as f (t) = e xt ei yt = e xt cos yt + ie xt sin yt. The real
and imaginary parts are differentiable everywhere with derivatives e xt (x cos yt −
y sin yt) and e xt (x sin yt + y cos yt) respectively. So
f (t) = xe xt (cos yt + i sin yt) + i ye xt (cos yt + i sin yt)
= (x + i y)e xt ei yt = aeat .
Chain rule
With the chain rule we can differentiate a composition f (g(t)) of two functions.
The function f (t), however, is defined on (a part of) R. Hence, in the composition
g(t) should also be a real-valued function. The chain rule then has the usual form
d
f (g(t)) = f (g(t))g (t).
dt
A consequence of this is:
d
[ f (t)]n = n[ f (t)]n−1 f (t)
dt
Left-hand limit
Right-hand limit
for n = 1, 2, . . ..
Now that the concepts continuity and differentiability of complex-valued functions have been introduced, we will proceed with the introduction of two classes
of functions that will play an important role in theorems on Fourier series, Fourier
integrals and Laplace transforms.
The first class consists of the so-called piecewise continuous functions. To start
with, we define in the usual manner the left-hand limit f (t−) and right-hand limit
f (t+) of a function at the point t:
f (t−) = lim f (t − h)
h↓0
and
f (t+) = lim f (t + h),
h↓0
provided these limits exist.
DEFINITION 2.3
Piecewise continuous
function
A function f (t) is called piecewise continuous on the interval [a, b] if f (t) is continuous at each point of (a, b), except possibly in a finite number of points t1 , t2 , . . . ,
tn . Moreover, f (a+), f (b−) and f (ti +), f (ti −) should exist for i = 1, . . . , n.
42
2 Mathematical prerequisites
A function f (t) is called piecewise continuous on R if f (t) is piecewise continuous
on each subinterval [a, b] of R.
One can show that a function f (t) which is piecewise continuous on an interval
[a, b] is also bounded on [a, b], that is to say: there exists a constant M > 0 such
that for all t in [a, b] one has | f (t) | ≤ M. Functions that possess a real or imaginary
part with a vertical asymptote in [a, b] are thus not piecewise continuous on [a, b].
Another example is the function f (t) = sin(1/t) for t = 0 and f (0) = 0. This
function is continuous everywhere except at t = 0. Since f (0+) does not exist, this
function is not piecewise continuous on [0, 1] according to our definition.
Note that the function value f (t) at a point t of discontinuity doesn't necessarily
have to equal f (t+) or f (t−).
A second class of functions to be introduced is the so-called piecewise smooth
functions. This property is linked to the derivative of the function. For a piecewise
continuous function, we will mean by f the derivative of f at all points where it
exists.
DEFINITION 2.4
Piecewise smooth function
A piecewise continuous function f (t) on the interval [a, b] is called piecewise
smooth if its derivative f (t) is piecewise continuous.
A function is called piecewise smooth on R if this function is piecewise smooth on
each subinterval [a, b] of R.
In figure 2.5 a graph is drawn of a real-valued piecewise smooth function.
t1
0
t2
t3
t4
t
FIGURE 2.5
A piecewise smooth function.
Left-hand derivative
Right-hand derivative
There are now two possible ways of looking at the derivative in the neighbourhood of a point: on the one hand as the limits f (t+) and f (t−); on the other
hand by defining a left-hand derivative f − (t) and a right-hand derivative f + (t) as
follows:
f − (t) = lim
h↑0
f (t + h) − f (t−)
,
h
f + (t) = lim
h↓0
f (t + h) − f (t+)
,
h
(2.20)
provided these limits exist. Note that in this definition of left-hand and right-hand
derivative one does not use the function value at t, since f (t) need not exist at the
point t. Often it is the case that f − (t) = f (t−) and f + (t) = f (t+). This holds
in particular for piecewise smooth functions, notably at the points of discontinuity
of f , as is proven in the next theorem.
2.3 Complex-valued functions
THEOREM 2.4
43
Let f (t) be a piecewise smooth function on the interval [a, b]. Then f + (a) =
f (a+), f − (b) = f (b−) and for all a < t < b one has, moreover, that f − (t) =
f (t−) and f + (t) = f (t+).
Proof
We present the proof for right-hand limits. The proof for left-hand limits is analogous. So let t ∈ [a, b) be arbitrary. Using the mean value theorem from calculus we will show that the existence of f (t+) implies that f + (t) exists and that
f (t+) = f + (t). Since f and f are both piecewise continuous, there exists an
h > 0 such that f and f have no point of discontinuity on (t, t + h]. Possibly f
has a discontinuity at t. If we now redefine f at t as f (t+), then f is continuous on
[t, t + h]. Moreover, f is differentiable on (t, t + h). According to the mean value
theorem there then exists a ξ ∈ (t, t + h) such that
f (t + h) − f (t+)
= f (ξ ).
h
Now let h ↓ 0, then ξ ↓ t. Since f (t+) = limξ ↓t f (ξ ) exists, it follows from
(2.20) that f + (t) exists and that f (t+) = f + (t).
When a function is not piecewise smooth, the left- and right-hand derivatives
will not always be equal to the left- and right-hand limits of the derivative, as the
following example shows.
EXAMPLE
Let the function f (t) be given by
t 2 sin(1/t)
0
f (t) =
for t = 0,
for t = 0.
The left- and right-hand derivatives of f (t) at t = 0, calculated according to (2.20),
exist and are equal to 0. However, if we first calculate the derivative f (t), then
f (t) =
2t sin(1/t) − cos(1/t)
0
for t = 0,
for t = 0.
It then turns out that f (0+) and f (0−) do not exist, since cos(1/t) has no left- or
right-hand limit at t = 0.
The theory of the Riemann integral and the improper Riemann integral for realvalued functions can easily be extended to complex-valued functions as well. For
complex-valued functions the (improper) Riemann integral exists on an interval if
and only if both the (improper) Riemann integral of the real part u(t) and the imaginary part v(t) exist on that interval. Moreover, one has
b
Definite integral
a
f (t) dt =
b
a
u(t) dt + i
b
v(t) dt.
(2.21)
a
Here a = −∞ or b = ∞ is also allowed. We recall that the value of an integral
does not change by altering the value of the function at the possible jump discontinuities. From (2.21) the following properties of definite integrals for complex-valued
functions immediately follow:
b
a
f (t) dt =
b
Re
a
b
a
f (t) dt =
f (t) dt,
b
a
Re f (t) dt
b
and
Im
a
f (t) dt =
b
a
Im f (t) dt.
2.4 Sequences and series
2.4
Sequences and series
2.4.1
45
Basic properties
The concept of an infinite series plays a predominant role in chapters 3 to 5 and will
also return regularly later on. We will assume that the reader is already acquainted
with the theory of sequences and series, as far as the terms of the sequence or the
series consist of real numbers. In this section the theory is extended to complex
numbers. In general, the terms of a sequence
(an )
with n = 0, 1, 2, . . .
will then be complex numbers, as is the case for the terms of a series
∞
an = a0 + a1 + · · · .
n=0
For limits of sequences of complex numbers we follow the same line as for limits of
complex-valued functions. Assuming that the concept of convergence of a sequence
of real numbers is known, we start with the following definition of convergence of
a sequence of complex numbers.
DEFINITION 2.5
Convergence of sequences
A sequence (an ) of complex numbers with u n = Re an and v n = Im an converges
if both the sequence of real numbers (u n ) and the sequence of real numbers (v n )
converge. Moreover, the limit of the sequence (an ) then equals
lim an = lim u n + i lim v n .
n→∞
EXAMPLE 2.14
n→∞
n→∞
Let the sequence (an ) be given by
an = n(ei/n − 1)
with n = 0, 1, 2, . . ..
Since ei/n = cos(1/n) + i sin(1/n) one has u n = Re an = n(cos(1/n) − 1)
and v n = Im an = n sin(1/n). Verify for yourself that limn→∞ u n = 0 and
limn→∞ v n = 1. Hence, the sequence (an ) converges and limn→∞ an = i.
Our next theorem resembles theorem 2.3 and can also be proven in the same way.
THEOREM 2.6
A sequence (an ) converges and has limit a if and only if limn→∞ | an − a | = 0.
EXAMPLE 2.15
For complex z one has
lim z n = 0
n→∞
if | z | < 1.
We know that for real r with −1 < r < 1 one has limn→∞ r n = 0. So if | z | < 1,
then limn→∞ z n = limn→∞ | z |n = 0. Using theorem 2.6 with a = 0 we
conclude that limn→∞ z n = 0.
Divergence of sequences
In the complex plane | an − a | is the distance from an to a. Theorem 2.6 states
that for convergent sequences this distance tends to zero as n tends to infinity.
A sequence diverges, or is called divergent, if the sequence does not converge.
All kinds of properties that are valid for convergent sequences of real numbers are
valid for convergent sequences of complex numbers as well. We will formulate the
following properties, which we immediately recognize from convergent sequences
of real numbers:
Let (an ) and (bn ) be two convergent sequences such that limn→∞ an = a and
limn→∞ bn = b. Then
46
2 Mathematical prerequisites
a
b
c
Partial sum
limn→∞ (αan + βbn ) = αa + βb
limn→∞ an bn = ab,
limn→∞ an /bn = a/b if b = 0.
for all α, β ∈ C,
Now that we know what a convergent sequence is, we can define convergence of
a series in the usual way. For this we use the partial sums sn of the sequence (an ):
n
sn =
ak = a0 + a1 + · · · + an .
k=0
DEFINITION 2.6
Convergence of a series
Sum of a series
EXAMPLE 2.16
Geometric series
A series ∞ an is called convergent if and only if the sequence of partial sums
n=0
(sn ) converges.
When s = limn→∞ sn , we call s the sum of the series. For a convergent series
the sum is also denoted by ∞ an , so s = ∞ an .
n=0
n=0
Consider the geometric series ∞ z n with ratio z ∈ C. The partial sum sn is
n=0
equal to sn = 1 + z + z 2 + · · · + z n . Note that this is a polynomial of degree
n. When z = 1, then we see by direct substitution that sn = n + 1. Hence, the
geometric series diverges for z = 1, since limn→∞ sn = ∞. Multiplying sn by the
factor 1 − z gives
(1 − z)sn = 1 + z + z 2 + · · · + z n − z(1 + z + z 2 + · · · + z n ) = 1 − z n+1 .
For z = 1 one thus has
sn =
1 − z n+1
.
1−z
For | z | < 1 we have seen that limn→∞ z n = 0 (see example 2.15); so then the
series converges with sum equal to 1/(1 − z). We write
∞
zn =
n=0
1
1−z
if | z | < 1.
Since for | z | ≥ 1 the sequence with terms z n does not tend to zero, the series
diverges for these values of z.
Just as for sequences, the convergence of a series can be verified on the basis of
the real and imaginary parts of the terms. For sequences we used this as a definition.
For series we formulate it as a theorem.
THEOREM 2.7
Let (an ) be a sequence of numbers and u n = Re an and v n = Im an . Then the series
∞
∞
∞
n=0 an converges if and only if both the series
n=0 u n and the series
n=0 v n
converge.
Proof
Let (sn ), (rn ) and (tn ) be the partial sums of the series with terms an , u n and v n
respectively. Note that sn = rn + itn . According to the definition of convergence of
a sequence, the sequence (sn ) converges if and only if both the sequence (rn ) and
the sequence (tn ) converge. This is precisely the definition of convergence for the
series with terms an , u n and v n .
From the preceding theorem we also conclude that for a convergent series with
terms an = u n + iv n one has
∞
n=0
an =
∞
n=0
un + i
∞
n=0
vn .
2.4 Sequences and series
47
For convergent series with complex terms, the same properties hold as for convergent series with real terms. Here we formulate the linearity property, which is a
direct consequence of definition 2.6 and the linearity property for series with real
terms.
THEOREM 2.8
∞
n=0 an and
Let
∞
∞
n=0 bn be convergent series with sum s and t respectively. Then
(αan + βbn ) = αs + βt
for all α, β ∈ C.
n=0
The next property formulates a necessary condition for a series to converge.
If the series
∞
n=0 an converges, then limn→∞ an = 0.
Proof
If the series
THEOREM 2.9
∞
n=0 an converges and has sum s, then
lim an = lim (sn − sn−1 ) = lim sn − lim sn−1 = s − s = 0.
n→∞
n→∞
n→∞
n→∞
As a consequence we have that limn→∞ an = 0 excludes the convergence of
the series. The theorem only gives a necessary and not a sufficient condition for
convergence. To show this, we consider the harmonic series.
EXAMPLE 2.17
Harmonic series
Series of the type ∞ 1/n p , with p a real constant, are called harmonic series.
n=1
From the theory of series with real terms it is known that for p > 1 the harmonic
series is convergent, while for p ≤ 1 the harmonic series is divergent. Hence, for
0 < p ≤ 1 we obtain a divergent series with terms that do tend to zero.
2.4.2
Absolute convergence and convergence tests
For practical applications one usually needs a strong form of convergence of a series. Convergence itself is not enough, and usually one requires in addition the
convergence of the series of absolute values, or moduli, of the terms.
DEFINITION 2.7
Absolute convergence
A series
verges.
∞
n=0 an is called absolutely convergent if the series
∞ |
n=0 an | con-
The series of the absolute values is a series with non-negative real terms. Convergence of series with non-negative terms can be verified using the following test,
which is known as the comparison test.
THEOREM 2.10
Comparison test
When (an ) and (bn ) are sequences of real numbers with 0 ≤ an ≤ bn for n =
0, 1, 2, . . ., and the series ∞ bn converges, then the series ∞ an converges
n=0
n=0
as well.
Proof
Let (sn ) be the partial sums of the series with terms an . Then sn+1 −sn = an+1 ≥ 0.
So sn+1 ≥ sn , which means that the sequence (sn ) is a non-decreasing sequence. It
is known, and this is based on fundamental properties of the real numbers, that such
a sequence converges whenever it has a upper bound. This means that there should
be a constant c such that sn ≤ c for all n = 0, 1, 2, . . .. For the sequence (sn ) this is
easy to show, since it follows from an ≤ bn that
sn = a0 + a1 + · · · + an ≤ b0 + b1 + · · · + bn
≤ b0 + b1 + · · · + bn + bn+1 + · · · .
48
2 Mathematical prerequisites
The sequence (sn ) apparently has as upper bound the sum of the series with terms
bn . This sum exists since it is given that this series converges. This proves the
theorem.
Absolutely convergent series with real terms are convergent. One can show this
as follows. Write an = bn − cn with bn = | an | and cn = | an | − an . It is
given that the series with terms bn converges. The terms cn satisfy the inequality
0 ≤ cn ≤ 2 | an |. According to the comparison test, the series with terms cn then
converges as well. Since an = bn − cn , the series with terms an thus converges. For
series with complex terms, the statement is a consequence of the next theorem.
THEOREM 2.11
Let (an ) be a sequence of numbers and let u n = Re an and v n = Im an . The series
∞
∞
∞
n=0 an converges absolutely if and only if both the series n=0 u n and n=0 v n
converge absolutely.
Proof
For the terms u n and v n one has the inequalities | u n | ≤ | an |, | v n | ≤ | an |. If
the series with the non-negative terms | an | converges, then we know from the comparison test that the series with, respectively, terms u n and v n converge absolutely.
Conversely, if the series with, respectively, terms u n and v n converge absolutely,
then the series with the non-negative terms | u n | + | v n | also converges. It then
2
follows from the inequality | an | = u 2 + v n ≤ | u n | + | v n | and again the comn
parison test that the series with terms | an | converges. This means that the series
with terms an converges absolutely.
THEOREM 2.12
An absolutely convergent series is convergent.
Proof
Above we sketched the proof of this statement for series with real terms. For series
with, in general, complex terms, the statement follows from the preceding theorem.
Specifically, when the series with complex terms is absolutely convergent, then the
series consisting of the real and the imaginary parts also converge absolutely. These
are series of real numbers and therefore convergent. Next we can apply theorem 2.7,
resulting in the convergence of the series with the complex terms.
Of course, introducing absolute convergence only makes sense when there are
convergent series which are not absolutely convergent. An example of this is the
series ∞ (−1)n /n. This series converges and has sum − ln 2 (see (2.26) with
n=1
t = 1), while the series of the absolute values is the divergent harmonic series with
p = 1 (see example 2.17).
We now present some convergence tests already known for series with real terms,
but which remain valid for series with complex terms.
THEOREM 2.13
If | an | ≤ bn for n = 0, 1, . . . and
∞
n=0 bn converges, then
∞
n=0 an converges.
Proof
According to the comparison test (theorem 2.10), the series with terms | an |
converges. The series with terms an is thus absolutely convergent and hence
convergent.
Of course, in order to use the preceding theorem, one should first have available a
convergent series with non-negative terms. Suitable candidates are the harmonic series with p > 1 and the geometric series with a positive ratio r satisfying 0 < r < 1
(see example 2.16), and all linear combinations of these two as well.
2.4 Sequences and series
EXAMPLE 2.18
49
Consider the series ∞ einω /(n 2 + n) where ω is a real constant. The series
n=1
converges absolutely since
1
1
einω
= 2
≤ 2
n2 + n
n +n
n
and the harmonic series with terms 1/n 2 converges.
A geometric series has the property that the ratio an+1 /an of two consecutive
terms is a constant. If, more generally, a sequence has terms with the property that
lim
n→∞
an+1
=L
an
for some L, then one may conclude, as for geometric series, that the series is absolutely convergent if L < 1 and divergent if L > 1. In the case L = 1, however, one
cannot draw any conclusion. We summarize this, without proof, in the next theorem.
THEOREM 2.14
D'Alembert's ratio test
EXAMPLE 2.19
Let (an ) be a sequence of terms unequal to zero with limn→∞ an+1 /an = L for
some L. Then one has:
a if L < 1, then the series with terms an converges absolutely;
b if L > 1, then the series with terms an diverges.
Consider the series ∞ z n /n p . Here z is a complex number and p an arbitrary
n=1
real constant. Put an = z n /n p , then
p
n p z n+1
n
an+1
|z| = |z|.
= lim
= lim
n→∞
n→∞ (n + 1) p z n
n→∞ n + 1
an
lim
Hence, the series is absolutely convergent for | z | < 1 and divergent for | z | > 1.
If | z | = 1, then no conclusions can be drawn from the ratio test. For p > 1 and
| z | = 1 we are dealing with a convergent harmonic series and so the given series
converges absolutely. For p ≤ 1 we are dealing with a divergent harmonic series
and so the series of absolute values diverges. From this we may not conclude that
the series itself diverges. Take for example p = 1 and z = −1, then one obtains the
series with terms (−1)n /n, which is convergent.
2.4.3
Series of functions
In the theory of Fourier series in part 2, and of the z-transform in chapter 18, we
will encounter series having terms an that still depend on a variable. The geometric
series in example 2.16 can again serve as an example. Other examples are
∞
z2
z3
zn
zn
=1+z+
+
+ ··· +
+ ···
n!
2!
3!
n!
n=0
∞
for z ∈ C,
cos 3t
cos nt
cos nt
cos 2t
+
+ ··· +
= cos t +
+ ···
4
9
n2
n2
n=1
for t ∈ R.
The first series is an example of a power series. The partial sums of this series are
polynomials in z, so functions defined on C. In section 2.5 we will study these more
closely. In this section we will confine ourselves to series of the second type, where
50
2 Mathematical prerequisites
the functions are defined on R or on a part of R. We will thus consider series of the
type
∞
f n (t).
n=0
Convergence of such a series depends, of course, on the value of t and then the
sum will in general depend on t as well. For the values of t for which the series
converges, the sum will be denoted by the function f (t). In this case we write
∞
f (t) =
f n (t)
n=0
Pointwise convergence
EXAMPLE 2.20
and call this pointwise convergence. For each value of t one has a different series
for which, in principle, one should verify the convergence. It turns out, however,
that in many cases it is possible to analyse the convergence for an entire interval.
Let f n (t) = t n . In example 2.16 it was already shown that ∞ t n converges for
n=0
| t | < 1, with sum 1/(1 − t). This means that the series ∞ f n (t) converges
n=0
on the interval (−1, 1) and that f (t) = 1/(1 − t). Outside this interval, the series
diverges.
One would like to derive properties of f (t) directly from the properties of the
functions f n (t), without knowing the function f (t) explicitly as a function of t.
One could wonder, for example, whether a series may be differentiated term-byterm, so whether f (t) =
f n (t) if f (t) =
f n (t). A simple example will show
that this is not always permitted.
EXAMPLE 2.21
Let, for example, f n (t) = sin(nt)/n 2 , then f n (t) = cos(nt)/n. So for each n > 0
the derivative exists for all t ∈ R. However, if we now look at ∞ f n (t) at t = 0,
n=1
then this equals ∞ 1/n, which is a divergent harmonic series, as we have seen
n=1
in example 2.17. Although all functions f n (t) are differentiable, f (t) is not.
One should also be careful with integration. When, for instance, the functions
f n (t) are integrable, then one would like to conclude from this that f (t) is also
integrable and that
f (t) dt =
f n (t) dt =
f n (t) dt .
This is not always the case, as our next example will show.
EXAMPLE 2.22
Let u n (t) = nte−nt for n = 0, 1, 2, . . . and let f n (t) = u n (t) − u n−1 (t) for
n = 1, 2, 3, . . . and f 0 (t) = 0. Then one has for the partial sums sn (t):
2
sn (t) = f 1 (t) + · · · + f n (t)
= u 1 (t) − u 0 (t) + u 2 (t) − u 1 (t) + · · · + u n (t) − u n−1 (t) = u n (t).
The sequence of partial sums converges and has limit
f (t) = lim sn (t) = lim nte−nt = 0.
2
n→∞
n→∞
On the interval (0, 1) one thus has, on the one hand,
1 ∞
0 n=0
f n (t) dt =
1
0
f (t) dt = 0,
2.5 Power series
51
while on the other hand
∞
1
n=0 0
f n (t) dt =
=
∞
1
n=1 0
∞
n=1
u n (t) − u n−1 (t) dt
1
= 1 (e − 1)
− 1 e−nt + 1 e−(n−1)t
2
2
2
0
2
2
∞
e−n = 1 .
2
n=1
These results are unequal, hence
1 ∞
0 n=0
f n (t) dt =
∞
n=0 0
1
f n (t) dt.
In order to define conditions such that properties like interchanging the order of
summation and integration are valid, one could for example introduce the notion
of uniform convergence. This is outside the scope of this book. We will therefore
always confine ourselves to pointwise convergence, and in the case when one of
the properties mentioned above is used, we will always state explicitly whether it is
allowed to do so.
EXERCISES
2.16
Use the comparison test to prove the convergence of:
∞
1
a
,
3+i
n
n=0
∞
b
c
2.17
sin n
,
n2
n=1
∞ −n(1+i)
e
.
n
n=1
Determine which of the following series converge. Justify each of your answers.
∞
(−i)n
a
,
n!
n=1
∞
b
c
2.18
2n + 1
,
3n + n
n=1
∞
n
.
(1 + i)n
n=1
Show that the following series of functions converges absolutely for all t:
∞
e2int
.
2n 4 + 1
n=0
2.5
Power series
As final subject of this chapter, we consider some properties of complex power
series. Power series were already introduced in section 2.4.3. These series have a
simple structure. Let us start with a definition.
52
2 Mathematical prerequisites
DEFINITION 2.8
Power series
A power series in the complex variable z and with complex coefficients c0 , c1 , c2 ,
. . . , is a series of the form
∞
cn z n = c0 + c1 z + c2 z 2 + · · · + cn z n + · · · .
n=0
Apparently, a partial sum sn = c0 + c1 z + c2 z 2 + · · · + cn z n is a polynomial
in z of degree at most n. The geometric series in example 2.16 is an example of
a power series. Other examples arise from so-called Taylor-series expansions of a
real-valued function f (t) at the real variable t = 0. Such a Taylor-series expansion
looks like this:
∞
f (n) (0) n
t .
n!
n=0
Here f (n) (0) is the value at t = 0 of the nth derivative. In this case we are dealing
with a real power series. For a large number of functions, the Taylor-series expansion at t = 0 is explicitly known and in the case of convergence the sum of the
Taylor-series often represents the function itself. Well-known examples are:
et =
∞
tn
t2
t3
=1+t +
+
+ ···
n!
2!
3!
n=0
sin t =
∞
(−1)n
t 2n+1
t3
t5
=t−
+
− ···
(2n + 1)!
3!
5!
(−1)n
t2
t4
t 2n
=1−
+
− ···
(2n)!
2!
4!
n=0
cos t =
for all t,
∞
n=0
ln(1 + t) =
∞
(−1)n+1
n=1
t2
t3
tn
=t−
+
− ···
n
2
3
(2.23)
for all t,
for all t,
for −1 < t ≤ 1.
(2.24)
(2.25)
(2.26)
The series above are power series in the real variable t. If we replace the real variable t by a complex variable z, then complex power series arise, for which we first
of all ask ourselves: for which complex z does the power series converge and, subsequently, what is the sum of that power series? If, for example, we replace the
real variable t in (2.23) by the complex variable z, then one can wonder if the series
converges for all z as well, and if its sum is then still equal to the exponential function e z . The answer is affirmative, but the treatment of functions defined on C will
be postponed until chapter 11. In this section we will only go into the question for
which values of z a power series converges. We first present some examples.
EXAMPLE 2.23
Given is the power series
∞
zn
z2
z3
=1+z+
+
+ ···.
n!
2!
3!
n=0
To investigate the values of z for which this series converges, we put an = z n /n!
and apply the ratio test:
|z|
n!
an+1
|z| =
.
=
an
(n + 1)!
n+1
2.5 Power series
53
We see that limn→∞ an+1 /an = 0 < 1 for all z. The series thus converges for
all z.
EXAMPLE 2.24
Given is the power series
∞
z2
(−1)n+1 n
z3
z4
z =z−
+
−
+ ···.
n
2
3
4
n=1
To investigate the values of z for which this series converges, we put an =
(−1)n+1 z n /n and apply the ratio test:
n
an+1
|z|.
=
an
n+1
We see that limn→∞ an+1 /an = | z | and so the series converges absolutely for
| z | < 1 and diverges for | z | > 1. For | z | = 1 no conclusion can be drawn from
the ratio test. For z = 1 we know from (2.26) (substitute t = 1) that the series
converges and has sum ln 2, while for z = −1 we know that the series diverges. For
all other values of z on the unit circle one can show, with quite some effort, that the
series converges.
EXAMPLE 2.25
Given is the power series
∞
n!z n = 1 + z + 2!z 2 + · · · .
n=0
To investigate the values of z for which this series converges, we put an = n!z n and
apply the ratio test:
an+1
(n + 1)!
| z | = (n + 1) | z | .
=
an
n!
We see that limn→∞ an+1 /an = ∞ for z = 0 and so the series diverges for all
z = 0 and it converges only for z = 0.
EXAMPLE 2.26
Given is the power series
∞
n 2 2n z n = 2z + 16z 2 + · · · .
n=0
To investigate the values of z for which this series converges, we put an = n 2 2n z n
and apply the ratio test:
an+1
=
an
n+1 2
| 2z | .
n
We see that limn→∞ an+1 /an = 2 | z | and so the series converges absolutely for
| z | < 1 and it diverges for | z | > 1 . If | z | = 1 , then | an | = n 2 and this sequence
2
2
2
does not tend to zero. Hence, on the circle with radius 1 the series diverges.
2
The previous examples suggest that for each power series there exists a number
R such that the power series converges absolutely for | z | < R and diverges for
| z | > R. This is indeed the case. The proof will be omitted. Usually one can find
this number R with the ratio test, as in the previous examples. Summarizing, we
now have the following.
54
2 Mathematical prerequisites
For a power series in z one of the following three statements is valid.
a The power series converges only for z = 0.
b There exists a number R > 0 such that the power series converges absolutely
for all z with | z | < R and diverges for | z | > R.
c The power series converges for all z.
Radius of convergence
Circle of convergence
The number R is called the radius of convergence. In case a we put R = 0 and in
case c we put R = ∞. The radii of convergence of the power series in examples
2.23 up to 2.26 are, respectively, R = ∞, R = 1, R = 0, R = 1 .
2
If R is the radius of convergence of a power series, then this power series has a
sum f (z) for | z | < R. In the complex plane the points with | z | = R form a circle
of radius R and with the origin as centre, and this is sometimes called the circle of
convergence. For example, the geometric series in example 2.16 has sum 1/(1 − z)
for | z | < 1.
Continuity and differentiability of functions on C will be defined in chapter 11.
As far as the technique of differentiation is concerned, there is no difference in
differentiating with respect to a complex variable or a real variable. Hence, the
derivative of 1/(1 − z) is equal to 1/(1 − z)2 . One can prove that the sum of a power
series is a differentiable function within the circle of convergence. Indeed, we have
the following theorem.
The power series ∞ cn z n and ∞ ncn z n−1 have the same radius of convern=0
n=1
gence R. Moreover, one has: if f (z) = ∞ cn z n for | z | < R, then f (z) =
n=0
∞
ncn z n−1 for | z | < R.
n=1
THEOREM 2.15
EXAMPLE 2.27
We know that
∞
nz n−1 =
n=1
∞
n
n=0 z = 1/(1 − z) for | z | < 1. Applying theorem 2.15 gives
1
(1 − z)2
for | z | < 1.
We can apply this theorem repeatedly (say k times) to obtain the following result:
∞
n(n − 1) · · · (n − k + 1)z n−k =
n=k
k!
.
(1 − z)k+1
Using binomial coefficients this can be written as
∞
∞
1
n n−k
n+k n
=
z
z =
k
k
(1 − z)k+1
n=k
n=0
for | z | < 1.
EXERCISES
∞
2.20
2n
n=0
2.19
n2 + 1
Determine the radius of convergence of the series
z 2n .
Determine the values of z for which the following series converges, and, moreover,
determine the sum for these values.
∞
1
(z − i)n .
1−i
n=0
2.21
2.22
∞
2n
n=0 z /n! satisfies f (z) = 2z f (z).
For which values of z does the series ∞ 2n z −n converge absolutely?
n=1
Show that the sum f (z) of
2.5 Power series
55
SUMMARY
Complex numbers play a fundamental role in the treatment of Fourier and Laplace
transforms. The functions that occur are mostly functions defined on (a part of) R
or on (a part of) the complex plane C, with function values being complex numbers.
Important examples are the time-harmonic functions eiωt with frequency ω and
defined on R, and rational functions defined on C.
Using complex numbers one can factorize polynomials entirely into linear factors. As a consequence, when allowing complex factors, the partial fraction expansion of a rational function (for which the degree of the numerator is smaller than
the degree of the denominator) will consist of fractions with numerators being just
constants and denominators being polynomials of degree one or powers thereof, depending on the multiplicity of the various zeros. If the rational function has real
coefficients, then one can also expand it as a sum of fractions with real coefficients
and having denominators which are (powers of) linear and/or quadratic polynomials.
The numerators associated with the quadratic denominators may then be linear.
The differential and integral calculus for complex-valued functions and for realvalued functions are very similar. If a complex-valued function f (t) = u(t) + iv(t)
with t ∈ R has a certain property, like continuity or differentiability, then this means
that both the real part u(t) and the imaginary part v(t) have this property. The
derivative f (t) of a complex-valued function equals u (t) + iv (t). As a result,
the existing rules for differentiation and integration of real-valued functions are also
valid for complex-valued functions. Classes of complex-valued functions that may
appear in theorems on Fourier and Laplace transforms are the class of piecewise
continuous functions and the class of piecewise smooth functions.
The theory of sequences and series of real numbers can easily be extended to a
theory of sequences and series of complex numbers. All kinds of properties, such
as convergence and absolute convergence of a series with terms an = u n + iv n ,
can immediately be deduced from the same properties for the series with real terms
u n and v n . Convergence tests, such as the ratio test, are the same for series with
complex terms and for series with real terms.
Just as real power series, complex power series have a radius of convergence R.
A power series in the complex variable z converges absolutely for | z | < R, that is
within a circle in the complex plane with radius R (the circle of convergence), and
diverges for | z | > R. Within the circle of convergence, the sum of a power series
can be differentiated an arbitrary number of times. The derivative can be determined
by differentiating the power series term-by-term.
SELFTEST
2.23
Determine the (complex) zeros and their multiplicities for the following polynomials
P(z):
a P(z) = z 3 − 1,
b P(z) = (z 2 + i)2 + 1,
c P(z) = z 5 + 8z 3 + 16z.
2.24
Determine the partial fraction expansion of
F(z) =
z2 + z − 2
.
(z + 1)3
56
2 Mathematical prerequisites
2.25
Determine the partial fraction expansion, into fractions with real coefficients, of
F(z) =
z 2 − 6z + 7
.
(z 2 − 4z + 5)2
2.26
2π
Calculate the integral 0 eit cos t dt.
2.27
Find out if the following series converge:
∞
2−i n
n
,
a
3
n=1
∞
b
2.28
n + in
.
n2
n=1
Given is the series of functions
∞
2−n int
e sin t.
n
n=1
Show that this series converges absolutely for all t.
2.29
2.30
Show that if the power series ∞ cn z n has radius of convergence R, then the
n=0
power series ∞ cn z 2n has radius of convergence R 1/2 .
n=0
a
Calculate the radius of convergence R of the power series
∞
(1 + i)2n n
z .
n+1
n=0
b
Let f (z) be the sum of this power series. Calculate z f (z) + f (z).
Part 2
Fourier series
INTRODUCTION TO PART 2
The Fourier series that we will encounter in this part are a tool to analyse numerous
problems in mathematics, in the natural sciences and in engineering. For this it is
essential that periodic functions can be written as sums of infinitely many sine and
cosine functions of different frequencies. Such sums are called Fourier series.
In chapter 3 we will examine how, for a given periodic function, a Fourier series
can be obtained, and which properties it possesses. In chapter 4 the conditions will
be established under which the Fourier series give an exact representation of the
periodic functions. In the final chapter the theory of the Fourier series is used to
analyse the behaviour of systems, as defined in chapter 1, and to solve differential
equations. The description of the heat distribution in objects and of the vibrations
of strings are among the oldest applications from which the theory of Fourier series
has arisen. Together with the Fourier integrals for non-periodic functions from part
3, this theory as a whole is referred to as Fourier analysis.
Jean-Baptiste Joseph Fourier (1768 – 1830) was born in Auxerre, France, as the
son of a tailor. He was educated by Benedictine monks at a school where, after
finishing his studies, he became a mathematics teacher himself. In 1794 he went to
Paris, where he became mathematics teacher at the Ecole Normale. He declined a
professorial chair offered to him by the famous Ecole Polytechnique in order to join
Napoleon on his expedition to Egypt. In 1789 he was appointed governor of part of
Egypt. Ousted by the English, he left Egypt again in 1801 and became prefect of
Grenoble. Here he started with heat experiments and their mathematical analysis.
Fourier's mathematical ideas were not entirely new, but were built on earlier work
by Bernoulli and Euler. Fourier was, however, the first to boldly state that any
function could be developed into a series of sine and cosine functions. At first, his
contemporaries refused to accept this, and publication of his work was held up for
several years by the members of the Paris Acad´ mie des Sciences. The problem was
e
that his ideas were considered to be insufficiently precise. And indeed, Fourier could
not prove that for an arbitrary function the series would always converge pointwise
to the function values. Dirichlet was one of the first to find proper conditions under
which a Fourier series would converge pointwise to a periodic function. For the
further development of Fourier analysis, additional fundamentally new results were
required, like set-theory and the Lebesgue integral, which was developed in the one
and a half centuries following Fourier.
Historically, Fourier's work has contributed enormously to the development of
mathematics. Fourier set down his work in a book on the theory of heat, Th´ orie ane
alytique de la chaleur, published in 1822. The heat or diffusion equation occurring
here, as well as the wave equation for the vibrating string, can be solved, under the
most frequently occurring additional conditions, using Fourier series. The methods
that were used turned out to be much more widely applicable. Thereafter, applying
Fourier series would produce fruitful results in many different fields, even though
58
the mathematical theory had not yet fully crystallized. By now, the Fourier theory
has become a very versatile mathematical tool. From the times of Fourier up to the
present day, research has been carried out in this field, both concrete and abstract,
and new applications are being developed.
CHAPTER 3
Fourier series: definition
and properties
INTRODUCTION
Many phenomena in the applications of the natural and engineering sciences are periodic in nature. Examples are the vibrations of strings, springs and other objects,
rotating parts in machines, the movement of the planets around the sun, the tides of
the sea, the movement of a pendulum in a clock, the voltages and currents in electrical networks, electromagnetic signals emitted by transmittters in satellites, light
signals transmitted through glassfibers, etc. Seemingly, all these systems operate in
complicated ways; the phenomena that can be observed often behave in an erratic
way. In many cases, however, they do show some kind of repetition. In order to
analyse these systems, one can make use of elementary periodic functions or signals
from mathematics, the sine and cosine functions. For many systems, the response
or behaviour can be completely calculated or measured, by exposing them to influences or inputs given by these elementary functions. When, moreover, these systems
are linear, then one can also calculate the response to a linear combination of such
influences, since this will result in the same linear combination of responses.
Hence, for the study of the aforementioned phenomena, two matters are of importance.
On the one hand one should look at how systems behave under influences that
can be described by elementary mathematical functions. Such an analysis will in
general require specific knowledge of the system being studied. This may involve
knowledge about how forces, resistances, and inertias influence each other in mechanical systems, how fluids move under the influence of external forces, or how
voltages, currents and magnetic fields are mutually interrelated in electrical applications. In this book we will not go into these analyses, but the results, mostly
in terms of mathematical formulations, will often be chosen as a starting point for
further considerations.
On the other hand it is of importance to examine if and how an arbitrary periodic
function can be described as a linear combination of elementary sine and cosine
functions. This is the central theme of the theory of Fourier series: determine the
conditions under which periodic functions can be represented as linear combinations
of sine and cosine functions. In this chapter we study such linear combinations (also
with infinitely many functions). These combinations are called Fourier series and
the coefficients that occur are the Fourier coefficients. We will also determine the
Fourier series and the Fourier coefficients for a number of standard functions and
treat a number of properties of Fourier series.
In the next chapter we will examine the conditions under which a Fourier series
gives an exact representation of the original function.
60
3.1 Trigonometric polynomials and series
61
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know what trigonometric polynomials and series are, and know how to determine
their coefficients
- know the definitions of the real and complex Fourier coefficients and Fourier series
- can determine the real and complex Fourier series for a given periodic function
- can calculate and interpret the spectrum of a periodic function
- can determine the Fourier series for some standard functions
- know and can apply the most important properties of Fourier series
- can develop a function on a given interval into a Fourier cosine or a Fourier sine
series.
3.1
Sinusoidal function
Fundamental frequency
Trigonometric polynomials and series
The central problem of the theory of Fourier series is, how arbitrary periodic functions or signals might be written as a series of sine and cosine functions. The sine
and cosine functions are also called sinusoidal functions. (See section 1.2.2 for a
description of periodic functions or signals and section 2.4.3 for a description of
series of functions.) In this section we will first look at the functions that can be
constructed if we start from the sine and cosine functions. Next we will examine
how, given such a function, one can recover the sinusoidal functions from which
it is build up. In the next section this will lead us to the definition of the Fourier
coefficients and the Fourier series for arbitrary periodic functions.
The period of periodic functions will always be denoted by T . We would like to
approximate arbitrary periodic functions with linear combinations of sine and cosine
functions. These sine and cosine functions must then have period T as well. One can
easily check that the functions sin(2π t/T ), cos(2π t/T ), sin(4π t/T ), cos(4πt/T ),
sin(6πt/T ), cos(6πt/T ) and so on all have period T . The constant function also
has period T . Jointly, these functions can be represented by sin(2πnt/T ) and
cos(2πnt/T ), where n ∈ N. Instead of 2π/T one often writes ω0 , which means
that the functions can be denoted by sin nω0 t and cos nω0 t, where n ∈ N. All these
functions are periodic with period T . In this context, the constant ω0 is called the
fundamental frequency: sin ω0 t and cos ω0 t will complete exactly one cycle on an
interval of length T , while all functions sin nω0 t and cos nω0 t with n > 1 will complete several cycles. The frequencies of these functions are thus all integer multiples
of ω0 . See figure 3.1, where the functions sin nω0 t and cos nω0 t are sketched for
n = 1, 2 and 3. Linear combinations, also called superpositions, of the functions
sin3ω 0t
sin2ω 0t
sinω 0t
1
1
0
–T /2
T /2
–1
t
–T /2
0
–1
FIGURE 3.1
The sinusoidal functions sin nω0 t and cos nω0 t for n = 1, 2 and 3.
T /2
t
cosω 0t
cos2ω 0t
cos3ω 0t
62
3 Fourier series: definition and properties
sin nω0 t and cos nω0 t are again periodic with period T . If in such a combination we
include a finite number of terms, then the expression is called a trigonometric polynomial. Besides the sinusoidal terms, a constant term may also occur here. Hence,
a trigonometric polynomial f (t) with period T can be written as
Trigonometric polynomial
f (t) = A + a1 cos ω0 t + b1 sin ω0 t + a2 cos 2ω0 t + b2 sin 2ω0 t
2π
+ · · · + an cos nω0 t + bn sin nω0 t
.
with ω0 =
T
In figure 3.2a some examples of trigonometric polynomials are shown with ω0 = 1
and so T = 2π. The polynomials shown are
f 1 (t) = 2 sin t,
f 2 (t) = 2(sin t − 1 sin 2t),
2
f 3 (t) = 2(sin t − 1 sin 2t + 1 sin 3t),
2
3
f 4 (t) = 2(sin t − 1 sin 2t + 1 sin 3t − 1 sin 4t).
2
3
4
a
b
f1 f2 f3
π
π
f4
0
0
–2π
π
–π
–π
2π t
–2π
π
–π
2π t
–π
FIGURE 3.2
Some trigonometric polynomials (a) and the sawtooth function (b).
Periodic extension
Trigonometric series
In figure 3.2b the sawtooth function is drawn. It is defined as follows. On the
interval (−T /2, T /2) = (−π, π) one has f (t) = t, while elsewhere the function is
extended periodically, which means that it is defined by f (t + kT ) = f (t) for all
k ∈ Z. The function f (t) is then periodic with period T and is called the periodic
extension of the function f (t) = t. The function values at the endpoints of the
interval (−T /2, T /2) are not of importance for the time being and are thus not
taken into account for the moment. Comparing the figures 3.2a and 3.2b suggests
that the sawtooth function, a periodic function not resembling a sinusoidal function
at all, can in this case be approximated by a linear combination of sine functions
only. The trigonometric polynomials f 1 , f 2 , f 3 and f 4 above, are partial sums of
the infinite series ∞ (−1)n−1 (2/n) sin nt. It turns out that as more terms are
n=1
being included in the partial sums, the approximations improve. When an infinite
number of terms is included, one no longer speaks of trigonometric polynomials,
but of trigonometric series. The most important aspect of such series is, of course,
how well they can approximate an arbitrary periodic function. In the next chapter
it will be shown that for a piecewise smooth periodic function it is indeed possible
to find a trigonometric series whose sum converges at the points of continuity and is
equal to the function.
At this point it suffices to observe that in this way a large class of periodic functions can be constructed, namely the trigonometric polynomials and series, all based
upon the functions sin nω0 t and cos nω0 t. All functions f which can be obtained
64
Orthogonal
3 Fourier series: definition and properties
On the basis of the last three equations it is said that the functions from the set
{sin nω0 t and cos nω0 t with n ∈ N} are orthogonal: the integral of a product of
two distinct functions over one period is equal to 0.
After this enumeration of results, we now return to (3.1) and try to determine
the unknown coefficients A, an and bn for a given f (t). To this end we multiply
the left-hand and right-hand side of (3.1) by cos mω0 t and then integrate over the
interval (−T /2, T /2). It then follows that
T /2
f (t) cos mω0 t dt
−T /2
=
T /2
A+
−T /2
=A
∞
(an cos nω0 t + bn sin nω0 t) cos mω0 t dt
n=1
T /2
−T /2
cos mω0 t dt +
∞
T /2
an
n=1
∞
+
−T /2
cos nω0 t cos mω0 t dt
T /2
bn
n=1
−T /2
sin nω0 t cos mω0 t dt.
In this calculation we assume, for the sake of convenience, that the integral of the
series may be calculated by integrating each term in the series separately. We note
here that in general this has to be justified. If we now use the results stated above,
then all the terms will equal 0 except for the term with cos nω0 t cos mω0 t, where n
equals m. The integral in this term has value T /2, and so
T /2
−T /2
f (t) cos mω0 t dt = am
T
,
2
or
am =
2 T /2
f (t) cos mω0 t dt
T −T /2
for m = 1, 2, . . ..
(3.2)
This means that for a given f (t), it is possible to determine am using (3.2). In an
analogous way an expression can be found for bm . Multiplying (3.1) by sin mω0 t
and again integrating over the interval (−T /2, T /2), one obtains an expression for
bm (also see exercise 3.2).
A direct integration of (3.1) over (−T /2, T /2) gives an expression for the constant A:
T /2
−T /2
f (t) dt =
=
T /2
−T /2
T /2
−T /2
A+
∞
(an cos nω0 t + bn sin nω0 t) dt
n=1
A dt = T A
and so
A=
1 T /2
f (t) dt.
T −T /2
The right-hand side of this equality is, up to a factor 2, equal to the right-hand side
of (3.2) for m = 0, because cos 0ω0 t = 1. Hence, instead of A one usually takes
a0 /2:
a0 = 2A =
2 T /2
f (t) dt.
T −T /2
3.2 Definition of Fourier series
65
All coefficients in (3.1) can thus be determined if f (t) is a given trigonometric polynomial or series. The calculations are summarized in the following two expressions,
from which the coefficients can be found for all functions in the class of trigonometric polynomials and series, in so far as these coefficients exist and interchanging the
order of summation and integration, mentioned above, is allowed:
an =
2 T /2
f (t) cos nω0 t dt
T −T /2
for n = 0, 1, 2, . . .,
(3.3)
bn =
2 T /2
f (t) sin nω0 t dt
T −T /2
for n = 1, 2, . . ..
(3.4)
In these equations, the interval of integration is (−T /2, T /2). This interval is precisely of length one period. To determine the coefficients an and bn , one can in
general integrate over any other arbitrary interval of length T . Sometimes the interval (0, T ) is chosen (also see exercise 3.4).
EXERCISES
3.1
Verify that all functions sin nω0 t and cos nω0 t with n ∈ N and ω0 = 2π/T have
period T .
3.2
Prove that if f (t) is a trigonometric polynomial with period T , then bn can indeed
be found using (3.4).
3.3
In (3.3) and (3.4) the an and bn are defined for, respectively, n = 0, 1, 2, . . . and
n = 1, 2, . . .. Why isn't it very useful to include b0 in these expressions?
3.4
Verify that we obtain the same values for an if we integrate over the interval (0, T )
in (3.3).
3.2
Definition of Fourier series
In the previous section we demonstrated how, starting from a collection of elementary periodic functions, one can construct new periodic functions by taking linear
combinations. The coefficients in this combination could be recovered using formulas (3.3) and (3.4). These formulas can in principle be applied to any arbitrary
periodic function with period T , provided that the integrals exist. This is an important step: the starting point is now an arbitrary periodic function. To it, we then
apply formulas (3.3) and (3.4), which were originally only intended for trigonometric polynomials and series. The coefficients an and bn thus defined are called the
Fourier coefficients. The series in (3.1), which is determined by these coefficients,
is called the Fourier series.
For functions that are piecewise smooth, the integrals in (3.3) and (3.4) exist.
One can even show that such a function is equal to the Fourier series in (3.1) at
the points of continuity. The proof of this is postponed until chapter 4. But at
present, we will give the formal definitions of the Fourier coefficients and the Fourier
series of a periodic function. In section 3.2.1 we define the Fourier series using
the trigonometric functions sin nω0 t and cos nω0 t, in accordance with (3.1). In
many cases it is easier to work with a Fourier series with functions einω0 t (the timeharmonic signals, as in section 1.2.2). This complex Fourier series is introduced in
section 3.2.2. Through Euler's formula, these two expressions for the Fourier series
are immediately related to each other.
66
3 Fourier series: definition and properties
3.2.1
Fourier series
If, for an arbitrary periodic function f , the coefficients an and bn , as defined by (3.3)
and (3.4), can be calculated, then these coefficients are called the Fourier coefficients
of the function f .
DEFINITION 3.1
Fourier coefficients
Let f (t) be a periodic function with period T and fundamental frequency ω0 =
2π/T , then the Fourier coefficients an and bn of f (t), if they exist, are defined by
an =
2 T /2
f (t) cos nω0 t dt
T −T /2
for n = 0, 1, 2, . . .,
(3.5)
bn =
2 T /2
f (t) sin nω0 t dt
T −T /2
for n = 1, 2, . . ..
(3.6)
In definition 3.1 the integration is over the interval (−T /2, T /2). One can, however, integrate over any arbitrary interval of length T . The only thing that matters is
that the length of the interval of integration is exactly one period (also see exercise
3.4).
In fact, in definition 3.1 a mapping or transformation is defined from functions to
number sequences. This is also denoted as a transformation pair:
f (t) ↔ an , bn .
Fourier transform
DEFINITION 3.2
Fourier series
One should pronounce this as: 'to the function f (t) belong the Fourier coefficients
an and bn '. This mapping is the Fourier transform for periodic functions. The function f (t) can be complex-valued. In that case, the coefficients an and bn will also
be complex. Using definition 3.1 one can now define the Fourier series associated
with a function f (t).
When an and bn are the Fourier coefficients of the periodic function f (t) with
period T and fundamental frequency ω0 = 2π/T , then the Fourier series of f (t) is
defined by
∞
a0
(an cos nω0 t + bn sin nω0 t).
+
2
n=1
(3.7)
We do emphasize here that for arbitrary periodic functions the Fourier series will
not necessarily converge for all t, and in case of convergence will not always equal
f (t). In chapter 4 it will be proven that for piecewise smooth functions the series
does equal f (t) at the points of continuity.
EXAMPLE 3.1
In section 3.1 it was suggested that the series ∞ (−1)n−1 (2/n) sin nt approxin=1
mates the sawtooth function f (t), given by f (t) = t for t ∈ (−π, π) and having
period 2π. We will now check that the Fourier coefficients of the sawtooth function
are indeed equal to the coefficients in this series. In the present situation we have
T = 2π, so ω0 = 2π/T = 1. The definition of Fourier coefficients can immediately be applied to the function f (t). Using integration by parts it follows for n ≥ 1
that
π
2 T /2
1 π
1
f (t) cos nω0 t dt =
t cos nt dt =
t (sin nt) dt
T −T /2
π −π
nπ −π
π
1
1
1
sin nt dt = 2 [cos nt]π = 0.
=
[t sin nt]π −
−π
−π
nπ
nπ −π
n π
an =
68
3 Fourier series: definition and properties
Amplitude
Initial phase
2
2
The factor an + bn is the amplitude of the nth harmonic, φn the initial phase.
Hence, the Fourier series can also be written as the sum of infinitely many harmonics, written exclusively in cosines. The amplitude of the nth harmonic tells us its
weight in the Fourier series. From the initial phase one can deduce how far the nth
harmonic is shifted relative to cos nω0 t.
EXAMPLE
Suppose that a function with period T = 2π has Fourier coefficients a1 = 1, a2 =
1/2, b2 = 1/2 and that all other coefficients are 0. Since ω0 = 2π/T = 1, the
Fourier series is then
cos t +
1
1
cos 2t + sin 2t.
2
2
The first harmonic is cos t, with amplitude 1 and initial phase 0. The amplitude of
√
√
the second harmonic is (1/2)2 + (1/2)2 = 1/2 = 2/2, while its initial phase
follows from tan φ2 = −1, so φ2 = −π/4. For the second harmonic we thus have
π
1
1√
1
2 cos 2t −
cos 2t + sin 2t =
.
2
2
2
4
3.2.2
Complex Fourier series
In many cases it is easier to work with another representation of the Fourier series.
One then doesn't use the functions sin nω0 t and cos nω0 t, but instead the functions
einω0 t . Euler's formula gives the connection between these functions, making it
possible to derive one formulation of the Fourier series from the other. According
to (2.11) one has
cos nω0 t =
einω0 t + e−inω0 t
2
and
sin nω0 t =
einω0 t − e−inω0 t
.
2i
If we substitute this into (3.7), it follows that
∞
a0
+
(an cos nω0 t + bn sin nω0 t)
2
n=1
=
∞
einω0 t + e−inω0 t
einω0 t − e−inω0 t
a0
+
− ibn
an
2
2
2
n=1
=
∞
a0
1
1
+
(an − ibn )einω0 t + (an + ibn )e−inω0 t
2
2
2
n=1
= c0 +
∞
cn einω0 t + c−n e−inω0 t =
∞
cn einω0 t .
n=−∞
n=1
Here the coefficients cn are defined as follows:
a
c0 = 0 ,
2
cn =
1
(an − ibn ),
2
c−n =
1
(an + ibn )
2
for n ∈ N.
(3.8)
Instead of a Fourier series with coefficients an and bn and the functions cos nω0 t
and sin nω0 t with n ∈ N, one can thus also construct, for a periodic function f (t),
a series with (complex) coefficients cn and time-harmonic functions einω0 t with
n ∈ Z. The coefficients cn are the complex Fourier coefficients. They can be
calculated from the coefficients an and bn using (3.8), but they can also be derived
directly from the function f (t). To this end, one should substitute for an and bn
3.2 Definition of Fourier series
69
in (3.8) the definitions (3.5) and (3.6) (see also exercise 3.5). This leads to the
following definition for the complex Fourier coefficients.
DEFINITION 3.3
Complex Fourier coefficients
Let f (t) be a periodic function with period T and fundamental frequency ω0 =
2π/T . Then the complex Fourier coefficients cn of f (t), whenever they exist, are
defined by
cn =
1 T /2
f (t)e−inω0 t dt
T −T /2
for n ∈ Z.
(3.9)
The complex Fourier coefficients have the prefix 'complex' since they've been
determined using complex exponentials, namely, the time-harmonic signals. This
prefix has thus nothing to do with the coefficients being themselves complex or not.
Like the Fourier coefficients from definition 3.1, the complex Fourier coefficients
from definition 3.3 can also be calculated with an integral over an interval that differs
from (−T /2, T /2), as long as the interval has length T .
The mapping defined by (3.9) will also be denoted by the transformation pair
f (t) ↔ cn .
Using the complex Fourier coefficients thus defined, we can now introduce the complex Fourier series associated with a periodic function f (t).
DEFINITION 3.4
Complex Fourier series
When cn are the complex Fourier coefficients of the periodic function f (t) with
period T and fundamental frequency ω0 = 2π/T , then the complex Fourier series
of f (t) is defined by
∞
cn einω0 t .
(3.10)
n=−∞
Hence, for periodic functions for which the complex Fourier coefficients exist, a
complex Fourier series exists as well. In chapter 4 it will be proven that for piecewise smooth functions the Fourier series converges to the function at the points of
continuity.
In (3.8) the complex Fourier coefficients were derived from the real ones. Conversely one can derive the coefficients an and bn from cn using
an = cn + c−n
and
bn = i(cn − c−n ).
(3.11)
Therefore, when determining the Fourier series one has a choice between the real
and the complex form. The coefficients can always be expressed in each other using
(3.8) and (3.11). From (3.5) and (3.6) it follows that for real periodic functions the
coefficients an and bn assume real values. From (3.8) it can then immediately be
deduced that cn and c−n are each other's complex conjugates:
c−n = cn
when f is real.
(3.12)
Since for real functions cn and c−n are each other's complex conjugates, we obtain
from (3.11) that
an = 2Re cn
and
bn = −2Im cn
when f is real.
(3.13)
In the next example we calculate for the sawtooth function, which we already encountered in section 3.1 and example 3.1, the complex Fourier coefficients in a
direct way. Moreover, we will verify that the coefficients an , bn and cn can indeed
be obtained from each other.
3.3 The spectrum of periodic functions
3.3
71
The spectrum of periodic functions
The periodic functions, or periodic signals, that we have considered so far, both
the real and the complex ones, were all defined for t ∈ R. Here the variable t is
often interpreted as a time-variable. We say that these functions are defined in the
time domain. For these functions the Fourier coefficients cn (or an and bn ) can be
determined. Through the Fourier series, each of these coefficients is associated with
a function of a specific frequency nω0 . The values of the Fourier coefficients tell us
the weight of the function with frequency nω0 in the Fourier series. For piecewise
smooth functions we will establish in chapter 4 that the Fourier series is equal to the
function. This means that these functions are completely determined by their Fourier
coefficients. Since the coefficients cn (or an and bn ) are associated with frequency
nω0 , we then say that the function f (t) is described by the Fourier coefficients in
the frequency domain. As soon as the values cn are known, the original function in
the time domain is also fixed.
In daily life as well, we often interpret signals in terms of frequencies. Sound and
light are quantities that are expressed in terms of frequencies, and we observe these
as pitch and colour.
The sequence of Fourier coefficients cn with n ∈ Z, which thus describe a function in the frequency domain, is called the spectrum of the function. Since n assumes
only integer values, the spectrum is called a discrete or a line spectrum. Often, not
the spectrum itself is given, but instead the amplitude spectrum | cn | and the phase
spectrum arg(cn ). Hence, the amplitude and phase spectrum are defined as soon as
the complex Fourier coefficients exist, also in the case when the function f (t) is
complex-valued. This definition of amplitude and phase is thus more general than
the one for the nth harmonic, which only existed in the case when f (t) was real.
Time domain
Frequency domain
Spectrum
Discrete spectrum
Line spectrum
Amplitude spectrum
Phase spectrum
EXAMPLE 3.3
Figure 3.4 shows the amplitude and phase spectrum of the sawtooth function, for
which we deduced in example 3.2 that the complex Fourier coefficients cn are given
by cn = (−1)n (i/n) for n = 0 and c0 = 0. The amplitude spectrum is thus given
by | cn | = 1/ | n | for n = 0 and | c0 | = 0, while the phase spectrum is given by
arg(cn ) = (−1)n (π/2) for n > 0 and by arg(cn ) = (−1)n−1 (π/2) for n < 0, and
is undefined for n = 0.
a
b
1
–3
–2
–1
0
π
2
|cn |
1
2
3
n
arg(cn)
0
n
–π
2
FIGURE 3.4
The amplitude spectrum (a) and phase spectrum (b) of the sawtooth function.
EXERCISES
3.10
Determine and sketch the amplitude and phase spectra of the functions from exercises 3.6 to 3.9.
72
3 Fourier series: definition and properties
3.4
Even function
Odd function
Fourier series for some standard functions
In the preceding sections the Fourier coefficients of the sawtooth function have been
determined. It will be convenient to know the Fourier series for some other standard
functions as well. Together with the properties, to be treated in the next section,
this will enable us to determine the Fourier series for quite a number of periodic
functions relatively easily. In this section we determine the Fourier series for a
number of functions. These standard functions and their Fourier coefficients are
also included in table 1 at the back of the book. The first two functions that will
be treated are even, that is, f (t) = f (−t). The third function will be odd, that is,
f (t) = − f (−t).
3.4.1
Periodic block function
The periodic block function
The periodic function pa,T (t) with period T > 0 and 0 ≤ a ≤ T and having value
1 for | t | ≤ a/2 ≤ T /2 and value 0 for a/2 < | t | ≤ T /2 is called the periodic
block function. Its graph is sketched in figure 3.5. The complex Fourier coefficients
1
0
–3T/2
–T
–T/2
–a/2
a/2
T/2
T
3T/2
t
FIGURE 3.5
The periodic block function pa,T (t).
of the periodic block function can be calculated for n = 0 using (3.9):
cn =
=
1 T /2
1 a/2 −inω0 t
1
pa,T (t)e−inω0 t dt =
e
dt =
T −T /2
T −a/2
T
2
T nω0
einω0 a/2 − e−inω0 a/2
2i
=
e−inω0 t
−inω0
a/2
−a/2
2 sin(nω0 a/2)
.
T
nω0
For n = 0 it follows that
c0 =
1 T /2
1 a/2
a
pa,T (t) dt =
1 dt = .
T −T /2
T −a/2
T
For a given value of a, the Fourier coefficients for n = 0 are thus equal to
2 sin(nω0 a/2)/T nω0 , which are precisely the values of 2 sin(ax/2)/T x evaluated
at x = nω0 , where n runs through the integers. In figure 3.6 the function f (x) =
2 sin(ax/2)/T x is drawn; for x = 0 the function is defined by limx→0 f (x) =
a/T = c0 . From this we can obtain the Fourier coefficients of the periodic block
function by evaluating the function values at nω0 for n ∈ Z. One thus has for the
periodic block function:
pa,T (t) ↔
2 sin(nω0 a/2)
.
T
nω0
Here one should take the value limx→0 2 sin(ax/2)/T x for n = 0.
(3.14)
76
3 Fourier series: definition and properties
a
T
4 sin2(ax/2)
ax 2T
π
a
–π
a
–4ω 0 –3ω 0 –2ω 0 –ω 0 0
ω0
2ω 0 3ω 0 4ω 0
x
FIGURE 3.10
Evaluating the function values of 4 sin2 (ax/2)/ax 2 T at x = nω0 for n ∈ Z gives
the Fourier coefficients of the periodic triangle function.
b Now determine the Fourier coefficients for a = T . What do these Fourier coefficients imply for the Fourier series?
3.12
Determine for a = T /2 the Fourier coefficients and the amplitude spectrum of the
periodic triangle function from section 3.4.2.
3.13
Determine the Fourier coefficients of the sawtooth function given by f (t) = 2t/T
on the interval (−T /2, T /2) and extended periodically elsewhere, and sketch the
amplitude and phase spectrum.
3.5
Properties of Fourier series
In the previous section Fourier series were determined for a number of standard
functions. In the same way one can, in principle, determine the Fourier series for
many more periodic functions. This, however, is quite cumbersome. By using a
number of properties of Fourier series one can determine in a relatively simple way
the Fourier series of a large number of periodic functions. These properties have
also been included in table 2 at the back of the book.
3.5.1
Linearity
Fourier coefficients of linear combinations of functions are equal to the same linear
combination of the Fourier coefficients of the individual functions. This property is
formulated in the following theorem.
THEOREM 3.1
Linearity of the Fourier
transform
When the complex Fourier coefficients of f (t) and g(t) are f n and gn respectively,
then one has for a, b ∈ C:
a f (t) + bg(t) ↔ a f n + bgn .
3.5 Properties of Fourier series
77
Proof
The proof of this theorem is a straightforward application of the linearity of integration. When cn denotes the Fourier coefficients of a f (t) + bg(t), then
cn =
=
EXAMPLE 3.5
1 T /2
(a f (t) + bg(t))e−inω0 t dt
T −T /2
b T /2
a T /2
f (t)e−inω0 t dt +
g(t)e−inω0 t dt = a f n + bgn .
T −T /2
T −T /2
With the linearity property one can easily determine the Fourier coefficients of linear
combinations of functions whose individual Fourier coefficients are already known.
Let f be the periodic function with period 6 as sketched in figure 3.11. The function
2
–3
f
0
3
t
–1
FIGURE 3.11
Periodic function as a combination of periodic block functions.
f is then equal to 2g − 3h, where g and h are periodic block functions, as defined
in section 3.4.1, with period 6 and, respectively, a = 4 and a = 2. Using (3.14)
or table 1 and applying theorem 3.1, it then follows that the Fourier coefficients are
given by
cn =
=
3.5.2
4 sin(nω0 4/2) 6 sin(nω0 2/2)
−
6
nω0
6
nω0
2 sin(2nπ/3) − 3 sin(nπ/3)
2 sin(n2π/3) sin(nπ/3)
−
=
.
3 n2π/6
n2π/6
nπ
Conjugation
The Fourier coefficients of the complex conjugate of f can be derived from the
Fourier coefficients of the function itself. How this can be done is the subject of our
next theorem.
THEOREM 3.2
Fourier coefficients of a
conjugate
When the Fourier coefficients of f (t) are equal to cn , then
f (t) ↔ c−n .
78
3 Fourier series: definition and properties
Proof
Since einω0 t = e−inω0 t , it follows by direct calculation of the Fourier coefficients
of f (t) that
1 T /2
1 T /2
f (t)e−inω0 t dt =
f (t)einω0 t dt
T −T /2
T −T /2
=
1 T /2
f (t)e−i(−n)ω0 t dt = c−n .
T −T /2
This property has a special consequence when f (t) is real. This is because we
then have f (t) = f (t). The Fourier coefficients of f (t) and f (t), whenever these
exist at least, must then also be equal and hence cn = c−n . This result has been
derived before, see (3.12). Furthermore, one has for the moduli that | cn | = | c−n |
and since the moduli of complex conjugates are the same (see (2.3)), this in turn
equals | c−n |. For a real function it thus follows that | cn | = | c−n |, which means
that the amplitude spectrum is even. We also know that the arguments of complex
conjugates are each other's opposite, and so arg(cn ) = arg(c−n ) = − arg(c−n ).
Hence, the phase spectrum is odd.
EXAMPLE
The standard functions treated in section 3.4 are all real. One can easily check that
the amplitude spectra are indeed even. For the sawtooth function one can check
moreover that the phase spectrum is odd, while the phase spectra of the periodic
block and triangle functions are zero, and so odd as well.
3.5.3
Shift in time
The standard functions treated in section 3.4 were all neatly 'centred' around t = 0.
From these one can, by a shift in time, obtain functions that are, of course, no longer
centred around t = 0. When the shift equals t0 , then the new function will be given
by f (t − t0 ). The Fourier coefficients of the shifted function can immediately be
obtained from the Fourier coefficients of the original function.
THEOREM 3.3
Shift in time
When cn are the Fourier coefficients of f (t), then
f (t − t0 ) ↔ e−inω0 t0 cn .
Proof
The Fourier coefficients of f (t − t0 ) can be calculated using the definition. In
this calculation we introduce the new variable τ = t − t0 and we integrate over
(−T /2, T /2) instead of ((−T /2) + t0 , (T /2) + t0 ), since this gives the same result:
1 T /2
f (t − t0 )e−inω0 t dt
T −T /2
1 (T /2)+t0
f (t − t0 )e−inω0 (t−t0 ) d(t − t0 )
= e−inω0 t0
T (−T /2)+t0
= e−inω0 t0
1 T /2
f (τ )e−inω0 τ dτ = e−inω0 t0 · cn .
T −T /2
It follows immediately from theorem 3.3 that the amplitude spectra of f (t) and
f (t − t0 ) are the same: |e−inω0 t0 cn | = | cn |. Hence, the amplitude spectrum of
80
3 Fourier series: definition and properties
A direct consequence of this theorem is that if f (t) is even, so f (t) = f (−t)
for all t, then cn = c−n . In this case the amplitude spectrum as well as the phase
spectrum are even. Furthermore, it follows from (3.8) that the coefficients bn of
the ordinary Fourier series are all 0. Thus, the Fourier series of an even function
contains only cosine terms. This result is easily understood: the sines are odd, while
the cosines are even. A series containing sine functions will never be even. When,
moreover, f (t) is real, then cn and c−n are each other's complex conjugate (see
(3.12)) and in that case the coefficients will be real as well.
EXAMPLE
The periodic block function and the periodic triangle function from sections 3.4.1
and 3.4.2 are even and real. The spectra are also even and real.
When f (t) is odd, so f (t) = − f (−t), it follows that cn = −c−n and so the
spectrum is odd. Since c−n = cn for f (t) real, the Fourier coefficients are purely
imaginary. The spectrum of a real and odd function is thus odd and purely imaginary. Moreover, in the case of an odd function it follows from (3.8) that the coefficients an are 0 and that the Fourier series consists of sine functions only.
EXAMPLE
The periodic sawtooth function is a real and odd function. The complex Fourier
coefficients are odd and purely imaginary, while the Fourier series contains only
sine functions.
EXERCISES
3.14
The periodic function f with period 4 is given by f (t) = 1 + | t | for | t | ≤ 1 and
f (t) = 0 for 1 < | t | < 2. Sketch the graph of the function and determine its
Fourier coefficients.
3.15
Determine the Fourier coefficients of the periodic function with period T defined by
f (t) = t on the interval (0, T ).
3.16
Let the complex-valued function f (t) = u(t) + iv(t) be given, where u(t) and v(t)
are real functions with Fourier coefficients u n and v n .
a Determine the Fourier coefficients of f (t) and of f (t).
b Suppose that f (t) is even, but not real. Will the Fourier coefficients of f (t) be
even and real then?
3.17
The amplitude spectrum of a function does not change when a shift in time is
applied. For which shifts does the phase spectrum remains unchanged as well?
3.18
In section 3.5.4 we derived that for even functions the ordinary Fourier series contains only cosine terms. Show that this also follows directly from (3.5) and (3.6).
3.6
Fourier sine series
Fourier cosine series
Fourier cosine and Fourier sine series
In section 3.5.4 we showed that the ordinary Fourier series of an even periodic function contains only cosine terms and that the Fourier series of an odd periodic function contains only sine terms. For the standard functions we have seen that the periodic block function and the periodic triangle function, which are even, do indeed
contain cosine terms only and that the sawtooth function, which is odd, contains
sine terms only. Sometimes it is desirable to obtain for an arbitrary function on the
interval (0, T ) a Fourier series containing only sine terms or containing only cosine
terms. Such series are called Fourier sine series and Fourier cosine series. In order
to find a Fourier cosine series for a function defined on the interval (0, T ), we extend
the function to an even function on the interval (−T, T ) by defining f (−t) = f (t)
for −T < t < 0 and subsequently extending the function periodically with period
3.6 Fourier cosine and Fourier sine series
Forced series development
EXAMPLE
81
2T . The function thus created is now an even function and its ordinary Fourier series
will contain only cosine terms, while the function is equal to the original function
on the interval (0, T ).
In a similar way one can construct a Fourier sine series for a function by extending
the function defined on the interval (0, T ) to an odd function on the interval (−T, T )
and subsequently extending it periodically with period 2T . Such an odd function
will have an ordinary Fourier series containing only sine terms.
Determining a Fourier sine series or a Fourier cosine series in the way described
above is sometimes called a forced series development.
Let the function f (t) be given by f (t) = t 2 on the interval (0, 1). We wish to obtain
a Fourier sine series for this function. We then first extend it to an odd function
on the interval (−1, 1) and subsequently extend it periodically with period 2. The
function and its odd and periodic extension are drawn in figure 3.13. The ordinary
1
1
0
1
–2
1
–1
2
–1
FIGURE 3.13
The function f (t) = t 2 on the interval (0, 1) and its odd and periodic extension.
Fourier coefficients of the function thus created can be calculated using (3.5) and
(3.6). Since the function is odd, all coefficients an will equal 0. For bn we have
bn =
0
1
2 T /2
f (t) sin nω0 t dt =
(−t 2 ) sin nπ t dt +
t 2 sin nπ t dt
T −T /2
−1
0
= 2
1
t 2 sin nπ t dt.
0
Applying integration by parts twice, it follows that
−2
nπ
2
=
nπ
bn =
1
2
2
[t sin nπt]1 − 2 2 [cos nπt]1
0
0
nπ
n π
n − 1)
2 2((−1)
2(cos nπ − 1)
− cos nπ =
− (−1)n .
2π 2
nπ
n
n2π 2
t 2 cos nπt
0
−
The Fourier sine series of f (t) = t 2 on the interval (0, 1) is thus equal to
∞
2
nπ
n=1
EXAMPLE
2((−1)n − 1)
− (−1)n sin nπt.
n2π 2
In this final example we will show that one can even obtain a Fourier cosine series
for the sine function on the interval (0, π). To this end we first extend sin t to an
even function on the interval (−π, π) and then extend it periodically with period
2π; see figure 3.14. The ordinary Fourier coefficients of the function thus created
can be calculated using (3.5) and (3.6). Since the function is even, all coefficients
3.6 Fourier cosine and Fourier sine series
83
SUMMARY
Trigonometric polynomials and series are, respectively, finite and infinite linear
combinations of the functions cos nω0 t and sin nω0 t with n ∈ N. They are all
periodic with period T = 2π/ω0 . When a trigonometric polynomial f (t) is given,
the coefficients in the linear combination can be calculated using the formulas
an =
2 T /2
f (t) cos nω0 t dt
T −T /2
for n = 0, 1, 2, . . .,
bn =
2 T /2
f (t) sin nω0 t dt
T −T /2
for n = 1, 2, . . ..
These formulas can be applied to any arbitrary periodic function, provided that the
integrals exist. The numbers an and bn are called the Fourier coefficients of the function f (t). Using these coefficients one can then form a Fourier series of the function f (t):
∞
a0
+
(an cos nω0 t + bn sin nω0 t).
2
n=1
Instead of a Fourier series with functions cos nω0 t and sin nω0 t one can also obtain
a complex Fourier series with the time-harmonic functions einω0 t . The complex
Fourier coefficients can then be calculated using
cn =
1 T /2
f (t)e−inω0 t dt
T −T /2
for n ∈ Z,
while the complex Fourier series has the form
∞
cn einω0 t .
n=−∞
These two Fourier series can immediately be converted into each other and, depending on the application, one may chose either one of these forms.
The sequence of Fourier coefficients (cn ) is called the spectrum of the function. This is usually split into the amplitude spectrum | cn | and the phase spectrum
arg(cn ).
For some standard functions the Fourier coefficients have been determined. Moreover, a number of properties were derived making it possible to find the Fourier
coefficients for far more functions and in a much simpler way than by a direct
calculation.
Even functions have Fourier series containing cosine terms only. Series like this
are called Fourier cosine series. Odd functions have Fourier sine series. When
desired, one can extend a function, given on a certain interval, in an even or an odd
way, so that they can be forced into a Fourier cosine or a Fourier sine series.
SELFTEST
3.23
The function f (t) is periodic with period 10 and is drawn on the interval (−5, 5) in
figure 3.15. Determine the ordinary and complex Fourier coefficients of f .
3.24
Show that when for a real function f the complex Fourier coefficients are real, f
has to be even, and when the complex Fourier coefficients are purely imaginary, f
has to be odd.
CHAPTER 4
The fundamental theorem of Fourier
series
INTRODUCTION
Chapter 3 has been a first introduction to Fourier series. These series can be associated with periodic functions. We also noted in chapter 3 that if the function satisfies
certain conditions, the Fourier series converges to the periodic function. What these
specific conditions should be has not been analysed in chapter 3. The conditions that
will be imposed in this book imply that the function should be piecewise smooth. In
this chapter we will prove that a Fourier series of a piecewise smooth periodic function converges pointwise to the periodic function. We stress here that this condition
is sufficient: when it holds, the series is pointwise convergent. This condition does
not cover all cases of pointwise convergence and is thus not necessary for convergence.
In the first section of this chapter we derive a number of properties of Fourier coefficients that will be useful in the second section, where we prove the fundamental
theorem. In the fundamental theorem we prove that for a piecewise smooth periodic
function the Fourier series converges to the function. In the third section we then
derive some further properties of Fourier series: product and convolution, Parseval's
theorem (which has applications in the analysis of systems and signals), and integration and differentiation of Fourier series. We end this chapter with the treatment
of Gibbs' phenomenon, which describes the convergence behaviour of the Fourier
series at a jump discontinuity. This is then also an appropriate occasion to introduce
the function Si(x), the sine integral. This function will re-appear in other chapters
as well.
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- can formulate Bessel's inequality and the Riemann–Lebesgue lemma
- can formulate and apply the fundamental theorem of Fourier series
- can calculate the sum of a number of special series using the fundamental theorem
- can determine the Fourier series of the product and the convolution of two periodic
functions
- can formulate and apply Parseval's theorem
- can integrate and differentiate Fourier series
- know the definition of the sine integral and know its limit
- can explain Gibbs' phenomenon∗ .
4.1
Bessel's inequality and Riemann–Lebesgue lemma
In chapter 3 we always assumed in the definition of the Fourier coefficients that
the integrals, necessary for the calculation of the coefficients, existed. As such,
86
4.1 Bessel's inequality and Riemann–Lebesgue lemma
87
this is not a very important problem: if the integrals do not exist, then the Fourier
coefficients do not exist and a further Fourier analysis is then impossible. In this
chapter we confine ourselves to piecewise continuous periodic functions and for
these it is easy to verify that the Fourier coefficients exist; see section 2.3.
As soon as the Fourier coefficients of a periodic function exist, it is, however, by
no means obvious that the Fourier series converges to the function. In this chapter
we will present conditions under which convergence is assured.
In the present section we will first treat some properties of the Fourier coefficients
that will be needed later on. First we show that the sum of the squares of the Fourier
coefficients of a piecewise continuous function is finite. This is called Bessel's
inequality. Next we show that the Fourier coefficients cn of a piecewise continuous
function tend to 0 as n → ±∞. This is called the Riemann–Lebesgue lemma and
is needed in the next section to prove the fundamental theorem of Fourier series.
THEOREM 4.1
Bessel's inequality
When cn are the Fourier coefficients of a piecewise continuous periodic function
f (t) with period T , then
∞
| cn |2 ≤
n=−∞
1 T /2
| f (t) |2 dt.
T −T /2
(4.1)
Proof
In the proof we use the partial sums of the Fourier series of f (t) with Fourier coefficients ck . We denote these by sn (t), so
n
sn (t) =
ck eikω0 t .
(4.2)
k=−n
For a fixed value of n with −n ≤ k ≤ n we now calculate
1 T /2
( f (t) − sn (t)) e−ikω0 t dt
T −T /2
T /2
1 n
1 T /2
f (t)e−ikω0 t dt −
cl
ei(l−k)ω0 t dt.
=
T −T /2
T l=−n
−T /2
The first integral in the right-hand side is precisely the definition of the Fourier
coefficient ck . The integrals in the sum in the right-hand side are all equal to 0 for
l = k. When l = k, the integrand is 1 and the integral T , so ultimately the sum
equals ck . We thus have
1 T /2
( f (t) − sn (t)) e−ikω0 t dt = ck − ck = 0
T −T /2
for −n ≤ k ≤ n.
Using this result we calculate the following integral:
T /2
−T /2
T /2
n
( f (t) − sn (t)) sn (t) dt =
ck
k=−n
−T /2
( f (t) − sn (t)) e−ikω0 t dt = 0.
If we now multiply f (t) − sn (t) in the integral in the left-hand side not by sn (t), but
by f (t) − sn (t), it follows that
T /2
−T /2
=
( f (t) − sn (t)) ( f (t) − sn (t)) dt =
T /2
−T /2
f (t) f (t) dt −
T /2
−T /2
T /2
−T /2
sn (t) f (t) dt
( f (t) − sn (t)) f (t) dt
88
4 The fundamental theorem of Fourier series
T /2
=
−T /2
T /2
=
−T /2
T /2
=
−T /2
| f (t) |2 dt −
T /2
n
ck
k=−n
| f (t) |2 dt −
T /2
n
ck
k=−n
| f (t) |2 dt −
eikω0 t f (t) dt
−T /2
−T /2
e−ikω0 t f (t) dt
n
ck T ck =
k=−n
T /2
−T /2
| f (t) |2 dt − T
n
| ck |2 .
k=−n
The first term in this series of equalities is greater than or equal to 0, since
( f (t) − sn (t)) ( f (t) − sn (t)) = | f (t) − sn (t) |2 ≥ 0. The last term must then
also be greater than or equal to 0, which means that
n
| ck |2 ≤
T
k=−n
T /2
−T /2
| f (t) |2 dt.
This inequality holds for any n ∈ N, while the right-hand side is independent of n.
One thus has
∞
| cn |2 = lim
n→∞
n=−∞
n
k=−n
| ck |2 ≤
1 T /2
| f (t) |2 dt.
T −T /2
If f (t) is a piecewise continuous function, then | f (t) |2 is one as well, and so
the right-hand side of inequality (4.1) is finite. In particular it then follows that the
2
series ∞
n=−∞ | cn | converges. Hence, we must have cn → 0 as n → ±∞. This
result is known as the Riemann–Lebesgue lemma.
THEOREM 4.2
Riemann–Lebesgue lemma
If f (t) is a piecewise continuous periodic function with Fourier coefficients cn , then
lim cn =
n→∞
lim cn = 0.
n→−∞
(4.3)
Theorem 4.2 can be interpreted as follows. In order to calculate the coefficients
cn , the function f (t) is multiplied by e−inω0 t and integrated over one period. For
increasing n, the frequency of the corresponding sine and cosine functions keeps
increasing. Now consider two consecutive intervals such that, for example, the sine
function is negative in the first interval and positive in the second. For increasing
n, and hence for ever smaller intervals, the value of f in the first and in the second
interval will differ less and less. Multiplied by first a negative and then a positive
sine function, the contributions to the integral will cancel each other better and better
for increasing n. In this way the coefficients will eventually converge to 0.
EXAMPLE
The periodic block function, introduced in section 3.4.1, is piecewise continuous.
The Fourier coefficients, which have also been calculated there, are equal to
2 sin(nω0 a/2)/T nω0 . The numerator ranges for increasing n between −1 and 1,
while the denominator tends to infinity as n → ∞. For the Fourier coefficients we
thus have
lim
2 sin(nω0 a/2)
= 0.
nω0
n→∞ T
The Fourier coefficients of the periodic block function thus tend to 0 as n → ∞.
Similarly one can check that the same is true for the Fourier coefficients of the
periodic triangle function and the sawtooth function from sections 3.4.2
and 3.4.3.
4.2 The fundamental theorem
89
EXERCISES
4.1
a Check for the periodic block function and the periodic triangle function from
section 3.4 whether or not they are piecewise smooth and whether or not the Fourier
coefficients cn exist.
2
b Show for the functions from part a that ∞
n=−∞ | cn | is convergent.
4.2
The ordinary Fourier coefficients of a piecewise continuous periodic function are an
and bn (see definition 3.1).
a Prove that limn→∞ an = limn→∞ bn = 0. To do so, start from theorem 4.2.
b Now assume that limn→∞ an = limn→∞ bn = 0. Use this to prove that
limn→±∞ cn = 0.
4.3
Check that the Fourier coefficients cn of the periodic sawtooth function tend to 0 as
n → ∞, in accordance with the Riemann–Lebesgue lemma.
4.2
The fundamental theorem
In chapter 3 we have noted more than once that at the points of continuity, the
Fourier series of a piecewise smooth periodic function is equal to that function. This
statement, formulated in the fundamental theorem of Fourier series, will be proven
in this section. This will involve pointwise convergence. Before we go into this, we
first introduce the so-called Dirichlet kernel. This is a function that will be needed
in the proof of the fundamental theorem. We will also deduce some properties of
the Dirichlet kernel.
DEFINITION 4.1
Dirichlet kernel
The Dirichlet kernel Dn (x) is defined by
n
Dn (x) =
e−ikω0 x = einω0 x + ei(n−1)ω0 x + ei(n−2)ω0 x + · · · + e−inω0 x .
k=−n
The Dirichlet kernel is a periodic function, which can be considered as a geometric series with 2n + 1 terms, starting with einω0 x and having ratio e−iω0 x . In
example 2.16 the sum of a geometric series was determined. From this it follows
that for e−iω0 x = 1 one has that
Dn (x) =
einω0 x (1 − e−iω0 (2n+1)x )
.
1 − e−iω0 x
(4.4)
Performing the multiplication in the numerator and multiplying numerator and denominator by eiω0 x/2 , it follows upon using (2.11) that
Dn (x) =
ei(n+1/2)ω0 x − e−i(n+1/2)ω0 x
sin((n + 1/2)ω0 x)
.
=
sin(ω0 x/2)
eiω0 x/2 − e−iω0 x/2
(4.5)
The above is valid for e−iω0 x = 1, that is to say, for ω0 x = k · 2π, or x = k · T with
2
k ∈ Z. If, however, we do have x = k · T , then e−ikω0 x = e−ik 2π = 1. From the
definition of the Dirichlet kernel it then immediately follows that Dn (k ·T ) = 2n +1
for k ∈ Z. Furthermore, it is easy to see that the Dirichlet kernel is an even function.
In figure 4.1 the graph of the Dirichlet kernel is sketched for n = 6. When n
increases, the number of oscillations per period increases. The peaks at the points
x = kT continue to exist and increase in value.
90
4 The fundamental theorem of Fourier series
13
D 6(x )
–T
T
0
x
FIGURE 4.1
The Dirichlet kernel for n = 6.
The integral of Dn (x) over one period is independent of n.
T /2 −ikω0 x
d x = 0 for k = 0, it follows that
−T /2 e
T /2
n
−T /2
Dn (x) d x =
T /2
k=−n −T /2
e−ikω0 x d x =
T /2
−T /2
In fact, since
1 d x = T.
(4.6)
Since Dn (x) is an even function, it moreover follows that
T /2
Dn (x) d x =
0
T
.
2
(4.7)
We have now established enough properties of the Dirichlet kernel to enable us
to formulate and prove the fundamental theorem. According to the fundamental
theorem, the Fourier series converges to the function at each point of continuity of a
piecewise smooth periodic function. At a point where the function is discontinuous,
the Fourier series converges to the average of the left- and right-hand limits at that
point. Hence, both at the points of continuity and at the points of discontinuity the
series converges to ( f (t+) + f (t−))/2. The fundamental theorem now reads as
follows.
THEOREM 4.3
Fundamental theorem of
Fourier series
Let f (t) be a piecewise smooth periodic function on R with Fourier coefficients cn .
Then one has for any t ∈ R:
∞
cn einω0 t =
n=−∞
1
( f (t+) + f (t−)) .
2
Proof
For the proof of this theorem we start from the partial sums sn (t) of the Fourier series
as defined in (4.2). Replacing ck by the integral defining the Fourier coefficients we
obtain
n
sn (t) =
k=−n
ck eikω0 t =
n
k=−n
1 T /2
f (u)e−ikω0 u du eikω0 t
T −T /2
n
1 T /2
=
f (u)
e−ikω0 (u−t) du.
T −T /2
k=−n
92
4 The fundamental theorem of Fourier series
For x = 0 the integrand of In (t) is not defined, since then the denominator of
Q(x) equals 0. However, since f (t) is a piecewise smooth function, the limits for
x → 0 of ( f (t + x) − f (t+))/x and ( f (t − x) − f (t−))/x, which occur in Q(x),
do exist according to theorem 2.4, and they equal f (t+) and f (t−) respectively.
Since the limit for x → 0 of x/ sin(ω0 x/2) exists as well, it follows that Q(x) is a
piecewise continuous function. Furthermore, we note that Q(x) is odd. And since
sin((n + 1 )ω0 x) is also odd, the integrand in (4.9) is an even function. We can
2
therefore extend the integration interval to [−T /2, T /2]. Using the trigonometric
identity sin(α + β) = sin α cos β + cos α sin β we then obtain:
In (t) =
T /2
1
Q(x) sin((n + 1 )ω0 x) d x
2
2T −T /2
=
T /2
1
Q(x) sin(nω0 x) cos(ω0 x/2) + cos(nω0 x) sin(ω0 x/2) d x
2T −T /2
=
1 2 T /2
·
Q(x) cos(ω0 x/2) sin(nω0 x) d x
4 T −T /2
+
1 2 T /2
·
Q(x) sin(ω0 x/2) cos(nω0 x) d x.
4 T −T /2
The two integrals in the final expression are precisely the formulas for the
ordinary Fourier coefficients of the function Q(x) cos(ω0 x/2) and the function
Q(x) sin(ω0 x/2); see definition 3.1. Since Q(x) is piecewise continuous, so are
the functions Q(x) cos(ω0 x/2) and Q(x) sin(ω0 x/2), and hence one can apply the
Riemann–Lebesgue lemma from theorem 4.2. We then see that indeed In (t) tends
to 0 as n → ∞. This completes the proof.
Having established the fundamental theorem, it is now a proven fact that Fourier
series of piecewise smooth functions converge. At the points of continuity of the
function, the Fourier series converges to the function value and at the points of discontinuity to the average of the left- and right-hand limits ( f (t+) + f (t−))/2. For
example, the Fourier series of the periodic block function from section 3.4.1 will
converge to the function with graph given by figure 4.2. At the points of discontinuity the function value is 1/2.
1
1
2
0
–3T/2
–T
–T/2 –a/2
a/2
T/2
T
3T/2
t
FIGURE 4.2
Limit of the Fourier series of the periodic block function.
From the fundamental theorem it immediately follows that if two periodic functions have the same Fourier series, then these functions must be equal at all points
of continuity. Moreover, it follows from the definition of the Fourier coefficients
that the values of the functions at the points of discontinuity have no influence on
4.2 The fundamental theorem
93
the coefficients. This is precisely the reason why we didn't pay any attention to the
values of functions at the points of discontinuity in chapter 3. In the end, the Fourier
series will always converge to the average of the left- and right-hand limit. Apart
from the points of discontinuity, functions with equal Fourier series are the same.
We formulate this result in the following uniqueness theorem.
THEOREM 4.4
Uniqueness theorem
Let f (t) and g(t) be piecewise smooth periodic functions with Fourier coefficients
f n and gn . If f n = gn for all n ∈ Z, then f (t) = g(t) at all points where f and g
are continuous.
From the fundamental theorem it follows that the Fourier series of a function converges pointwise to the function at the points of continuity. Some general remarks
can be made about the rate of convergence. If we compare the Fourier coefficients of
the periodic block function and the periodic triangle function from sections 3.4.1 and
3.4.2, then we observe that the coefficients of the discontinuous block function decrease proportional to 1/n, while the coefficients of the continuous triangle function
decrease proportional to 1/n 2 . The feature that Fourier coefficients of continuous
functions decrease more rapidly compared to discontinuous functions is true in general. If the derivative is continuous as well, then the Fourier coefficients decrease
even more rapidly. As higher derivatives are continuous as well, the Fourier coefficients decrease ever more rapidly. Hence, the smoother the function, the smaller
the contribution of high frequency components in the Fourier series. We summarize
this in the following statements.
a If the function f (t) is piecewise continuous, then the Fourier coefficients tend
to zero (this is the Riemann–Lebesgue lemma).
b If f (t) is continuous and f (t) piecewise continuous, then the Fourier coefficients decrease as 1/n, so limn→±∞ ncn = 0.
c If f (t) and its derivatives up to the (k − 1)th order are continuous and f (k) (t)
is piecewise continuous, then the Fourier coefficients decrease as 1/n k , so
limn→±∞ n k cn = 0.
These statements will not be proven here (but see the remarks following theorem
4.10). From these statements it follows that in comparison to smooth functions, for
less smooth functions one needs more terms from the series in order to achieve the
same degree of accuracy. The statements also hold in the opposite direction: the
faster the Fourier coefficients decrease, the smoother the function. An example is
the following result, which will be used in chapter 7 and is stated without proof here.
THEOREM 4.5
Let a sequence of numbers cn be given for which ∞
n=−∞ | cn | < ∞. Then the
series ∞
cn einω0 t converges to the continuous function f (t) having Fourier
n=−∞
coefficients cn .
For functions with discontinuities it is the case that – no matter how many terms
one includes – in a small neighbourhood left and right of a discontinuity, any approximation will 'overshoot' the function value on one side and 'undershoot' the
function value on the other side. It is even the case that the values of these overshoots
are a fixed percentage of the difference | f (t+) − f (t−) |. These overshoots do get
closer and closer to the point of discontinuity, as more terms are being included.
This curious phenomenon is called Gibbs' phenomenon and will be discussed in
section 4.4.2.
An important side result of the fundamental theorem is the fact that sums can be
calculated for many of the series for which, up till now, we could only establish the
convergence using the tests from chapter 2. Below we present some examples of
such calculations.
4.3 Further properties of Fourier series
99
for n = 0 and c0 = 1 . The squares of these coefficients are sin2 (nπ/2)/n 2 π 2 for
2
n = 0 and 1 for n = 0. Multiplied by 2 these are exactly the Fourier coefficients of
4
the periodic triangle function from section 3.4.2 for a = 1, ω0 = π and T = 2.
4.3.2
Parseval's identity
In this subsection we will show that for a large class of functions, Bessel's inequality
from section 4.1 is in fact an equality. Somewhat more general is the following
Parseval identity, which has important applications in the analysis of signals and
systems.
THEOREM 4.8
Parseval's identity
Let f (t) and g(t) be piecewise smooth periodic functions with Fourier coefficients
f n and gn . Then
∞
1 T /2
f (t)g(t) dt =
f n gn .
T −T /2
n=−∞
(4.13)
Proof
According to theorem 4.6, the Fourier coefficients of the product h of two func∞
tions f and g are given by h k =
n=−∞ f n gk−n . In particular this holds for
the Fourier coefficient h 0 , for which, moreover, one has by definition that h 0 =
T /2
(1/T ) −T /2 f (t)g(t) dt. Combining these facts, it follows that
∞
1 T /2
f (t)g(t) dt = h 0 =
f n g−n .
T −T /2
n=−∞
Instead of the function g we now take the conjugate function g. According to theorem 3.2 the Fourier coefficients of g are g−n , proving (4.13).
It is now just a small step to prove that the Bessel inequality is an equality for
piecewise smooth functions. In order to do so, we take g(t) in theorem 4.8 equal to
the function f (t). The Fourier coefficients of f (t) will be denoted by cn again. We
then obtain
∞
1 T /2
1 T /2
| f (t) |2 dt =
f (t) f (t) dt =
cn cn
T −T /2
T −T /2
n=−∞
=
∞
| cn |2 .
(4.14)
n=−∞
In section 1.2.3 the power P of a periodic time-continuous signal was defined as
T /2
(1/T ) −T /2 | f (t) |2 dt. If f (t) is a piecewise smooth periodic function, then according to (4.14) the power can also be calculated using the Fourier coefficients:
Power of piecewise smooth
periodic function
P=
4.3.3
∞
1 T /2
| f (t) |2 dt =
| cn |2 .
T −T /2
n=−∞
Integration
Using Parseval's identity from the previous subsection, one can derive a result concerning the relationship between the Fourier coefficients of the integral of a periodic
function and the Fourier coefficients of the periodic function itself. We thus want
4.3 Further properties of Fourier series
101
one may integrate its Fourier series term-by-term. The resulting series converges to
the integral of the function.
EXAMPLE 4.5
Consider the periodic function f with period 2 for which f (t) = 1 for −1 ≤ t <
0 and f (t) = −1 for 0 ≤ t < 1. One can easily check (see exercise 3.22, if
necessary) that the Fourier coefficients of this function are given by c0 = 0 and
t
cn = ((−1)n − 1)/inπ. Furthermore, −T /2 f (τ ) dτ is a periodic triangle function
with period 2 and a = 1. According to section 3.4.2 its Fourier coefficients are equal
to (1 − (−1)n )/n 2 π 2 , which indeed equals cn /inπ . The zeroth Fourier coefficient
of this periodic triangle function equals 1/2. From (4.10) it immediately follows that
indeed
−
∞
4 ∞
1
cn (−1)n
1 − (−1)n
1
=
= 2
= .
inπ
2
n2π 2
π k=1 (2k − 1)2
n=−∞
n=−∞
∞
n=0
4.3.4
n=0
Differentiation
In section 4.3.3 we have seen that the Fourier coefficients of the integral of a piecewise smooth periodic function with c0 = 0 can be derived quite easily from the
Fourier coefficients of the function itself. The function could in fact be integrated
term-by-term and the resulting series converged to the integral of the function. In
this section we investigate under which conditions the term-by-term derivative of the
Fourier series of a function converges to the derivative of the function itself. Termby-term differentiation leads less often to a convergent series. It is not hard to understand the reason for this. In section 4.3.3 we have seen that integrating a Fourier
series corresponds to a division of the nth term by a factor proportional to n, improving the rate of convergence of the series. However, differentiating a Fourier series
corresponds to a multiplication of the nth term by a factor proportional to n, and this
will diminish the rate of convergence.
EXAMPLE 4.6
Consider the sawtooth function f (t) = 2t/T for −T /2 < t < T /2 from section
3.4.3. We have seen that the Fourier coefficients are equal to i(−1)n /nπ and so the
n inω0 t /nπ. The sawtooth function has
Fourier series is equal to ∞
n=−∞ i(−1) e
discontinuities at t = ±T /2, ±3T /2, . . .. If we differentiate the Fourier series of
the sawtooth function term-by-term, then we find
∞
n=−∞
i(−1)n inω0 einω0 t /nπ =
∞
(−1)n+1 ω0 einω0 t /π.
n=−∞
This series does not converge, since the terms in the series do not tend to 0 for
n → ∞.
It turns out that continuity of the periodic function is an important condition for
the term-by-term differentiability. We formulate this in the following theorem.
THEOREM 4.10
Differentiation of Fourier
series
Let f (t) be a piecewise smooth periodic continuous function with Fourier coefficients cn and for which f (t) is piecewise smooth as well. Then
∞
1
( f (t+) + f (t−)) =
inω0 cn einω0 t .
2
n=−∞
(4.16)
102
4 The fundamental theorem of Fourier series
Proof
Since f (t) is piecewise smooth, f (t) has a convergent Fourier series. Let cn be
the Fourier coefficients of f (t), then
cn =
1 T /2
f (t)e−inω0 t dt.
T −T /2
Since f is continuous, one can apply integration by parts. We then find:
cn =
1
T
f (t)e−inω0 t
T /2
−T /2
+ inω0
1 T /2
f (t)e−inω0 t dt.
T −T /2
The first term in the right-hand side is 0 for all n, since f (t) is periodic, so
f (−T /2) = f (T /2), and e−inω0 T /2 = einω0 T /2 . The second term is, by definition, up to a factor inω0 , equal to cn and hence
cn = inω0 cn .
If we now apply the fundamental theorem to the function f it follows that
∞
∞
1
( f (t+) + f (t−)) =
cn einω0 t =
inω0 cn einω0 t .
2
n=−∞
n=−∞
In order to derive the equality cn = inω0 cn in the proof of theorem 4.10, we did
not use the assumption that f (t) was piecewise smooth. Hence, this equality also
holds when f (t) is piecewise continuous. If we now apply the Riemann–Lebesgue
lemma to the function f (t), it follows that limn→±∞ cn = limn→±∞ inω0 cn = 0.
This proves statement b about the rate of convergence from section 4.2. By applying
this repeatedly, statement c follows (see also exercise 4.20).
EXERCISES
4.10
Equation (4.14) has been stated for complex Fourier coefficients. Give the equation
if one uses the ordinary Fourier coefficients.
4.11
An application of Parseval's identity can be found in electronics. Suppose that in
an electric circuit the periodic voltage v(t) gives rise to a periodic current i(t), both
with period T and both piecewise smooth. Let v n and i n be the Fourier coefficients
of v(t) and i(t) respectively. Show that for the average power P over one period
(see section 1.2.3) one has
P=
∞
n=−∞
4.12
vn in =
∞
v n i −n .
n=−∞
Consider the periodic block function f (t) with period π and for some a ∈ R with
0 < a < π, and the periodic block function g(t) with period π and some b ∈ R
with 0 < b < π (see section 3.4.1). Assume that a ≤ b.
a Use Parseval's identity to show that
∞
a(π − b)
sin na sin nb
.
=
2
2
n
n=1
b
Choose a = b = π/2 and derive (4.10) again.
4.3 Further properties of Fourier series
4.13
103
Consider the periodic triangle function from section 3.4.2 for a = T /2.
a Use (4.14) to obtain that
∞
1
1
1
π4
1
.
= 1 + 4 + 4 + 4 + ··· =
96
(2n + 1)4
3
5
7
n=0
b Let S = ∞ 1/n 4 . Split this into a sum over the even and a sum over the odd
n=1
positive integers and then show that S = π 4 /96 + S/16. Conclude that
∞
π4
1
=
.
90
n4
n=1
4.14
Let f (t) be the periodic function with period 2 defined for −1 < t ≤ 1 by
f (t) =
a
t2 + t
−t 2 + t
for −1 < t ≤ 0,
for 0 < t ≤ 1.
Use the results from exercise 4.9 to show that
∞
π6
1
=
.
6
960
(2k + 1)
k=0
b
Use part a and the method of exercise 4.13b to show that
∞
1
π6
.
=
6
945
k
k=1
4.15
Let f (t) be a piecewise smooth periodic function with period T and Fourier series
∞
inω0 t , where c = 0. Show that for −T /2 ≤ a ≤ b one has
0
n=−∞ cn e
b
a
f (t) dt =
∞
cn
(einω0 b − einω0 a ).
inω0
n=−∞
n=0
4.16
Let f (t) be a piecewise smooth periodic function with period T and with Fourier
coefficients an and bn , where a0 = 0. Show that for −T /2 ≤ a ≤ b one has
b
a
4.17
f (t) dt =
∞
1
(an (sin nω0 b − sin nω0 a) − bn (cos nω0 b − cos nω0 a)) .
nω0
n=1
Consider the periodic function f (t) with period 2π defined for −π < t ≤ π by
f (t) =
−1
1
for −π < t ≤ 0,
for 0 < t ≤ π.
a Determine the ordinary Fourier coefficients of f and give the Fourier series of
f.
b Integrate the series from part a over [−π, t] and determine the resulting constant
using (4.10).
c Show that the function represented by the series from part b is the periodic
function with period 2π given by g(t) = | t | − π for −π < t ≤ π.
d Use exercise 3.6 to determine in a direct way the ordinary Fourier series of the
function g from part c and use this to verify the result from part b.
104
4 The fundamental theorem of Fourier series
4.18
Consider the periodic function f (t) with period 2 from exercise 4.14. Use the integration theorem to show that
g(t) =
∞
∞
4
1
1
4
+ 2
eπint ,
2
4
π n=−∞ (2n + 1)
π n=−∞ (2n + 1)4
where g(t) is the periodic function with period 2 given for −1 < t ≤ 1 by
g(t) = 1 | t |3 − 1 t 2 + 1 .
3
2
6
Finally calculate the Fourier coefficient c0 of g(t) and check your answer using
exercise 4.13a.
4.19
Show that theorem 4.10 reads as follows, when formulated in terms of the ordinary Fourier coefficients. Let f (t) be a piecewise smooth periodic continuous function with ordinary Fourier coefficients an and bn and for which f (t) is piecewise
smooth. Then
1 ( f (t+) + f (t−)) =
2
∞
(nω0 bn cos nω0 t − nω0 an sin nω0 t).
n=1
4.20
Let f (t) be a periodic continuous function with piecewise smooth continuous derivative f (t). Show that limn→±∞ n 2 cn = 0.
4.21
In the example prior to theorem 4.10 we saw that the Fourier series of the periodic
sawtooth function f (t) with period T , given by f (t) = 2t/T for −T /2 < t ≤ T /2,
could not be differentiated term-by-term. However, now consider the even function
g(t) with period T given by f (t) = 2t/T for 0 ≤ t ≤ T /2.
a Determine the Fourier cosine series of the function g(t).
b Differentiate the series from part a term-by-term. Verify that the result is the
Fourier sine series of the function g (t) and that this series converges to g (t) for all
t = nT /2 (n ∈ Z). How can this result be reconciled with theorem 4.10?
4.22
a Determine the Fourier series of the periodic function with period 2π defined for
−π < t ≤ π by
f (t) =
0
sin t
for −π < t ≤ 0,
for 0 < t ≤ π.
b Verify that we can differentiate f from part a by differentiating its Fourier series, except at t = nπ (n ∈ Z). Describe the function that is represented by the
differentiated series.
4.23
Formulate the convolution theorem (theorem 4.7) for the ordinary Fourier coefficients.
4.24
Let f be the periodic block function with period 2 and a = 1, so f (t) = 1 for
| t | ≤ 1 and f (t) = 0 for 1 < | t | ≤ 1. Let g be the periodic triangle function with
2
2
period 2 and a = 1 , so g(t) = 1 − 2 | t | for | t | ≤ 1 and g(t) = 0 for 1 < | t | ≤ 1.
2
2
2
a Show that f 1 ∗ f 2 is even when both f 1 and f 2 are even periodic functions with
period T .
4.4 The sine integral and Gibbs' phenomenon
105
b Show that f ∗ g is the even periodic function with period 2 which for 0 ≤ t ≤ 1
is given by
− 1 t 2 + 1 for 0 ≤ t ≤ 1 ,
2
4
2
( f ∗ g)(t) =
1 (t − 1)2 for 1 < t ≤ 1.
2
2
c Determine the Fourier series of ( f ∗g)(t) and verify that it converges to ( f ∗g)(t)
for all t ∈ R.
d Verify the constant in the Fourier series of ( f ∗ g)(t) by calculating the zeroth
Fourier coefficient in a direct way.
4.4
The sine integral and Gibbs' phenomenon
The fundamental theorem of Fourier series was formulated for piecewise smooth
functions. According to this theorem, the series converges pointwise to the function.
Possible points of discontinuity were excluded here. At these points, the series converges to the average value of the left- and right-hand limits of the function. Towards
the end of section 4.2 we already noted that in a small neighbourhood of a discontinuity, the series will approximate the function much slower. This had already been
observed by Wilbraham in 1848, but his results fell into oblivion. In 1898 the physicist Michaelson published an article in the magazine Nature, in which he doubted
the fact that 'a real discontinuity (of a function f ) can replace a sum of continuous
curves' (i.e., the terms in the partial sums sn (t)). This is because Michaelson had
constructed a machine which calculated the nth partial sum of the Fourier series of
a function up to n = 80. In a small neighbourhood of a discontinuity, the partial
sums sn (t) did not behave as he had expected: the sums continued to deviate and the
largest deviation, the so-called overshoot of sn (t) relative to f (t), did not decrease
with increasing n. In figure 4.3 this is illustrated by the graphs of the partial sums
approximating the periodic block function for different values of n. We see that the
overshoots get narrower with increasing n, but the magnitude remains the same. In
Overshoot
a
b
c
FIGURE 4.3
Partial sums of the periodic block function.
a letter to Nature from 1899, Gibbs explained this phenomenon and showed that
sn (t) will always have an overshoot of about 9% of the magnitude of the jump at the
discontinuity. We will investigate this so-called Gibbs' phenomenon more closely
for the periodic block function. Before we do so, we first introduce the sine integral,
a function that will be needed to determine Gibbs' phenomenon quantitatively. The
sine integral will be encountered in later chapters as well.
4.4 The sine integral and Gibbs' phenomenon
107
The first and last term are equal. If we solve for I ( p), we obtain
I ( p) =
1
.
1 + p2
Next we integrate I ( p) over the interval (0, ∞) to obtain
∞
0
I ( p) dp =
∞
0
1
π
dp = [arctan p]∞ = .
0
2
1 + p2
(4.18)
On the other hand we find by interchanging the order of integration that
∞
0
I ( p) dp =
=
∞
0
0
∞
0
∞ sin t
t
e− pt sin t dp dt =
∞
0
e− pt
sin t dt
−t p=0
p=∞
dt.
(4.19)
We state without proof that interchanging the order of integration is allowed. With
the equality of the right-hand sides of (4.18) and (4.19) the theorem is proved.
4.4.2
Gibbs' phenomenon∗
In this section we treat Gibbs' phenomenon. It is a rather technical treatment, which
does not result in any specific new insight into Fourier series. This section may
therefore be omitted without any consequences for the study of the remainder of the
book.
We treat Gibbs' phenomenon using the periodic block function. Since it will
result in much simpler formulas, we will not start from the periodic block function
as defined in section 3.4.1, but instead from the periodic function f (t) defined on
the interval (−T /2, T /2) by
1
for 0 < t < 1 T ,
2
2
f (t) =
− 1 for − 1 T < t < 0.
2
2
This odd function has a Fourier sine series whose coefficients have been calculated
in exercise 3.22. The partial sums sn (t) of the Fourier series are
n
sn (t) =
2
sin((2k − 1)ω0 t).
(2k − 1)π
k=1
(4.20)
The graph of the partial sum for n = 12 is drawn in figure 4.5. In it, Gibbs' phenomenon is clearly visible again: immediately next to a discontinuity of the function
f (t), the partial sum overshoots the values of f (t). We will now calculate the overshoot, that is, the magnitude of the maximum difference between the function and
the partial sums immediately next to the discontinuity. By determining the derivative of the partial sum, we can find out where the maximum difference occurs, and
subsequently calculate its value. Differentiating sn (t) gives
n
sn (t) =
n
2
4
ω0 cos((2k − 1)ω0 t) =
cos((2k − 1)ω0 t).
π
T
k=1
k=1
(4.21)
In order to determine the zeros of the derivative, we rewrite the last sum. For this
we use the trigonometric identity sin α − sin β = 2 sin((α − β)/2) cos((α + β)/2).
4.4 The sine integral and Gibbs' phenomenon
109
integral). The value of this integral was given in section 4.4.1 and hence
lim sn
n→∞
π
2nω0
1 n π sin((2k − 1)π/2n)
n→∞ π
n (2k − 1)π/2n
k=1
= lim
1 π sin x
1
d x = 1.852 . . . = 0.589 . . . .
π 0
x
π
This establishes the value at the first maximum next to the jump. Since the jump
has magnitude 1, the overshoot of the function value 0.5 is approximately 9 % of
the jump. Since the additional contribution for large values of n gets increasingly
smaller, this overshoot will remain almost constant with increasing n. Furthermore
we see that the value of t where the extremum is attained is getting closer and closer
to the point of discontinuity.
In this section we studied Gibbs' phenomenon using the periodic block function.
However, the phenomenon occurs in a similar way for other piecewise smooth functions having points of discontinuity. There is always an overshoot of the partial sums
immediately to the left and to the right of the points of discontinuity, with a value
approximately equal to 9 % of the magnitude of the jump. As more terms are being
included in the partial sums, the extrema are getting closer and closer to the point of
discontinuity.
=
EXERCISES
4.25
Use the definition to verify that the sine integral Si(x) is an odd function.
4.26
The sine integral can be considered as an approximation of the function f (x) given
by f (x) = π/2 for x > 0 and f (x) = −π/2 for x < 0.
a What is the smallest value of x > 0 for which Si(x) has a maximum?
b What is the value of Si(x) at the first maximum and what percentage of the jump
at x = 0 of the function f (x) does the overshoot amount to?
4.27∗
Consider the periodic function with period T given by f (t) = 2t/T − 1 for 0 <
t ≤ T . Let sn (t) be the nth partial sum of the Fourier series of f .
a Show that f arises from the sawtooth function from section 3.4.3 by a shift over
T /2. Next determine sn (t).
b Show that
2
sn (t) = − (Dn (t) − 1),
T
where Dn is the Dirichlet kernel from definition 4.1. Subsequently determine the
value of t for which sn (t) has its first extreme value immediately to the left of the
discontinuity at t = 0.
c Calculate the magnitude of the extremum from part b and show that the overshoot is again approximately equal to 9 % of the magnitude of the jump at t = 0.
SUMMARY
When a periodic function is piecewise continuous, its Fourier coefficients, as defined in chapter 3, exist. According to Bessel's inequality, the sum of the squares of
the moduli of the Fourier coefficients of a periodic piecewise continuous function
is finite. From this, the lemma of Riemann–Lebesgue follows, which states that the
Fourier coefficients tend to 0 for n → ±∞. The fundamental theorem of Fourier
series has been formulated for piecewise smooth periodic functions. For such functions the Fourier series converges to the function at the points of continuity, and to
110
4 The fundamental theorem of Fourier series
the average of the left- and right-hand limit at the points of discontinuity. If f is a
piecewise smooth periodic function with period T , then one has, according to the
fundamental theorem,
∞
n=−∞
cn einω0 t = 1 ( f (t+) + f (t−)) ,
2
where
cn =
1 T /2
f (t)e−inω0 t dt
T −T /2
and
ω0 =
2π
.
T
As functions are smoother, meaning that higher derivatives do not contain discontinuities, the convergence of the Fourier series to the function is faster. Using the
fundamental theorem, further properties of Fourier series were derived, such as series for products and convolutions, term-by-term differentiation and integration of
Fourier series, and Parseval's identity
∞
1 T /2
f (t)g(t) dt =
f n gn ,
T −T /2
n=−∞
where f n and gn are the Fourier coefficients of f and g.
While the Fourier series of a piecewise smooth function with discontinuities does
converge at each point of continuity to the function value, the series always has an
overshoot of about 9 % of the magnitude of the jump immediately next to a point
of discontinuity. This phenomenon is called Gibbs' phenomenon. As more terms
are included in the partial sums, the overshoot shifts closer and closer to the point
of discontinuity. For the analysis of this phenomenon we defined the sine integral
Si(x), having as properties Si(π) = 1.852 . . . and limx→∞ Si(x) = π/2.
SELFTEST
4.28
Let f (t) be the odd periodic function with period 2π defined for 0 ≤ t < π by
2 t for 0 ≤ t < π ,
π
2
f (t) =
π
1
for ≤ t < π .
2
a For which values of t ∈ R does the Fourier series of f (t) converge? What is the
sum for those values of t for which there is convergence?
b Determine the Fourier series of f and verify the fundamental theorem of Fourier
series for t = 0 and t = π.
c Can one differentiate the function f by differentiating the Fourier series termby-term? If not, explain. If so, give the function that is represented by the differentiated series.
d Can one integrate the function f over [−π, t] by integrating the Fourier series
term-by-term? If not, explain. If so, give the function that is represented by the
integrated series.
4.29
Let a ∈ R with 0 < a ≤ π/2. Use the periodic block function pa,π and the periodic
triangle function qa,π to show that
∞
sin3 na
a2
(3π − 4a).
=
8
n3
n=1
4.4 The sine integral and Gibbs' phenomenon
111
Finally evaluate
∞
(−1)n
.
(2n + 1)3
n=0
4.30
Let f (t) be the periodic function f (t) = | sin t | with period 2π.
a Find the Fourier series of f and verify that it converges to f (t) for all t ∈ R.
b Show that
∞
1
n=1
4n 2 − 1
c
=
1
2
∞
and
1 π
(−1)n
= − .
2
4
4n 2 − 1
n=1
Show that
∞
1
π2
1
− .
=
16
2
(4n 2 − 1)2
n=1
4.31
Let f be the periodic sawtooth function with period 2 given by f (t) = t for −1 <
t ≤ 1.
a Show that f 1 ∗ f 2 is an even function if both f 1 and f 2 are odd periodic functions
with period T .
b Show that f ∗ f is the even periodic function with period 2 which for 0 ≤ t ≤ 1
is given by ( f ∗ f )(t) = − 1 t 2 +t − 1 . (Hint: how can one express f for 1 < t ≤ 2?
2
3
Now split the integral in two parts.)
c Prove that for all t ∈ R we have
2 ∞ 1
cos nπt.
( f ∗ f )(t) = − 2
π n=1 n 2
d Show that for t = 0, the result from part c is equivalent to Parseval's identity
(4.14) for the function f .
e Verify that term-by-term differentiation of ( f ∗ f )(t) is allowed for 0 < | t | ≤ 1
and describe for all t ∈ R the function that is represented by the differentiated series.
f Determine in a direct way the zeroth Fourier coefficient of ( f ∗ f )(t) and verify
the answer using the result from part c. Next, verify that term-by-term integration
over [−1, t] is allowed and describe for all t ∈ R the function that is represented by
the integrated series.
CHAPTER 5
Applications of Fourier series
INTRODUCTION
Applications of Fourier series can be found in numerous places in the natural sciences as well as in mathematics itself. In this chapter we confine ourselves to two
kinds of applications, to be treated in sections 5.1 and 5.2. Section 5.1 explains how
Fourier series can be used to determine the response of a linear time-invariant system to a periodic input. In section 5.2 we discuss the applications of Fourier series
in solving partial differential equations, which often occur when physical processes,
such as heat conduction or a vibrating string, are described mathematically.
The frequency response, introduced in chapter 1 using the response to the periodic time-harmonic signal eiωt with frequency ω, plays a central role in the calculation of the response of a linear time-invariant system to an arbitrary periodic
signal. Specifically, a Fourier series shows how a periodic signal can be written as
a superposition of time-harmonic signals with frequencies being an integer multiple
of the fundamental frequency. By applying the so-called superposition rule for linear time-invariant systems, one can then easily find the Fourier series of the output.
This is because the sequence of Fourier coefficients, or the line spectrum, of the output arises from the line spectrum of the input by a multiplication by the frequency
response at the integer multiples of the fundamental frequency.
For stable systems which can be described by ordinary differential equations,
which is almost any linear time-invariant system occurring in practice, we will see
that the frequency response can easily be derived from the differential equation.
The characteristic polynomial of the differential equation of a stable system has
no zeros on the imaginary axis, and hence there are no periodic eigenfunctions.
As a consequence, the response to a periodic signal is uniquely determined by the
differential equation. If there are zeros iω of the characteristic polynomial on the
imaginary axis, then a periodic input may lead to resonance. For the corresponding
frequencies ω, the frequency response is meaningless.
In the second and final section we treat applications of Fourier series in solving partial differential equations by separation of variables. This method is explained systematically in the case when the functions have one time-variable and
one position-variable. We limit ourselves to simple examples of initial and boundary
value problems, with the partial differential equation being either the onedimensional diffusion or heat equation, or the one-dimensional wave equation.
113
114
5 Applications of Fourier series
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- can express the line spectrum of the response of an LTC-system to a periodic input
in terms of the frequency response and the line spectrum of the input
- know what eigenfunctions and eigenfrequencies are for a system described by a
differential equation, and know the relevance of the zeros of the corresponding
characteristic polynomial
- know when the periodic response of an LTC-system to a periodic input is uniquely
determined by the differential equation, and know what causes resonance
- can determine the frequency response for stable LTC-systems described by a differential equation
- can use separation of variables and Fourier series to determine in a systematic way
a formal solution of the one-dimensional heat equation and the one-dimensional
wave equation, under certain conditions.
5.1
LTC-system
Linear time-invariant systems with periodic input
In this section we will occupy ourselves with the determination of the response of
a linear time-invariant continuous-time system (LTC-system for short) to a periodic
input. Calculating the response as a function of time or, put differently, determining
the response in the time domain, is often quite difficult. If, however, we have the
line spectrum of the input at our disposal, so if we know the sequence of Fourier
coefficients, then it will turn out that by using the frequency response of the LTCsystem it is easy to determine the line spectrum of the output. Apparently it is easy
to calculate the response in the frequency domain.
The frequency response of an LTC-system was introduced in chapter 1 by the
property
eiωt → H (ω)eiωt .
That is to say, the response to the time-harmonic signal eiωt of frequency ω is equal
to H (ω)eiωt . We assume that for periodic inputs u(t) one has that
u(t) =
∞
u n einω0 t ,
(5.1)
n=−∞
where ω0 = 2π/T and u n is the sequence of Fourier coefficients, or line spectrum,
of u(t). When u(t) is a piecewise smooth function, then we know from the fundamental theorem of Fourier series that (5.1) holds everywhere if we assume that
u(t) = (u(t+) + u(t−))/2 at the points of discontinuity of u(t). In the present
chapter this will always be tacitly assumed. One now has the following theorem.
THEOREM 5.1
Let y(t) be the response of a stable LTC-system to a piecewise smooth and periodic
input u(t) with period T , fundamental frequency ω0 and line spectrum u n . Let H (ω)
be the frequency response of the system. Then y(t) is again periodic with period T
and the line spectrum yn of y(t) is given by
yn = H (nω0 )u n
for n = 0, ±1, ±2, . . ..
(5.2)
Proof
Let the line spectrum u n of the input u(t) be given. Then (5.1) holds, which represents u(t) as a superposition of the time-harmonic signals einω0 t with frequencies
nω0 . If only a finite number of Fourier coefficients u n are unequal to zero, then
u(t) is a finite linear combination of time-harmonic signals. Because of the linearity
5.1 Linear time-invariant systems with periodic input
Superposition rule
115
of the system, in calculating the response it is sufficient first to determine the responses to the time-harmonic signals and then to take the same linear combination
of the responses.
We now assume that for LTC-systems this property may be extended to infinite
linear combinations of time-harmonic signals, so to series such as Fourier series
from (5.1). We then say that for LTC-systems the superposition rule holds (also see
chapter 10). This means that first one can determine the response to a time-harmonic
signal with frequency nω0 using the frequency response. Because of the stability of
the LTC-system we know that H (ω) is defined for all values of ω (see section 1.3.3).
On the basis of the superposition rule one thus has that
∞
y(t) =
u n H (nω0 )einω0 t .
n=−∞
We see that y(t) is periodic with period T and, moreover, that the line spectrum yn
of the response satisfies (5.2).
EXAMPLE 5.1
For a stable LTC-system the frequency response is given by
H (ω) =
1
−ω2 + 3iω + 2
.
Consider the periodic input u(t) with period 2π given on the interval (−π, π) by
u(t) = t. The line spectrum yn of the response y(t) satisfies (5.2). One has
2
H (nω0 ) = 1/(2 + 3niω0 − n 2 ω0 ). The line spectrum u n of u(t) can be obtained
by a direct calculation of the Fourier coefficients. The result is: u n = (−1)n i/n for
n = 0, u 0 = 0. Hence,
(−1)n i
for n = 0,
2
n(2 + 3inω0 − n 2 ω0 )
yn =
0
for n = 0.
Systems that can be realized in practice are often described by differential equations. Well-known examples are electrical networks and mechanical systems. We
will now examine how for such systems one can determine the frequency response.
5.1.1
Systems described by differential equations
In chapter 1 systems described by differential equations were briefly introduced. For
such systems the relation between an input u(t) and the corresponding output y(t) is
described by an ordinary differential equation with constant coefficients of the form
am
dm y
d m−1 y
dy
m + am−1 dt m−1 + · · · + a1 dt + a0 y
dt
= bn
Order of a differential
equation
dnu
d n−1 u
du
+ b0 u
+ bn−1 n−1 + · · · + b1
dt n
dt
dt
(5.3)
with n ≤ m. Here a0 , a1 , . . . , am and b0 , b1 , . . . , bn are constants with am = 0 and
bn = 0. The number m is called the order of the differential equation. An electric
network with one source, a voltage generator or a current generator, and furthermore
consisting of resistors, capacitors and inductors, can be considered as a system. The
voltage of the voltage generator or the current from the current generator is then an
input, with the output being, for example, the voltage across a certain element in the
116
5 Applications of Fourier series
network or the current through a specific branch. One can derive the corresponding
differential equation from Kirchhoff's laws and the voltage–current relations for the
separate elements in the network. For a resistor of resistance R this is Ohm's law:
v(t) = Ri(t), where i(t) is the current through the resistor and v(t) the voltage
across the resistor. For a capacitor with capacitance C and an inductor with selfinductance L these relations are, respectively,
v(t) =
1 t
i(τ ) dτ
C −∞
and
v(t) = L
d
i(t).
dt
The theory of networks is not a subject of this book and thus we shall not occupy
ourselves with the derivation of the differential equations. Hence, for the networks
in the examples we will always state the differential equation explicitly. Readers
with knowledge of network theory can derive these for themselves; others should
consider the differential equation as being given and describing an LTC-system.
The same assumptions will be made with respect to mechanical systems. Here
the differential equations follow from Newton's laws and the force–displacement
relations for the mechanical components such as masses, springs and dampers. If
we denote the force by F and the displacement by x, then one has for a spring with
spring constant k that F(t) = kx(t), for a damper with damping constant c that
F(t) = cd x/dt, and for a mass m that F(t) = md 2 x/dt 2 . In these formulas the
direction of the force and the displacement have not been taken into account.
The formulas for the mechanical systems are very similar to the formulas for the
electrical networks when we replace a voltage v(t) by a force F(t) and a charge
t
Q(t) = −∞ i(τ ) dτ by a displacement. The latter means that a current i(t) is
replaced by a velocity d x/dt. The formula for the spring is then comparable to the
formula for the capacitor, the formula for the damper with the one for the resistor
and the formula for the mass with the inductor. This is listed in the following table.
electric network
mechanical system
v(t) = Q(t)/C (capacitor)
F(t) = kx(t) (spring)
v(t) = Rd Q/dt (resistor)
F(t) = cd x/dt (damper)
v(t) = Ld 2 Q/dt 2 (inductor)
F(t) = md 2 x/dt 2 (mass)
In chapter 1 we already noted that LTC-systems which are equal in a mathematical sense can physically be realized in different ways. Hence, mathematically a
mechanical network can be the same as an electric network.
For an LTC-system described by a differential equation, the frequency response
H (ω) can easily be obtained. To this end we introduce the polynomials
A(s) = am s m + am−1 s m−1 + · · · + a1 s + a0 ,
B(s) = bn s n + bn−1 s n−1 + · · · + b1 s + b0 .
Characteristic polynomial
The polynomial A(s) is called the characteristic polynomial of differential equation
(5.3). The following theorem shows how one can obtain H (ω) for those values of ω
for which A(iω) = 0.
THEOREM 5.2
Let an LTC-system be described by differential equation (5.3) and have characteristic polynomial A(s). Then one has for all ω with A(iω) = 0:
H (ω) =
B(iω)
.
A(iω)
(5.4)
5.1 Linear time-invariant systems with periodic input
117
Proof
In order to find the frequency response, we substitute the input u(t) = eiωt into
(5.3). The response y(t) is then of the form H (ω)eiωt . Since the derivative of eiωt
is iωeiωt , substitution into (5.3) leads to A(iω)H (ω)eiωt = B(iω)eiωt . From this
it follows that H (ω) = B(iω)/A(iω), which proves the theorem.
It is natural to examine the problems that may arise if A(iω) = 0 for a certain
value of ω, in other words, if the characteristic polynomial has zeros on the imaginary axis. In order to do so, we will study in some more detail the solutions of
differential equations with constant coefficients.
When the input u(t) is known, the right-hand side of (5.3) is known and from the
theory of ordinary differential equations we then know that several other solutions
y(t) exist. In fact, when one solution y(t) of (5.3) is found, then all solutions can
be obtained by adding to y(t) an arbitrary solution x(t) of the differential equation
am
Homogeneous solution
Eigenfunction
Particular solution
dm x
d m−1 x
dx
+ a0 = 0.
+ am−1 m−1 + · · · + a1
dt m
dt
dt
(5.5)
This differential equation is called the homogeneous differential equation corresponding to equation (5.3) and a solution of equation (5.5) is called a homogeneous
solution or eigenfunction. These are thus the solutions with u(t) being the nullsignal. Of course, the null-function x(t) = 0 for all t satisfies the homogeneous
differential equation. This homogeneous solution will be called the trivial homogeneous solution or trivial eigenfunction.
We will say that the general solution y(t) of (5.3) can be written as a particular
solution added to the general homogeneous solution:
general solution = particular solution + general homogeneous solution.
The general homogeneous solution can easily be determined using the characteristic
equation. This equation arises by substituting x(t) = est into (5.5), where s is a
complex constant. From example 2.11 it follows that this is the result: (am s m +
am−1 s m−1 + · · · + a1 s + a0 )est = 0. Since est = 0 one has
Characteristic equation
A(s) = am s m + am−1 s m−1 + · · · + a1 s + a0 = 0.
(5.6)
To each zero s of the characteristic polynomial corresponds the homogeneous solution est . More generally, one can show that to a zero s with multiplicity k there also
correspond k distinct homogeneous solutions, namely
est , test , . . . , t k−1 est .
Fundamental homogeneous
solution
Eigenfrequency
Now the sum of the multiplicities of the distinct zeros of the characteristic polynomial is equal to the degree m of the polynomial, which is the order of the differential
equation. So in this way there is a total of m distinct homogeneous solutions that
correspond to the zeros of the characteristic polynomial. We call these solutions
the fundamental homogeneous solutions, since it follows from the theory of ordinary differential equations with constant coefficients that the general homogeneous
solution can be written as a linear combination of the homogeneous solutions corresponding to the zeros of the characteristic polynomial.
To a zero iω on the imaginary axis corresponds the time-harmonic fundamental
solution eiωt of (5.5) with period T = 2π/ | ω | if ω = 0 and an arbitrary period
if ω = 0. In this case the value ω is called an eigenfrequency. Note that a timeharmonic fundamental solution with period T also has period 2T , 3T , etc. So when
there are zeros of the characteristic polynomial on the imaginary axis, then there
exist non-trivial periodic eigenfunctions. One can show that the converse is also
118
5 Applications of Fourier series
true, that is: if a non-trivial periodic eigenfunction with period T exists, then the
characteristic polynomial has a zero lying on the imaginary axis and corresponding
to this a time-harmonic fundamental solution having period T as well.
Hence, when u(t) is a periodic input with period T and when, moreover, there
exist non-trivial eigenfunctions having this same period, then a periodic solution
of equation (5.3) with period T will certainly not be unique. Possibly, periodic
solutions of (5.3) will not even exist. We will illustrate this in the next example.
EXAMPLE 5.2
Given is the differential equation
y + 4y = u.
The characteristic equation is s 2 + 4 = 0 and it has zeros 2i and −2i on the imaginary axis. To these correspond the fundamental solutions e2it and e−2it . Hence, the
general homogeneous solution is x(t) = αe2it +βe−2it for arbitrary complex α and
β. So all eigenfunctions are periodic with period π. Now when u(t) = 4 cos 2t is a
periodic input with period π, then there is no periodic solution with period π. This
is because one can show by substitution that y(t) = t sin 2t is a particular solution
of the given differential equation for this u. Note that this solution is not bounded.
The general solution is thus
y(t) = t sin 2t + αe2it + βe−2it .
Resonance
Since all homogeneous solutions are bounded, while the particular solution is not,
none of the solutions of the differential equation will be bounded, let alone periodic.
The periodic input u(t) gives rise to unbounded solutions here. This phenomenon is
called resonance.
The preceding discussion has given us some insight into the problems that may
arise when the characteristic polynomial has zeros on the imaginary axis. When
eigenfunctions with period T occur, periodic inputs with period T can cause resonance. In the case when there are no eigenfunctions with period T , the theory of
ordinary differential equations with constant coefficients states that for each periodic
u(t) in (5.3), having period T and an nth derivative which is piecewise continuous,
there exists a periodic solution y(t) with period T as well. This solution is then
uniquely determined, since there are no periodic homogeneous solutions with period T . The solution y(t) can then be determined using the frequency response. We
will illustrate this in our next example.
EXAMPLE 5.3
Consider the differential equation
y + 3y + 2y = cos t.
Note that the right-hand side is periodic with period 2π . The corresponding homogeneous differential equation is
x + 3x + 2x = 0.
The characteristic equation is s 2 + 3s + 2 = 0 and has zeros s = −1 and s =
−2. There are thus no zeros on the imaginary axis and hence there are no periodic
eigenfunctions, let alone periodic eigenfunctions with period 2π. As a consequence,
there is exactly one periodic solution y(t) with period 2π . This can be determined as
follows. We consider the differential equation as an LTC-system with input u(t) =
cos t. Applying (5.4) gives the frequency response of the system:
H (ω) =
1
.
2 + 3iω − ω2
5.1 Linear time-invariant systems with periodic input
119
Since u(t) = (eit + e−it )/2, it is a linear combination of time-harmonic signals
whose response can be found using the frequency response calculated above: eit →
H (1)eit and e−it → H (−1)e−it . Hence,
cos t →
eit
e−it
H (1)eit + H (−1)e−it
=
+
2
2(1 + 3i) 2(1 − 3i)
1
=
(cos t + 3 sin t).
10
In general we will consider stable systems. When a stable system is described
by a differential equation of type (5.3), then one can prove (see chapter 10) that the
real parts of the zeros of the characteristic polynomial are always negative. In other
words, the zeros lie in the left-half plane of the complex plane and hence, none of
the zeros lie on the imaginary axis. Then A(iω) = 0 for all ω and on the basis
of theorem 5.2 H (ω) then exists for all ω. Using theorem 5.1 one can thus determine
the line spectrum of the response for any periodic input.
Systems like electrical networks and mechanical systems are described by differential equations of the form (5.3) with the coefficients a0 , a1 , . . . , am and b0 , b1 ,
. . . , bn being real numbers. The response to a real input is then also real. In chapter 1, section 1.3.4, we have called these systems real systems. A sinusoidal signal
A cos(ωt + φ0 ) with amplitude A and initial phase φ0 , for which the frequency ω is
equal to an eigenfrequency of the system is then an eigenfunction. If the sinusoidal
signal is not an eigenfunction, then A(iω) = 0 and so H (ω) exists. In section 1.3.4
we have seen that the response of a real system to the sinusoidal signal then equals
A | H (ω) | cos(ωt + φ0 +
(ω)).
The amplitude is distorted with the factor | H (ω) |, which is the amplitude response
of the system, and the phase is shifted over (ω) = arg H (ω), which is the phase
response of the system. Now an arbitrary piecewise smooth, real and periodic input
can be written as a superposition of these sinusoidal signals:
u(t) =
∞
An cos(nω0 t + φn ).
n=0
On the basis of the superposition rule, the response y(t) of a stable LTC-system is
then equal to
y(t) =
∞
An | H (nω0 ) | cos(nω0 t + φn +
(nω0 )).
n=0
Filter
EXAMPLE 5.4
Depending on the various frequencies contained in the input, amplitude distortions
and phase shifts will occur.
If H (nω0 ) = 0 for certain values of n, then we will say that the frequency nω0
is blocked by, or will not pass, the system. Designing electrical networks with
frequency responses that meet specific demands, or, put differently, designing filters,
is an important part of network theory. Examples are low-pass filters, blocking
almost all frequencies above a certain limit, high-pass filters, blocking almost all
frequencies below a certain limit, or more general, band-pass filters, through which
only frequencies in a specific band will pass.
We close this section with some examples.
In figure 5.1 an electrical network is drawn, which is considered as an LTC-system
with input the voltage across the voltage generator and output the voltage across the
120
5 Applications of Fourier series
i (t )
R
+
u (t )
y (t )
C
_
FIGURE 5.1
An RC-network.
capacitor. The corresponding differential equation reads as follows:
RC y (t) + y(t) = u(t).
The characteristic equation RCs +1 = 0 has one negative zero. The system is stable
and has frequency response
H (ω) =
1
.
iω RC + 1
Now let the sinusoidal input with frequency ω and initial phase φ0 be given by
u(t) = a cos(ωt + φ0 ). In order to determine the response, we need the modulus
and argument of the frequency response. Using the notation RC = τ one obtains
that
| H (ω) | =
1
=
iωτ + 1
1
,
1 + ω2 τ 2
arg(H (ω)) = − arg(iωτ + 1) = − arctan(ωτ ).
The response of the system to u(t) is thus
y(t) =
a
1 + ω2 τ 2
cos(ωt − arctan(ωτ ) + φ0 ).
As we can see, the amplitude becomes small for large ω. One could consider this
network as a low-pass filter.
EXAMPLE 5.5
In figure 5.2 a simple mechanical mass–spring system is drawn. An external force,
V
m
F(t ) = u(t)
A
W
0
y
FIGURE 5.2
A simple mechanical mass–spring system.
the input u(t), acts on a mass m, which can move in horizontal direction to the left
and to the right over a horizontal plane. A spring V connects m with a fixed point A.
Furthermore, a frictional force W acts on m. The displacement y(t) is considered
as output. The force that the spring exerts upon the mass m equals −cy(t) (c is
5.1 Linear time-invariant systems with periodic input
121
the spring constant). The frictional force is equal to −αy (t). The corresponding
differential equation is then
u(t) = my (t) + αy (t) + cy(t),
having as characteristic equation ms 2 +αs +c = 0. For α > 0, the roots are real and
negative or they have a negative real part −α/2m. We are then dealing with a stable
√
system. If, however, α = 0, then the system has an eigenfrequency ωr = c/m and
a periodic eigenfunction with frequency ωr , and so the system is no longer stable.
The response to the periodic input cos(ωr t) is not periodic. Resonance will then
occur.
As a final example we treat an application of Parseval's identity for periodic
functions.
EXAMPLE 5.6
The electric network from figure 5.3 is considered as an LTC-system with input
the voltage u(t) across the voltage generator and output the voltage y(t) across the
√
resistor. In this network the quantities C, L, R satisfy the relation R = L/C. The
L
+
u (t )
C
–
C
R
y (t )
L
FIGURE 5.3
Electric network from example 5.6.
relation between the input u(t) and the corresponding output y(t) is given by
y − (2/RC)y + (1/LC)y = u − (1/LC)u.
The frequency response follows immediately from (5.4). If we put α = 1/RC, then
√
it follows from R = L/C that α 2 = 1/LC and so
H (ω) =
iω + α
(iω)2 − α 2
=
.
2 − 2iαω + α 2
iω − α
(iω)
If u(t) is a periodic input with period T and line spectrum u n , then, owing to
| a + ib | = | a − ib |, the amplitude spectrum | yn | of the output is equal to
| yn | = | H (nω0 )u n | =
inω0 + α
inω0 + α
| un | = | un | .
un =
inω0 − α
inω0 − α
Apparently, the amplitude spectrum is not altered by the system. This then has
consequences for the power of the output. Applying Parseval's identity for periodic
functions, we can calculate the power P of y(t) as follows:
P=
All-pass system
∞
∞
1 T
1 T
| y(t) |2 dt =
| yn |2 =
| u n |2 =
| u(t) |2 dt.
T 0
T 0
n=−∞
n=−∞
We see that the power of the output equals the power of the input. Systems having
this property are called all-pass systems (see also chapter 10).
122
5 Applications of Fourier series
EXERCISES
5.1
A stable LTC-system is described by a differential equation of the form (5.3). Let
x(t) be an eigenfunction of the system. Show that limt→∞ x(t) = 0.
5.2
For an LTC-system the relation between an input u(t) and the corresponding response y(t) is described by y + y = u. Let u(t) be the periodic input with period
2π, which is given on the interval (−π, π) by
u(t) =
for | t | < π/2,
for | t | > π/2.
1
0
Calculate the line spectrum of the output.
5.3
For an LTC-system the frequency response H (ω) is given by
H (ω) =
1
0
for | ω | ≤ π ,
for | ω | > π .
a Can the system be described by an ordinary differential equation of the form
(5.3)? Justify your answer.
b We apply the periodic signal of exercise 5.2 to the system. Calculate the power
of the response.
5.4
To the network of example 5.6 we apply the signal u(t) = | sin t | as input. Calculate
π
the integral (1/π) 0 y(t) dt of the corresponding output y(t). This is the average
value of y(t) over one period.
5.5
For an LTC-system the relation between an input u(t) and the output y(t) is described by the differential equation
y + 2y + 4y = u + u.
a
b
5.6
Which frequencies do not pass through the system?
Calculate the response to the input u(t) = sin t + cos2 2t.
Given is the following differential equation:
2
y + ω0 y = u
with | ω0 | = 0, 1, 2, . . ..
Here u(t) is the periodic function with period 2π given by
u(t) =
t +π
−t + π
for −π < t < 0,
for 0 < t < π .
Does the differential equation have a unique periodic solution y(t) with period 2π ?
If so, determine its line spectrum.
5.2
Partial differential equations
In this section we will see how Fourier series can be applied in solving partial differential equations. For this, we will introduce a method which will be explained
systematically by using a number of examples wherein functions u(x, t) occur, depending on a time-variable t and one position-variable x. However, this method can
also be applied to problems with two, and often also three or four, position-variables.
Here we will confine ourselves to the simple examples of the one-dimensional heat
equation and the one-dimensional wave equation.
5.2 Partial differential equations
5.2.1
123
The heat equation
In Fourier's time, around 1800, heat conduction was already a widely studied phenomenon, from a practical as well as from a scientific point of view. In the industry
the phenomenon was important in the use of metals for machines, while in science
heat conduction was an issue in determining the temperature of the earth's interior,
in particular its variations in the course of time. The first problem that Fourier (1761
- 1830) addressed in his book Th´ orie analytique de la chaleur from 1822 was the
e
determination of the temperature T in a solid as function of the position variables
x, y, z and the time t. From physical principles he showed that the temperature
T (x, y, z, t) should satisfy the partial differential equation
∂T
=k
∂t
Heat equation
Diffusion equation
EXAMPLE 5.7
∂2T
∂2T
∂2T
+
+ 2
∂x2
∂ y2
∂z
.
Here k is a constant, whose value depends on the material of the solid. This equation is called the heat equation. The same equation also plays a role in the diffusion
of gases and liquids. In that case the function T (x, y, z, t) does not represent temperature, but the local concentration of the diffusing substances in a medium where
the diffusion takes place. The constant is in that case the diffusion coefficient and
the equation is then called the diffusion equation. We now look at the equation in a
simplified situation.
Consider a thin rod of length L with a cylinder shaped cross-section and flat ends.
The ends are kept at a temperature of 0◦ C by cooling elements, while the sidesurface (the mantle of the cylinder) is insulated, so that no heat flows through it. At
time t = 0 the temperature distribution in the longitudinal direction of the rod (the
x-direction; see figure 5.4) is given by a function f (x). So for fixed value of x, the
temperature in a cross-section of the rod is the same everywhere. This justifies a
description of the problem using only x as a position variable. The variables y and z
can be omitted, and so we can consider the temperature as function of x and t only:
T = T (x, t). The preceding equation then changes into
∂T
∂2T
=k 2
∂t
∂x
for 0 < x < L and t > 0.
We call this partial differential equation the one-dimensional heat equation.
In the remainder of this chapter we will denote the function that should satisfy a
partial differential equation by u(x, t). For the partial derivatives we introduce the
following frequently used notation:
∂u
= ux ,
∂x
∂ 2u
= uxx ,
∂x2
∂u
= ut ,
∂t
∂ 2u
= u tt .
∂t 2
(5.7)
With this notation the one-dimensional heat equation looks like this:
u t = ku x x
Boundary condition
(5.8)
Since the temperature at both ends is kept at 0◦ C for all time, starting from t = 0,
one has the following two boundary conditions:
u(0, t) = 0,
Initial condition
for 0 < x < L and t > 0.
u(L , t) = 0
for t ≥ 0.
(5.9)
Finally we formulate the situation for t = 0 as an initial condition:
u(x, 0) = f (x)
for 0 ≤ x ≤ L.
(5.10)
124
5 Applications of Fourier series
T (x,t)
f (x )
L
x
0
0°
0°
t
FIGURE 5.4
Thin rod with temperature distribution f (x) at t = 0.
Linear homogeneous
condition
Formal solution
Step 1
Here f (x) is a piecewise smooth function. This situation is shown in figure 5.4. The
partial differential equation (5.8) is an example of a linear homogeneous equation.
That is to say, when two functions u(x, t) and v(x, t) satisfy this equation, then so
does any linear combination of these two functions. In particular the null-function
satisfies the equation. The boundary conditions (5.9) have the same property (verify
this). This is why these conditions are also called linear homogeneous conditions.
Constructing a solution of equation (5.8) satisfying conditions (5.9) and (5.10) will
consist of three steps. In the first two steps we will be using separation of variables to construct a collection of functions which satisfy (5.8) as well as the linear
homogeneous conditions (5.9). To this end we must solve a so-called eigenvalue
problem, which will take place in the second step. Next we construct in the third
step, by means of an infinite linear combination of the functions from this collection,
or, put differently, by superposition, a solution which also satisfies the inhomogeneous condition (5.10). In this final step the Fourier series enter, and so we will have
to deal with all kinds of convergence problems. If we ignore these problems during
the construction, then it is said that we have obtained a formal solution, for which,
in fact, one still has to show that it actually is a solution, or even a unique solution.
Separation of variables
Using separation of variables we will construct a collection of functions satisfying
the partial differential equation (5.8) and the linear homogeneous conditions (5.9)
and having the form
u(x, t) = X (x)T (t),
(5.11)
where X (x) is a function of x only and T (t) is a function of t only. If we substitute
(5.11) into (5.8), then we obtain the relation
X (x)T (t) = k X (x)T (t).
After a division by k X (x)T (t) the variables x and t are separated:
T (t)
X (x)
=
.
kT (t)
X (x)
5.2 Partial differential equations
125
Here the left-hand side is a function of t and independent of x, while the right-hand
side is a function of x and independent of t. Therefore, an equality can only occur
if both sides are independent of x and t, and hence are equal to a constant. Call this
constant c, then
T (t)
X (x)
=
= c.
kT (t)
X (x)
The constant c is sometimes called the constant of separation. We thus have
X (x) − cX (x) = 0,
(5.12)
T (t) − ckT (t) = 0.
(5.13)
Substitution of (5.11) into the linear homogeneous boundary conditions (5.9) gives
u(0, t) = X (0)T (t) = 0 and u(L , t) = X (L)T (t) = 0 for t ≥ 0. We are not
interested in the trivial solution T (t) = 0 and so X (0) = 0 and X (L) = 0. Together
with (5.12) this leads to the problem
X − cX = 0
Eigenvalue
Eigenfunction
Step 2
for 0 < x < L, X (0) = 0, X (L) = 0,
(5.14)
where X (x) and c are to be determined. First we will solve problem (5.14) and
subsequently (5.13). Problem (5.14) obviously has the trivial solution X (x) = 0,
which is of no interest. We are therefore interested in those values of c for which
there exists a non-trivial solution X (x). These values are called eigenvalues and
the corresponding non-trivial solutions eigenfunctions. Determining the eigenvalues
and their corresponding eigenfunctions is the second step in our solution method.
Calculating eigenvalues and eigenfunctions
When we try to solve problem (5.14) we have to distinguish two cases, namely c = 0
and c = 0.
a For c = 0 equation (5.14) becomes X = 0, which has general solution X (x) =
αx + β. From the boundary conditions it follows that X (0) = β = 0 and X (L) =
αL = 0. Hence, β = α = 0. We then obtain the trivial solution and this means that
c = 0 is not an eigenvalue.
b For c = 0 the characteristic equation s 2 − c = 0 corresponding to (5.14) has
two distinct roots s1 and s2 with s2 = −s1 . Note that these roots may be complex.
The general solution is then
X (x) = αes1 x + βe−s1 x .
The first boundary condition X (0) = 0 gives α + β = 0, so β = −α. Next we
obtain from the second boundary condition X (L) = 0 the equation
α(es1 L − e−s1 L ) = 0.
For α = 0 we get the trivial solution again. So we must have es1 L − e−s1 L = 0,
implying that e2s1 L = 1. From this it follows that s1 = inπ/L, where n is an integer
2
and n = 0. This gives us eigenvalues c = s1 = −(nπ/L)2 . The corresponding
eigenfunction X (x) is X (x) = 2iα sin(nπ x/L). However, since α is arbitrary, we
can say that eigenvalue c = −(nπ/L)2 corresponds to the eigenfunction
X n (x) = sin(nπ x/L),
where we may now assume that n is a positive integer.
For c = −(nπ/L)2 the first-order differential equation (5.13) has the fundamental solution
2
Tn (t) = e−(nπ/L) kt .
126
5 Applications of Fourier series
Fundamental solution
We have thus found the following collection of fundamental solutions satisfying
equation (5.8) and conditions (5.9):
2 2
2
u n (x, t) = Tn (t)X n (x) = e−n π kt/L sin(nπ x/L)
Step 3
for n = 1, 2, . . ..
(5.15)
Superposition of fundamental solutions
Since (5.8) is a linear homogeneous differential equation and (5.9) are linear homogeneous conditions, it is possible to take linear combinations of fundamental
solutions, or, as it is often put by scientists, to form new solutions of (5.8) and (5.9)
by superpositions. Each finite linear combination a1 u 1 + a2 u 2 + · · · + an u n of
fundamental solutions also satisfies (5.8) and (5.9). However, in general we cannot
expect that a suitable finite linear combination will give us a solution which also
satisfies the remaining inhomogeneous condition. We therefore try an infinite linear
combination of fundamental solutions, still called a superposition of fundamental
solutions. It has the form
u(x, t) =
∞
2 2
2
An e−n π kt/L sin(nπ x/L).
(5.16)
n=1
If this superposition is to satisfy the inhomogeneous condition (5.10), then u(x, t)
should be equal to the function f (x) for t = 0 and hence
u(x, 0) =
∞
An sin(nπ x/L) = f (x)
for 0 ≤ x ≤ L.
n=1
The coefficients An can thus be found by determining the Fourier sine series of
f (x). The result is
An =
2 L
f (x) sin(nπ x/L) d x.
L 0
Substitution of these coefficients in (5.16) finally gives us a formal solution of the
heat conduction problem. Since we required f (x) to be piecewise smooth, and we
also assumed that f (x) equals the average value of the left- and right-hand limit
at jumps, f (x) is equal to the sum of its Fourier sine series on the interval [0, L].
Hence, u(x, 0) = f (x). It it also easy to see that for x = 0 and x = L the sum of
the series in (5.16) equals 0, since all terms are 0 then. The homogeneous conditions
are thus also satisfied. To show, however, that the series (5.16) also converges for
other values of x and t > 0, and that its sum u(x, t) satisfies differential equation
(5.8), requires a detailed analysis of the convergence of the series in (5.16). We
will content ourselves here with stating that in the case when f (x) is piecewise
smooth, one can prove that u(x, t) found in this way is indeed a solution of the heat
conduction problem that we have posed, and even that it is the unique solution.
In the preceding example the temperature at both ends of the rod was kept at 0◦ C.
This resulted in linear homogeneous conditions for the heat conduction problem.
We will now look at what happens with the temperature of a rod whose ends are
insulated.
EXAMPLE 5.8
Insulation of the ends can be expressed mathematically as u x (0, t) = 0 and
u x (L , t) = 0. The heat conduction problem for this rod is thus as follows:
u t = ku x x
u x (0, t) = 0, u x (L , t) = 0
u(x, 0) = f (x)
for 0 < x < L, t > 0,
for t ≥ 0,
for 0 ≤ x ≤ L.
5.2 Partial differential equations
127
Going through the three steps from example 5.7 again, we find a difference in step
1: instead of problem (5.14) we obtain the problem
X − cX = 0
X (0) = 0, X (L) = 0.
for 0 ≤ x ≤ L,
For c = 0 we find that X (x) is a constant. So now c = 0 is an eigenvalue
as well, with eigenfunction a constant. For c = 0 we again find the eigenvalues c = −(nπ/L)2 with corresponding eigenfunctions X n (x) = cos(nπ x/L) for
n = 1, 2, . . .. Hence, the eigenfunctions are
X n (x) = cos(nπ x/L)
for n = 0, 1, 2, 3, . . ..
Note that n now starts from n = 0. The remainder of the construction of the collection of fundamental solutions is entirely analogous to the previous example. The
result is the collection of fundamental solutions
2 2
2
u n (x, t) = e−n π kt/L cos(nπ x/L)
for n = 0, 1, 2, . . ..
Superposition of the fundamental solutions in step 3 gives
u(x, t) = 1 A0 +
2
∞
2 2
2
An e−n π kt/L cos(nπ x/L).
n=1
Since u(x, 0) = f (x) for 0 ≤ x ≤ L, the coefficients An are equal to
An =
2 L
f (x) cos(nπ x/L) d x
L 0
for n = 0, 1, 2, 3, . . ..
These Fourier coefficients arise by determining the Fourier cosine series of f (x).
Substitution of these coefficients in the series for u(x, t) then gives a formal solution
again. Since f (x) is piecewise smooth, one can prove once more that this is a unique
solution.
If we look at the solution for t → ∞, then all terms in the series for u(x, t) tend
to 0, except for the term A0 /2. One can indeed prove that limt→∞ u(x, t) = A0 /2.
It is then said that in the stationary phase the temperature no longer depends on t
and is everywhere equal to the average temperature
1 L
A0
=
f (x) d x
2
L 0
at t = 0 on the interval [0, L]. Figure 5.5 illustrates this result.
5.2.2
The wave equation
We now consider the example of a vibrating string of length L and having fixed ends.
Just as in the heat conduction problem, we will not discuss the physical arguments
needed to derive the wave equation.
EXAMPLE 5.9
The equation describing the vertical displacement u(x, t) of a vibrating string is
u tt = a 2 u x x
Wave equation
for 0 < x < L, t > 0.
(5.17)
Here a is a constant which is related to the tension in the string. This equation is
called the wave equation. Since the ends of the string are fixed, one has the following
boundary conditions:
u(0, t) = 0,
u(L , t) = 0
for t ≥ 0.
(5.18)
128
5 Applications of Fourier series
FIGURE 5.5
Temperature distribution in a rod with insulated ends.
u
u (x, 0)
L
x
0
t
FIGURE 5.6
Displacement of a vibrating string.
In figure 5.6 we see the displacement of the string at time t = 0. We thus have the
initial condition
u(x, 0) = f (x)
for 0 ≤ x ≤ L.
(5.19)
Moreover, it is given that at t = 0 the string has no initial velocity. So as a second
initial condition we have
u t (x, 0) = 0
for 0 ≤ x ≤ L.
(5.20)
130
5 Applications of Fourier series
Step 3
Superposition
Superposition of the fundamental solutions gives the series
u(x, t) =
∞
An cos(nπat/L) sin(nπ x/L).
(5.22)
n=1
By substituting the remaining initial condition (5.19), a Fourier series arises:
∞
u(x, 0) = f (x) =
An sin(nπ x/L)
for 0 ≤ x ≤ L.
n=1
For the Fourier coefficients An one thus has
An =
2 L
f (x) sin(nπ x/L) d x.
L 0
The series (5.22) together with these coefficients give the formal solution of our
problem. One can indeed show that the formal solution thus obtained is the unique
solution, provided that f (x) is piecewise smooth.
In many cases the assumption of an initial velocity u t (x, 0) = 0 is artificial. This
assumption made it possible for us to find a simple solution for Tn (t). For a string
which is struck from its resting position, one takes as initial conditions u(x, 0) = 0
and u t (x, 0) = g(x). When both the initial displacement and the initial velocity
are unequal to 0, then we are dealing with a problem with two inhomogeneous conditions. As a consequence, the functions Tn (t) will contain sine as well as cosine
terms. We must then determine the coefficients in two distinct Fourier series. For
detailed results we refer to Fourier series, transforms and boundary value problems
by J. Hanna and J.H. Rowland, pages 228 – 233. In the same book, pages 219 – 227,
one can also find the derivation of the wave equation and the heat equation, as well
as the verification that the formal solutions constructed above are indeed solutions,
which moreover are unique.
EXERCISES
5.7
A thin rod of length L with insulated sides has its ends kept at 0◦ C. The initial
temperature is
u(x, 0) =
x
L−x
for 0 ≤ x ≤ L/2,
for L/2 ≤ x ≤ L.
Show that the temperature u(x, t) is given by the series
u(x, t) =
4L ∞ (−1)n −(2n+1)2 π 2 kt/L 2
e
sin((2n + 1)π x/L).
π 2 n=0 (2n + 1)2
5.8
Both ends and the sides of a thin rod of length L are insulated. The initial temperature of the rod is u(x, 0) = 3 cos(8π x/L). Write down the heat equation for
this situation and determine the initial and boundary conditions. Next determine the
temperature u(x, t).
5.9
For a thin rod of length L the end at x = L is kept at 0◦ C, while the end at x = 0
is insulated (as well as the sides). The initial temperature of the rod is u(x, 0) =
7 cos(5π x/2L). Write down the heat equation for this situation and determine the
initial and boundary conditions. Next determine the temperature u(x, t).
5.2 Partial differential equations
5.10
131
Determine the solution of the following initial and boundary value problem:
ut = u x x
u x (0, t) = 0,
u(x, 0) =
u(2, t) = 0
1
2−x
for 0 < x < 2, t > 0,
for t ≥ 0,
for 0 < x < 1,
for 1 ≤ x < 2.
5.11
A thin rod of length L has initial temperature u(x, 0) = f (x). The end at x = 0 is
kept at 0◦ C and the end at x = L is insulated (as well as the sides). Write down the
heat equation for this situation and determine the initial and boundary conditions.
Next determine the temperature u(x, t).
5.12
Determine the displacement u(x, t) of a string of length L, with fixed ends and
initial displacement u(x, 0) = 0.05 sin(4π x/L). At time t = 0 the string has no
initial velocity.
5.13
A string is attached at the points x = 0 and x = L and has as initial displacement
f (x) =
0.02x
0.02(L − x)
for 0 < x < L/2,
for L/2 ≤ x < L.
At time t = 0 the string has no initial velocity. Write down the corresponding initial
and boundary value problem and determine the solution. One could call this the
problem of the 'plucked string': the initial position is unequal to 0 and the string is
pulled at the point x = L/2, while the initial velocity is equal to 0.
5.14
A string is attached at the points x = 0 and x = 2 and has as initial displacement
u(x, 0) = 0. The initial velocity is
u t (x, 0) = g(x) =
0.05x
0.05(2 − x)
for 0 < x < 1,
for 1 < x < 2.
Write down the corresponding initial and boundary value problem and determine
the solution. This problem could be called the problem of the 'struck string': the
initial position is equal to 0, while the initial velocity is unequal to 0, and the string
is struck at the midpoint.
5.15
Determine the solution of the following initial and boundary value problem, where
k is a constant:
ut = a2u x x
u x (0, t) = 0,
u t (x, 0) = 0,
u x (π, t) = 0
u(x, 0) = kx
for 0 < x < π , t > 0,
for t > 0,
for 0 < x < π .
SUMMARY
In this chapter Fourier series were first applied to determine the response of an LTCsystem to a periodic input. Here the frequency response, introduced in chapter 1,
played a central role. It determines the response to a time-harmonic input. Since the
input can be represented as a superposition of time-harmonic signals, using Fourier
series, one can easily determine the line spectrum of the output by applying the
superposition rule. This line spectrum is obtained by multiplying the line spectrum
of the input with the values of the frequency response at the integer multiples of the
fundamental frequency of the input:
yn = H (nω0 )u n .
132
5 Applications of Fourier series
Here yn is the line spectrum of the output y(t), u n the line spectrum of the input
u(t), H (ω) the frequency response, and ω0 the fundamental frequency of the input.
For real inputs and real systems the properties of the time-harmonic signals are taken
over by the sinusoidal signals.
Systems occurring in practice are often described by differential equations of the
form
am
dm y
d m−1 y
dnu
d n−1 u
+ am−1 m−1 + · · · + a0 y = bn n + bn−1 n−1 + · · · + b0 u.
dt m
dt
dt
dt
In order to determine a periodic solution y(t) for a given periodic signal u(t), it is
important to know whether or not there are any periodic eigenfunctions. These are
periodic solutions of the homogeneous differential equation arising from the differential equation above by taking the right-hand side equal to 0. Periodic eigenfunctions correspond to zeros s = iω of the characteristic polynomial A(s) =
am s m + am−1 s m−1 + · · · + a0 , and these lie on the imaginary axis. When the
period of a periodic input coincides with the period of a periodic eigenfunction,
then resonance may occur, that is, for a given u(t) the differential equation does not
have periodic solutions, but instead unbounded solutions.
When the differential equation describes a stable system, then all zeros of A(s)
lie in the left-half plane and the frequency response is then for all ω equal to
H (ω) =
B(iω)
,
A(iω)
with B(s) = bn s n + bn−1 s n−1 + · · · + b0 . For real systems this means that there are
no sinusoidal eigenfunctions, that is, no sinusoidal signals with an eigenfrequency.
Secondly, Fourier series were applied in solving the one-dimensional heat
equation
u t = ku x x ,
and the one-dimensional wave equation
u tt = a 2 u x x .
Using the method of separation of variables, and solving an eigenvalue problem,
one can obtain a collection of fundamental solutions satisfying the partial differential equation under consideration, as well as the corresponding linear homogeneous
conditions, but not yet the remaining inhomogeneous condition(s). By superposition of the fundamental solutions one can usually construct a formal solution which
also satisfies the inhomogeneous condition(s). In most cases the formal solution is
the solution of the problem being posed. In the superposition of the fundamental solutions lies the application of Fourier series. The fundamental solutions describe, in
relation to one or several variables, sinusoidal functions with frequencies which are
an integer multiple of a fundamental frequency. This fundamental frequency already
emerges when one calculates the eigenvalues. The superposition is then a Fourier
series whose coefficients can be determined by using the remaining inhomogeneous
condition(s).
SELFTEST
5.16
For the frequency response of an LTC-system one has
H (ω) = (1 − e−2iω )2 .
5.2 Partial differential equations
a
b
c
5.17
133
Is the response to a real periodic input real again? Justify your answer.
Calculate the response to the input u(t) = sin(ω0 t).
What is the response to a periodic input with period 1?
For an LTC-system the relation between an input u(t) and the corresponding output
y(t) is described by the differential equation y + 4y + 4y = u. Let u(t) be the
periodic input with period 2π, given on the interval (−π, π) by
u(t) =
πt + t 2
πt − t 2
for −π < t < 0,
for 0 < t < π .
Determine the first harmonic of the output y(t).
5.18
For an LTC-system the relation between an input u(t) and the output y(t) is described by the differential equation y + y + 4y + 4y = u + u.
a Does the differential equation determine the periodic response to a periodic input uniquely? Justify your answer.
b Let u(t) = cos 3t and y(t) the corresponding output. Calculate the power of the
output.
5.19
A thin rod of length L has constant positive initial temperature u(x, 0) = u 0 for
0 < x < L. The ends are kept at 0◦ C. The so-called heat-flux through a crosssection of the rod at position x0 (0 < x0 < L) and at time t > 0 is by definition
equal to −K u x (x0 , t). Show that the heat-flux at the midpoint of the rod (x0 = L/2)
equals 0.
5.20
Consider a thin rod for which one has the following equations:
u t = ku x x
u(0, t) = 0,
u(x, 0) =
a
0
u(L , t) = 0
for 0 < x < L, t > 0,
for t ≥ 0,
for 0 ≤ x ≤ L/2,
for L/2 < x ≤ L,
where a is a constant.
a Determine the solution u(x, t).
b Two identical iron rods, each 20 cm in length, have their ends put against each
other. Both of the remaining ends, at x = 0 and at x = 40 cm, are kept at 0◦ C. The
left rod has a temperature of 100◦ C and the right rod a temperature of 0◦ C. Calculate for k = 0.15 cm2 s−1 the temperature at the boundary layer of the two rods,
10 minutes after the rods made contact, and show that this value is approximately
36◦ C.
c Calculate approximately how many hours it will take to reach a temperature
of 36◦ C at the boundary layer, when the rods are not made of iron, but concrete
(k = 0.005 cm2 s−1 ).
5.21
Given is the following initial and boundary value problem:
u tt = a 2 u x x
u(0, t) = 0, u(L , t) = 0
u(x, 0) = sin(π x/L)
u t (x, 0) = 7 sin(3π x/L)
for 0 < x < L, t > 0,
for t > 0,
for 0 < x < L,
for 0 < x < L.
Show that the first two steps of the method described in section 5.2 lead to the
collection of fundamental solutions
u n (x, t) = (An sin(nπat/L) + Bn cos(nπat/L)) sin(nπ x/L),
and subsequently determine the formal solution which is adjusted to the given initial
displacement and initial velocity.
Part 3
Fourier integrals and distributions
INTRODUCTION TO PART 3
In part 2 we have developed the Fourier analysis for periodic functions. To a periodic
function f (t) we assigned for each n ∈ Z a Fourier coefficient cn ∈ C. Using these
Fourier coefficients we then defined the Fourier series, and under certain conditions
on the function f this Fourier series converged to the function f . Schematically this
can be represented as follows:
periodic function f (t) with period T and frequency ω0 = 2π/T
↓
Fourier coefficients cn =
1 T /2
f (t)e−inω0 t dt for n ∈ Z
T −T /2
↓
∞
Fourier series f (t) =
cn einω0 t .
n=−∞
Often, however, we have to deal with non-periodic phenomena. In part 3 we now
set up a similar kind of Fourier analysis for these non-periodic functions. To a nonperiodic function we will assign for each ω ∈ R a number F(ω) ∈ C. Instead of a
sequence of numbers cn , we thus obtain a function F(ω) defined on R. The function
F(ω) is called the Fourier transform of the non-periodic function f (t). Next, the
Fourier series is replaced by the so-called Fourier integral: instead of a sum over
n ∈ Z we take the integral over ω ∈ R. As for the Fourier series, this Fourier
integral will represent the original function f again, under certain conditions on f .
The scheme for periodic function will in part 3 be replaced by the following scheme
for non-periodic functions:
non-periodic function f (t)
↓
Fourier transform F(ω) =
∞
−∞
f (t)e−iωt dt for ω ∈ R
↓
Fourier integral f (t) =
∞
1
F(ω)eiωt dω.
2π −∞
136
We will start chapter 6 by showing that the transition from the Fourier series to
the Fourier integral can be made quite plausible by taking the limit T → ∞ of the
period T in the theory of the Fourier series. Although this derivation is not mathematically correct, it does result in the right formulas and in particular it will show
us precisely how the Fourier transform F(ω) should be defined. Following the formal definition of F(ω), we first calculate a number of standard examples of Fourier
transforms. Next, we treat some fundamental properties of Fourier transforms.
In chapter 7 the fundamental theorem of the Fourier integral is proven: a function
f (t) can be recovered from its Fourier transform through the Fourier integral (compare this with the fundamental theorem of Fourier series from chapter 4). We finish
the theory of the Fourier integral by deriving some important additional properties
from the fundamental theorem, such as Parseval's identities for the Fourier integral.
A fundamental problem in the Fourier analysis of non-periodic functions is the
fact that for very elementary functions, such as the constant function 1, the Fourier
transform F(ω) does not exist (we will show this in chapter 6). In physics it turned
out that useful results could be obtained by a symbolic manipulation with the Fourier
transform of such functions. Eventually this led to new mathematical objects, called
'distributions'. Distributions form an extension of the concept of a function, just as
the complex numbers form an extension of the real numbers. And just as the complex numbers, distributions have become an indispensable tool in the applications of
Fourier analysis in, for example, systems theory and (partial) differential equations.
In chapter 8 distributions are introduced and some basic properties of distributions
are treated. The Fourier transform of distributions is examined in chapter 9.
Just as in part 2, the Fourier analysis of non-periodic functions and distributions
is applied to the theory of linear systems and (partial) differential equations in the
final chapter 10.
CHAPTER 6
Fourier integrals: definition
and properties
INTRODUCTION
We start this chapter with an intuitive derivation of the main result for Fourier integrals from the fundamental theorem of Fourier series. A mathematical rigorous
treatment of the results obtained is postponed until chapter 7. In the present chapter
the Fourier integral will thus play a minor role. First we will concentrate ourselves
on the Fourier transform of a non-periodic function, which will be introduced in
section 6.2, motivated by our intuitive derivation. After discussing the existence of
the Fourier transform, a number of frequently used and often recurring examples are
treated in section 6.3. In section 6.4 we prove some basic properties of Fourier transforms. Subsequently, the concept of a 'rapidly decreasing function' is discussed in
section 6.5; in fact this is a preparation for the distribution theory of chapters 8 and
9. The chapter closes with the treatment of convolution and the convolution theorem
for non-periodic functions.
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know the definition of the Fourier transform
- can calculate elementary Fourier transforms
- know and can apply the properties of the Fourier transform
- know the concept of rapidly decreasing function
- know the definition of convolution and know and can apply the convolution
theorem.
6.1
An intuitive derivation
In the introduction we already mentioned that in order to make the basic formulas of the Fourier analysis of non-periodic function plausible, we use the theory of
Fourier series. We do emphasize that the derivation in this section is mathematically not correct. It does show which results are to be expected later on (in chapter
7). It moreover motivates the definition of the Fourier transform of a non-periodic
function.
So let us start with a non-periodic function f : R → C which is piecewise
smooth (see chapter 2 for 'piecewise smooth'). For an arbitrary T > 0 we now
consider the function f T (t) on the interval (−T /2, T /2) obtained from f by taking
f equal to 0 outside this interval. The function values at the points −T /2 and
T /2 are of no importance to us and are left undetermined. Next we extend f T (t)
periodically to R. See figure 6.1. In this way we obtain a function with period T
to which we can apply the theory of Fourier series. Since this periodic function
coincides with the original function f on (−T /2, T /2), one thus has, according to
138
140
6 Fourier integrals: definition and properties
G (ω)
–5 ∆ω
–∆ω
0 ∆ω 2∆ω
FIGURE 6.2
1 ∞
1
An approximation of 2π −∞ G(ω) dω by 2π
7∆ω
∞
n=−∞ G(n ω) ·
ω
ω.
which will be a good approximation for
∞
1
G(ω) dω.
2π −∞
(6.4)
This is illustrated in figure 6.2. Formulas (6.2) and (6.4) combined suggest that for
T → ∞ the identity (6.1) will transform into
f (t) =
∞
1
2π −∞
∞
−∞
f (τ )e−iωτ dτ eiωt dω.
(6.5)
Formula (6.5) can be interpreted as follows: when a function f (τ ) is multiplied by
a factor e−iωτ and is then integrated over R, and when subsequently the resulting
function of ω is multiplied by eiωt and then again integrated over R, then, up to a
factor 2π, the original function f will result.
This important result will return in chapter 7 as the so-called fundamental theorem of the Fourier integral and it will also be proven there. This is because the
intuitive derivation given here is incorrect in two ways. First, (6.3) is only an approximation of the right-hand side of (6.1) since G(n ω) is an integral over R instead
of over (−T /2, T /2), as was the case in the right-hand side of (6.1). Furthermore,
the right-hand side of (6.3) may certainly not be considered as an approximating
Riemann sum of the integral in (6.4). This is because if we consider the integral in
(6.4) as an improper Riemann integral, then we recall that by definition this equals
b
1
G(ω) dω.
a→−∞,b→∞ 2π a
lim
Hence, the integral over R is not at all defined through Riemann sums, but using the
limit above. There is thus no real justification why (6.3) should transform into (6.4)
for T → ∞. In chapter 7 we will see, however, that the important result (6.5) is
indeed correct for a large class of functions.
6.2
The Fourier transform
Motivated by the results from the previous section we now define, for a function
f : R → C, a new function F(ω) (for ω ∈ R) by the inner integral in (6.5).
6.2 The Fourier transform
DEFINITION 6.1
Fourier transform or
spectrum
141
For a given function f : R → C the function F(ω) (for ω ∈ R) is defined by
F(ω) =
∞
−∞
f (t)e−iωt dt,
(6.6)
provided the integral exists as an improper Riemann integral. The function F(ω) is
called the Fourier transform or spectrum of f (t).
Fourier transform
Time domain
Frequency domain
Amplitude spectrum
Phase spectrum
Energy spectrum
Spectral energy density
EXAMPLE 6.1
Unit step function
Heaviside function
Sometimes F(ω) is called the spectral density of f (t). The mapping assigning
the new function F(ω) to f (t) is called the Fourier transform. We will sometimes
write (F f )(ω) instead of F(ω), while the notation (F f (t))(ω) will also be useful,
though not very elegant. In the mathematical literature F(ω) is usually denoted
by fˆ(ω). Often, f (t) will represent a function depending on time, while F(ω)
usually depends on frequency. Hence, it is said that f (t) is defined in the time
domain and that F(ω) is defined in the frequency domain. Since e−iωt is a complexvalued function, F(ω) will in general be a complex-valued function as well, so
F : R → C. Often F(ω) is then split up into a real part and an imaginary part. One
also regularly studies | F(ω) |, the so-called amplitude spectrum (sometimes called
spectral amplitude density) of f , and arg F(ω), the so-called phase spectrum of f .
Finally, in signal theory one calls | F(ω) |2 the energy spectrum or spectral energy
density of f (t).
The definition of F(ω) closely resembles the one given for the Fourier coefficients cn . Here, however, we take ω ∈ R instead of n ∈ Z and in addition we
integrate the function f (t) over R instead of over a bounded interval (of length one
period). The fact that we integrate over R causes lots of problems. For the function
f (t) = 1 on (−T /2, T /2) for example, one can determine the Fourier coefficients,
while for the function f (t) = 1 on R the Fourier transform F(ω) does not exist.
The function introduced in our next example, re-appearing quite regularly, also has
no Fourier transform.
Let (t) be the function defined by
(t) =
1
0
for t ≥ 0,
otherwise.
See figure 6.3. This function is called the unit step function or Heaviside function.
The Fourier transform of (t) does not exist, because
∞
−∞
(t)e−iωt dt =
∞
0
e−iωt dt = lim
A
A→∞ 0
e−iωt dt.
Since (e−iωt ) = −iωe−iωt (see example 2.11), it follows that
∞
−∞
(t)e−iωt dt =
A
1
i
=
lim e−iωt
0
−iω A→∞
ω
⑀(t )
1
0
FIGURE 6.3
The unit step function (t).
lim e−iω A − 1 .
A→∞
1
t
142
6 Fourier integrals: definition and properties
However, the limit lim A→∞ e−iω A does not exist, since lim A→∞ sin Aω (and also
lim A→∞ cos Aω) does not exist.
Next we introduce an important class of functions for which F(ω) will certainly
exist.
DEFINITION 6.2
Absolutely integrable
∞
A function f : R → C is called absolutely integrable (on R) if −∞ | f (t) | dt
exists as an improper Riemann integral.
When f (t) is absolutely integrable and F(ω) is as in (6.6), then
| F(ω) | =
=
∞
−∞
∞
−∞
f (t)e−iωt dt ≤
∞
−∞
f (t)e−iωt dt
| f (t) | e−iωt dt,
and since e−iωt = 1 it then follows from definition 6.2 that
| F(ω) | ≤
∞
−∞
| f (t) | dt < ∞.
(6.7)
This shows that F(ω) exists when f (t) is absolutely integrable. Not all functions
that we will need are absolutely integrable. This explains the somewhat weaker
formulation of definition 6.1.
On the basis of the intuitive result (6.5) we expect that for each t ∈ R the function
value f (t) can be recovered from the spectrum F(ω) using the formula
f (t) =
Fourier integral
Discrete or line spectrum
Continuous spectrum
∞
1
F(ω)eiωt dω.
2π −∞
(6.8)
The right-hand side of (6.8) is called the Fourier integral of f . Formula (6.8) does
however pose some problems. Even when f (t) is absolutely integrable, F(ω) need
not be absolutely integrable. In section 6.3.1 we will see an example. On the other
hand we can still make sense of (6.8), even if F(ω) has no improper Riemann integral! And when the integral in (6.8) does exist (in some sense or other), then its
value is not necessarily equal to f (t). We will return to all of these problems in
chapter 7. In the meantime we will agree to call the right-hand side of (6.8) the
Fourier integral of f (t), provided that it exists. In chapter 7 it will be shown that,
just as for Fourier series, the Fourier integral does indeed exist for a large class of
functions and that (6.8) is valid.
If we now look at the Fourier series of a periodic function again, then we see
that only integer multiples n ω of the frequency ω = 2π/T occur (we write ω
here, instead of ω0 as used in part 2). In this case the spectrum is a function on Z,
which is a so-called discrete set. Therefore, it is said that a periodic function has a
discrete or line spectrum (see also section 3.3). However, in the Fourier integral all
frequencies ω occur, since we integrate over R. Hence, a non-periodic function leads
to a continuous spectrum. Note, though, that this does not mean that the function
F(ω) is a continuous function, but only that F(ω) depends on a continuous variable
ω.
The transition from a discrete to a continuous spectrum can be illustrated quite
nicely with the same sort of process as in section 6.1. Let f : R → C be piecewise
smooth and zero outside an interval (−T /2, T /2) (using the terminology of section
6.1 we thus have f T (t) = f (t)). Then the corresponding F(ω) certainly exists,
since we only integrate over the bounded interval (−T /2, T /2). If we now extend
f periodically, precisely as we did in section 6.1, then we can determine the Fourier
6.3 Some standard Fourier transforms
147
Closely related to this example is the function g(t) defined by
e−at
0
g(t) =
for t ≥ 0,
otherwise,
where a ∈ C with Re a > 0. A convenient way of expressing g(t) uses the unit step
function from example 6.1, since g(t) = (t)e−at . As before one quickly shows for
g(t) that
G(ω) =
a − iω
1
.
= 2
a + iω
a + ω2
The function G(ω) is now complex-valued. For a > 0 the function G(ω) has real
part a/(a 2 +ω2 ) = F(ω)/2 and imaginary part ω/(a 2 +ω2 ). Note that ω/(a 2 +ω2 )
is not improper Riemann integrable.
6.3.4
The Gauss function
To conclude, we determine the spectrum F(ω) of the function f (t) = e−at for
fixed a > 0. The function f (t) is called the Gauss function. In order to determine
F(ω) directly from the definition, one would need a part of complex function theory
which falls outside the scope of this book. There is, however, a clever trick to find
F(ω) in an indirect manner. To do so, we will assume the following fact:
2
Gauss function
∞
−∞
e−x d x =
2
√
π
(6.15)
(for a proof see for example Fourier analysis by T.W. K¨ rner, Lemma 48.10). As
o
a matter of fact, (6.15) also shows immediately that the function f (t) is absolutely
integrable. Since f (t) = f (−t) it follows as in section 6.3.2 that
F(ω) =
∞
−∞
∞
2
e−at e−iωt dt = 2
0
e−at cos ωt dt.
2
We will now determine the derivative of F(ω). In doing so, we assume that the
differentiation may be carried out within the integral. It would lead us too far to even
formulate the theorem that would justify this step. The result of the differentiation
with respect to ω within the integral is as follows:
F (ω) = −2
∞
0
te−at sin ωt dt.
2
Integrating by parts we obtain
1 ∞ −at 2
sin ωt dt
e
a 0
∞
1 −at 2
ω ∞ −at 2
=
e
sin ωt
−
e
cos ωt dt.
0
a
a 0
F (ω) =
The first term in this difference is equal to 0, while the second term equals −ωF(ω)/
2a. Hence we obtain for F(ω) the (differential) equation
F (ω) = (−ω/2a)F(ω).
If we now divide left-hand and right-hand sides by F(ω), then
ω
F (ω)
=− ,
F(ω)
2a
that is,
d
ω
ln | F(ω) | = − .
dω
2a
148
6 Fourier integrals: definition and properties
d
But also dω (−ω2 /4a) = −ω/2a and thus apparently ln | F(ω) | = −ω2 /4a + C
for an arbitrary constant C. It then follows that
2
| F(ω) | = eC e−ω /4a .
This in fact states that F(ω) = De−ω /4a , where D is an arbitrary constant (note
that eC is always positive). If we substitute ω = 0, then we see that D = F(0).
√
Now change to the variable x = t a in (6.15), then it follows that
2
D = F(0) =
∞
∞
√ 2√
2
1
e−at dt = √
e−(t a) a dt =
a −∞
−∞
π
.
a
The spectrum of the Gauss function is thus given by
π −ω2 /4a
e
.
a
F(ω) =
(6.16)
This is again a continuous real-valued function with limω→±∞ F(ω) = 0. It is
2
quite remarkable that apparently the spectrum of e−at is of the same form as
2
the original function. For a = 1/2 one has in particular that (Fe−t /2 )(ω) =
√
√
2 /2
2πe−ω , so up to the factor 2π exactly the same function. In figure 6.8 the
Gauss function and its spectrum are drawn.
b
a
√π/a
1
(F e –at 2 )(ω)
e –at
0
2
1
t
0
1
ω
FIGURE 6.8
The Gauss function (a) and its spectrum (b).
For the moment this concludes our list of examples. The most important results
have been included in table 3. In the next section some properties of the Fourier
transform are established, enabling us, among other things, to calculate more Fourier
transforms.
EXERCISES
6.2
Consider for fixed a ∈ C with Re a > 0 the function g(t) defined by g(t) =
(t)e−at (also see section 6.3.3).
a Show that for the spectrum G(ω) one has: G(ω) = (a − iω)/(a 2 + ω2 ).
b Take a > 0. Show that the Fourier integral for the imaginary part of G(ω) (and
hence also for G(ω) itself) does not exist as an improper integral.
c Verify that for the limit a → 0 the function g(t) transforms into (t), while
G(ω) for ω = 0 transforms into −i/ω. This seems to suggest that (t) has the function −i/ω as its spectrum. This, however, contradicts the result from example 6.1.
We will return to this in chapters 8 and 9 on distribution theory.
6.4 Properties of the Fourier transform
6.3
Determine the Fourier transform of the function
cos t
0
f (t) =
6.4
for −π/2 ≤ t ≤ π/2,
otherwise.
Determine the spectrum G(ω) and verify that G(ω) is continuous when g(t) is given
by
|t |
0
g(t) =
6.5
149
for −1 ≤ t ≤ 1,
otherwise.
Let a > 0 be fixed. Determine the spectrum F(ω) of the function
for 0 ≤ t ≤ a/2,
1
−1 for −a/2 ≤ t < 0,
f (t) =
0
otherwise.
a
Show that F(ω) is continuous at ω = 0.
b
6.4
Properties of the Fourier transform
A large number of properties that we have established for Fourier series will return
in this section. Often the proofs are slightly more difficult, since in Fourier analysis
on R we always have to deal with improper (Riemann) integrals. The theory of improper integrals has quite a few difficult problems and we will not always formulate
the precise theorems that will be needed. In section 6.3.4, for example, an improper
integral and differentiation were 'interchanged', without formulating a theorem or
giving conditions justifying this. A more frequently occurring problem is reversing
the order of integration in a repeated integral. For example, even when
∞
b
∞
a
f (x, y) d x dy
∞
∞
and
a
b
f (x, y) dy d x
both exist, then still they do not necessarily have the same value (see exercise 6.6).
Theorems on interchanging the order of integration will not be presented here. The
interested reader can find such theorems in, for example, Fourier analysis by T.W.
K¨ rner, Chapters 47 & 48. When we interchange the order of integration in the
o
proof of a theorem, we will always give sufficient conditions in the theorem such
that the interchanging is allowed.
After these preliminary remarks we now start examining the properties of the
Fourier transform.
6.4.1
Linearity
Linear combinations of functions are carried over by the Fourier transform into the
same linear combination of the Fourier transforms of these functions. We formulate
this linearity property in a precise manner in the following theorem.
THEOREM 6.1
Linearity of the Fourier
transform
Let f (t) and g(t) be two functions with Fourier transforms F(ω) and G(ω) respectively. Then a F(ω) + bG(ω) is the Fourier transform of a f (t) + bg(t).
Proof
This theorem follows immediately from the linearity of integration:
(a f 1 (t) + b f 2 (t)) dt = a
f 1 (t) dt + b
f 2 (t) dt
150
6 Fourier integrals: definition and properties
for arbitrary functions f 1 and f 2 and a, b ∈ C. Now take f 1 (t) = f (t)e−iωt and
f 2 (t) = g(t)e−iωt .
Because of this property, the Fourier transform is called a linear transformation.
6.4.2
Conjugation
For the complex conjugate of a function one has the following theorem.
THEOREM 6.2
Spectrum of the complex
conjugate
Let f (t) be a function with spectrum F(ω). Then the spectrum of the function f (t)
is given by F(−ω).
Proof
This result follows immediately from the properties of definite integrals of complexvalued functions (see section 2.3):
(F f (t))(ω) =
∞
−∞
f (t)e−iωt dt =
∞
−∞
f (t)eiωt dt = F(−ω).
When in particular the function f (t) is real-valued, so f (t) = f (t), then theorem
6.2 implies that F(−ω) = F(ω), or F(−ω) = F(ω).
6.4.3
Shift in the time domain
For a given function f (t) and a fixed a ∈ R, the function f (t − a) is called the
function shifted over a. There is simple relationship between the spectra of these
two functions.
THEOREM 6.3
Shift in the time domain
Let f (t) be a function with spectrum F(ω). Then one has for any fixed a ∈ R that
(F f (t − a))(ω) = e−iωa F(ω).
Proof
By changing to the new variable τ = t − a one obtains
(F f (t − a))(ω) =
∞
f (t − a)e−iωt dt =
−∞
−iωa F(ω).
= e
Phase factor
∞
−∞
f (τ )e−iω(τ +a) dτ
So when a function is shifted over a in the time domain, its spectrum is multiplied
by the factor e−iωa . Note that this only changes the phase spectrum and not the
amplitude spectrum. The factor e−iωa is called a phase factor.
6.4.4
Shift in the frequency domain
For a shift in the frequency domain there is a result similar to that of section 6.4.3.
THEOREM 6.4
Shift in the frequency domain
Let f (t) be a function with spectrum F(ω).
(F eiat f (t))(ω) = F(ω − a).
Then one has for a ∈ R that
Proof
F(ω − a) =
∞
−∞
f (t)e−i(ω−a)t dt =
∞
−∞
eiat f (t)e−iωt dt = (Feiat f (t))(ω).
152
6 Fourier integrals: definition and properties
Fourier cosine transform
One now calls the integral in the right-hand side (so without the factor 2) the Fourier
cosine transform of the even function f (t); we denote it by Fc (ω), so F(ω) =
2Fc (ω). Since cos(−ωt) = cos ωt, the function F(ω) is in this case an even
function as well. This also follows from (6.18) since F(−ω) = (F f (−t))(ω) =
(F f (t))(ω) = F(ω). If, in addition, we know that f (t) is real-valued, then it follows from (6.19) that F(ω) is also real-valued. We already saw this in sections 6.3.1
and 6.3.2 for the functions pa and qa . Hence, for an even and real-valued function
f (t) we obtain that F(ω) is also even and real-valued.
When a function f (t) is only defined for t > 0, then one can calculate the Fourier
cosine transform of this function using the integral in (6.19). This is then in fact the
ordinary Fourier transform (up to a factor 2) of the function that arises by extending
f to an even function on R; for t < 0 one thus defines f by f (t) = f (−t). The
value at t = 0 is usually taken to be 0, but this is hardly relevant.
There are similar results for odd functions. If g : R → C is odd and if we use
the fact that sin ωt = (eiωt − e−iωt )/2i, then we obtain, just as in the case of even
functions,
G(ω) = −2i
Fourier sine transform
EXAMPLE
∞
0
g(t) sin ωt dt.
(6.20)
One then calls the integral in the right-hand side (so without the factor −2i) the
Fourier sine transform of the odd function g(t); we denote it by G s (ω), so G(ω) =
−2i G s (ω). Now G(ω) is an odd function. Again this also follows from (6.18). If,
in addition, we know that g(t) is real-valued, then it follows from (6.20) that G(ω)
can only assume purely imaginary values.
If a function g(t) is only defined for t > 0, then one can calculate its Fourier sine
transform using the integral in (6.20). This is then the ordinary Fourier transform
(up to a factor −2i) of the function that arises by extending g to an odd function on
R; for t < 0 one defines g by g(t) = −g(−t) and g(0) = 0.
Consider for t > 0 the function f (t) = e−at , where a > 0 is fixed. Extend this
function to an even function on R. Then the Fourier transform of f equals the
function 2a/(a 2 + ω2 ). This is because the even extension of f is the function
e−a| t | and the Fourier transform of this function has been determined in section
6.3.3. Because of the factor 2, the Fourier cosine transform is given by a/(a 2 + ω2 ).
In exercise 6.14 you will be asked to determine the Fourier transform of the odd
extension of f (for the case a = 1).
6.4.7
Selfduality
The selfduality derived in this section is a preparation for the distribution theory
in chapters 8 and 9. First we observe the following. Until now we have always
regarded f (t) as a function of the time t and F(ω) as a function of the frequency
ω. In fact f and F are just two functions from R to C for which the name of the
variable is irrelevant.
THEOREM 6.6
Selfduality
Let f (t) and g(t) be piecewise smooth and absolutely integrable functions with
spectra F(ω) and G(ω) respectively. Then
∞
−∞
f (x)G(x) d x =
∞
−∞
F(x)g(x) d x.
6.4 Properties of the Fourier transform
153
Proof
From the definition of spectrum it follows that
∞
−∞
f (x)G(x) d x =
=
∞
−∞
∞
∞
f (x)
∞
−∞ −∞
−∞
g(y)e−i x y dy d x
f (x)g(y)e−i x y d yd x.
We mention without proof that under the conditions of theorem 6.6 one may interchange the order of integration. We then indeed obtain
∞
−∞
f (x)G(x) d x =
6.4.8
∞
−∞
∞
g(y)
−∞
f (x)e−i x y d x dy =
∞
−∞
g(y)F(y) dy.
Differentiation in the time domain
Important for the application of Fourier analysis to, for example, differential equations is the relation between differentiation and Fourier transform. In this section we
will see how the spectrum of a derivative f can be obtained from the spectrum of
f . In particular it will be assumed that f (t) is continuously differentiable, so f (t)
exists on R and is continuous.
THEOREM 6.7
Differentiation in time
domain
Let f (t) be a continuously differentiable function with spectrum F(ω) and assume
that limt→±∞ f (t) = 0. Then the spectrum of f (t) exists and (F f )(ω) =
iωF(ω).
Proof
Since f is continuous it follows from integration by parts that
B
lim
A→−∞,B→∞ A
=
lim
f (t)e−iωt dt
f (t)e−iωt
B
+
B
lim
iω
A
A→−∞,B→∞
A→−∞,B→∞
A
= lim f (B)e−iωB − lim f (A)e−iω A + iωF(ω),
B→∞
A→−∞
f (t)e−iωt dt
where in the last step we used that F(ω) exists. Since limt→±∞ f (t) = 0, it follows
that lim B→∞ f (B)e−iωB = 0 and lim A→−∞ f (A)e−iω A = 0. Hence, (F f )(ω)
exists and we also see immediately that (F f )(ω) = iωF(ω).
Of course, theorem 6.7 can be applied repeatedly, provided that the conditions are
satisfied in each case. If, for example, f is twice continuously differentiable (so f
is now a continuous function) and both limt→±∞ f (t) = 0 and limt→±∞ f (t) =
0, then theorem 6.7 can be applied twice and we obtain that (F f )(ω) =
(iω)2 F(ω) = −ω2 F(ω). In general one has: if f is m times continuously differentiable and limt→±∞ f (k) (t) = 0 for each k = 0, 1, 2, . . . , m − 1 (where f (k)
denotes the kth derivative of f and f (0) = f ), then
(F f (m) )(ω) = (iω)m F(ω).
(6.21)
We will use the Gauss function to illustrate theorem 6.7; it is the only continuously
differentiable function among the examples in section 6.3.
EXAMPLE 6.2
The derivative of the function f (t) = e−at is given by the continuous function
2
2
f (t) = −2ate−at . Moreover, limt→±∞ e−at = 0. Theorem 6.7 can thus be
2
6.4 Properties of the Fourier transform
6.4.10
155
Integration
Finally we will use the differentiation rule in the time domain to derive a rule for
integration in the time domain.
THEOREM 6.9
Integration in time domain
Let f (t) be a continuous and absolutely integrable function with spectrum F(ω).
t
Assume that limt→∞ −∞ f (τ ) dτ = 0. Then one has for ω = 0 that
F
t
−∞
f (τ ) dτ (ω) =
F(ω)
.
iω
Proof
t
Put g(t) = −∞ f (τ ) dτ . Since f is continuous, it follows that g is a continuously
differentiable function. Moreover, g (t) = f (t) (fundamental theorem of calculus). Since, according to our assumptions, limt→±∞ g(t) = 0, theorem 6.7 can
now be applied to the function g. One then obtains that (F f )(ω) = (F g )(ω) =
iω(F g)(ω) and so the result follows by dividing by iω for ω = 0.
∞
∞
Note that limt→∞ g(t) = −∞ f (τ ) dτ = 0. But −∞ f (τ ) dτ is precisely
F(0), and so the conditions of theorem 6.9 apparently imply that F(0) = 0.
6.4.11
Continuity
We want to mention one final result which is somewhat separate from the rest of
section 6.4. It is in agreement with a fact that can easily be observed: all spectra
from section 6.3 are continuous functions on R.
THEOREM 6.10
Continuity of spectra
Let f (t) be an absolutely integrable function. Then the spectrum F(ω) is a continuous functions on R.
Since theorem 6.10 will not be used in the future, we will not give a proof (see
for example Fourier analysis by T.W. K¨ rner, Lemma 46.3). For specific functions,
o
like the functions from section 6.3, the theorem can usually be verified quite easily.
The function sin t/t is not absolutely integrable (this was noted in section 6.3.1).
In exercise 7.5 we will show that the spectrum of this function is the discontinuous
function π p2 (ω).
EXERCISES
6.6
In this exercise we will show that interchanging the order of integration is not always allowed. To do so, we consider the function f (x, y) on {(x, y) ∈ R2 | x > 0
∞ ∞
and y > 0} given by f (x, y) = (x−y)/(x+y)3 . Show that 1 ( 1 f (x, y) d x) dy
∞ ∞
and 1 ( 1 f (x, y) dy) d x both exist, but are unequal.
6.7
Use the linearity property to determine the spectrum of the function f (t) =
3e−2| t | + 2iqa (t), where qa (t) is the triangle function from (6.12).
6.8
Use the modulation theorem to determine the spectrum of the function f (t) =
e−7| t | cos πt.
6.9
a Let F(ω) be the spectrum of f (t). What then is the spectrum of f (t) sin at
(a ∈ R)?
b Determine the spectrum of
f (t) =
sin t
0
for −π ≤ t ≤ π,
otherwise.
156
6 Fourier integrals: definition and properties
6.10
Let b ∈ R and a ∈ C with Re a > 0 be fixed. Determine the Fourier transform of
the function f (t) = (t)e−at cos bt and g(t) = (t)e−at sin bt.
6.11
Prove the scaling property from theorem 6.5 for c < 0.
6.12
∞
Verify that for an odd function f (t) one has F(ω) = −2i 0 f (t) sin ωt dt.
6.13
a Let f (t) be a real-valued function and assume that the spectrum F(ω) is an even
function. Show that F(ω) has to be real-valued.
b Let f (t) be a real-valued function. Show that | F(ω) | is even.
6.14
For t > 0 we define the function f (t) by f (t) = e−t . This function is extended
to an odd function on R, so f (t) = −et for t < 0 and f (0) = 0. Determine the
spectrum of f .
6.15
Consider for a > 0 fixed the function
f (t) =
a
b
1
0
for 0 < t ≤ a,
for t > a.
Determine the Fourier cosine transform of f .
Determine the Fourier sine transform of f .
6.16
Determine in a direct way, that is, using the definition of F(ω), the spectrum of the
function t pa (t), where pa (t) is given by (6.10). Use this to check the result from
example 6.3 in section 6.4.9.
6.17
Consider the Gauss function f (t) = e−at .
a Use the differentiation rule in the frequency domain to determine the spectrum
of t f (t).
b Note that t f (t) = − f (t)/2a. Show that the result in part a agrees with the
result from example 6.2, which used the differentiation rule in the time domain.
6.18
Give at least three functions f whose Fourier transform F(ω) is equal to k f (ω),
where k is a constant.
6.19
Determine the spectrum of the function (t)te−at (for a ∈ C with Re a > 0).
2
6.5
Rapidly decreasing functions
In this section the results on differentiation in the time domain from section 6.4.8
are applied in preparation for chapters 8 and 9 on distributions. To this end we will
introduce a collection of functions V which is invariant under the Fourier transform,
that is to say: if f (t) ∈ V , then F(ω) ∈ V . We recall that, for example, for the
absolutely integrable functions this is not necessarily the case: if f (t) is absolutely
integrable, then F(ω) is not necessarily absolutely integrable. If we now look at the
differentiation rules from sections 6.4.8 and 6.4.9, then we see that differentiation
in one domain corresponds to multiplication in the other domain. We thus reach the
conclusion that we should introduce the collection of so-called rapidly decreasing
functions. It will turn out that this collection is indeed invariant under the Fourier
transform. Let us write f (t) ∈ C ∞ (R) to indicate that f can be differentiated
arbitrarily many times, that is, f (k) (t) exists for each k ∈ N. It is also said that f is
infinitely differentiable.
DEFINITION 6.3
Rapidly decreasing function
A function f : R → C in C ∞ (R) is called rapidly decreasing if for each m and
n ∈ N the function t n f (m) (t) is bounded on R, that is to say, there exists a constant
M > 0 such that |t n f (m) (t)| < M for all t ∈ R.
6.5 Rapidly decreasing functions
157
In this definition the constant M will of course depend on the value of m and n.
The term 'rapidly decreasing' has a simple explanation: for large values of n ∈ N
the functions | t |−n decrease rapidly for t → ±∞ and from definition 6.3 it follows
for m = 0 that | f (t) | < M | t |−n for each n ∈ N. Hence, the function f (and even
all of its derivatives) has to decrease quite rapidly for t → ±∞.
The collection of all rapidly decreasing functions will be denoted by S(R) or
simply by S. For f ∈ S it follows immediately from the definition that c f ∈ S as
well for an arbitrary constant c ∈ C. And since
t n ( f + g)(m) (t) = t n f (m) (t) + t n g (m) (t) ≤ t n f (m) (t) + t n g (m) (t) ,
it also follows that f + g ∈ S whenever f ∈ S and g ∈ S.
Now this is all quite nice, but are there actually any functions at all that belong to
S, besides the function f (t) = 0 (for all t ∈ R)?
THEOREM 6.11
The Gauss function f (t) = e−at (a > 0) belongs to S.
2
Proof
First note that f is infinitely differentiable because f (t) = (h ◦ g)(t), where g(t) =
−at 2 and h(s) = es are infinitely differentiable functions. From the chain rule it
then follows that f ∈ C ∞ (R). By repeatedly applying the product rule and the
2
chain rule again, it follows that (e−at )(m) is a finite sum of terms of the form
2
2
ct k e−at (k ∈ N and c a constant). Hence, t n (e−at )(m) is also a finite sum of
2
terms of the form ct k e−at for each m and n ∈ N, and we must show that this is
2
bounded on R. It now suffices to show that t k e−at is bounded on R for arbitrary
k ∈ N, since then any finite sum of such terms is bounded on R. The boundedness
2
2
of t k e−at on R follows immediately from the fact that t k e−at is a continuous
2
function with limt→±∞ t k e−at = 0.
By multiplication and differentiation one can obtain new functions in S: for
f (t) ∈ S one has t n f (t) ∈ S for each n ∈ N and even (t n f (t))(m) ∈ S for
each m and n ∈ N. For the proof of the latter statement one has to apply the product
rule repeatedly again, resulting in a sum of terms of the form t k f (l) (t), which all
belong to S again because f ∈ S. The same argument shows that the product f · g
also belongs to S when f, g ∈ S.
THEOREM 6.12
S is invariant under Fourier
transform
If f (t) ∈ S, then F(ω) ∈ S.
Proof
First we have to show that F(ω) ∈ C ∞ (R). But since f ∈ S, it follows that
t p f (t) ∈ S for each p ∈ N and so, according to the remark above, t p f (t) ∈ S
is absolutely integrable for each p ∈ N. The differentiation rule in the frequency
domain can now be applied arbitrarily often. Hence, F(ω) ∈ C ∞ (R) and (6.22)
holds. We still have to show that ωn F (m) (ω) is bounded on R for every m, n ∈ N.
According to (6.22) one has
ωn F (m) (ω) = ωn (F(−it)m f (t))(ω).
(6.23)
We now want to apply the differentiation rule in the time domain repeatedly. In
order to do so, we first note that t m f (t) ∈ S, which implies that this function is
at least n times continuously differentiable. Also (t m f (t))(k) ∈ S for each k =
0, 1, . . . , n − 1 and so we certainly have limt→±∞ (t m f (t))(k) = 0. Hence, the
differentiation rule in the time domain can indeed be applied repeatedly, and from
(6.21) it then follows that
(iω)n (F g(t))(ω) = (F(g(t))(n) )(ω),
(6.24)
158
6 Fourier integrals: definition and properties
where g(t) = (−it)m f (t). Combining equations (6.23) and (6.24) shows that
(iω)n F (m) (ω) is the Fourier transform of (g(t))(n) = ((−it)m f (t))(n) , which is
absolutely integrable. The boundedness of (iω)n F (m) (ω), and so of ωn F (m) (ω),
then follows from the simple relationship (6.7), applied to the function (g(t))(n) .
This proves that F(ω) ∈ S.
Let us take the Gauss function f (t) = e−at ∈ S (a > 0) as an example. From
(6.16) it indeed follows that F(ω) ∈ S. This is because F(ω) has the same form as
f (t).
2
EXERCISES
6.20
Indicate why e−a| t | (a > 0) and (1 + t 2 )−1 do not belong to S.
6.21
Consider the Gauss function f (t) = e−at (a > 0).
2
a Verify that for arbitrary k ∈ N one has limt→±∞ t k e−at = 0.
b Determine the first three derivatives of f and verify that these are a finite
sum of terms of the form ct l f (t) (l ∈ N and c a constant). Conclude that
limt→±∞ f (k) (t) = 0 for k = 1, 2, 3.
6.22
Let f and g belong to S. Show that f · g ∈ S.
2
6.6
Convolution
Convolution of periodic functions has been treated in chapter 4. In this section we
study the concept of convolution for non-periodic functions. Convolution arises, for
example, in the following situation. Let f (t) be a function with spectrum F(ω).
Then F(ω) is a function in the frequency domain. Often, such a function is multiplied by another function in the frequency domain. One should think for instance of
a (sound) signal with a spectrum F(ω) containing undesirable high or low frequencies, which we then send through a filter in order to remove these frequencies. We
may remove, for example, all frequencies above a fixed frequency ω0 by multiplying F(ω) by the block function p2ω0 (ω). In general one will thus obtain a product
function F(ω)G(ω) in the frequency domain. Now the question is, how this alters
our original signal f (t). In other words: which function has as its spectrum the
function F(ω)G(ω)? Is this simply the product function f (t)g(t), where g(t) is a
function with spectrum G(ω)? A very simple example shows that this cannot be the
case. The product of the block function pa (t) (now in the time domain) with itself,
that is, the function ( pa (t))2 , is just pa (t) again. However, the product of (F pa )(ω)
(see (6.11)) with itself is 4 sin2 (aω/2)/ω2 , and this does not equal (F pa )(ω). We
do recognize, however, the Fourier transform of the triangle function qa (t), up to
a factor a (see (6.13)). Which operation can turn two block functions into a triangle function? The solution of this problem, in the general case as well, is given by
convolution.
DEFINITION 6.4
Convolution
The convolution product (or convolution for short) of two functions f and g, denoted
by f ∗ g, is defined by
( f ∗ g)(t) =
∞
−∞
f (τ )g(t − τ ) dτ
for t ∈ R,
provided the integral exists.
Before we discuss the existence of the convolution, we will first of all verify that
( pa ∗ pa )(t) indeed results in the triangle function qa (t), up to the factor a.
CHAPTER 7
The fundamental theorem of the
Fourier integral
INTRODUCTION
Now that we have calculated a number of frequently used Fourier transforms and
have been introduced to some of the properties of Fourier transforms, it is time to
return to the Fourier integral
∞
1
F(ω)eiωt dω.
2π −∞
It is quite reasonable to expect that, analogous to the Fourier series, the Fourier integral will in general be equal to f (t). In section 6.1 this has already been derived
intuitively from the fundamental theorem of Fourier series. Therefore, we start this
chapter with a proof of this crucial result, which we will call the fundamental theorem of the Fourier integral. It shows that the function f (t) can be recovered from
its spectrum F(ω) through the Fourier integral. We should note, however, that the
integral should not be interpreted as an ordinary improper Riemann integral, but as
a so-called 'Cauchy principal value'.
Using the fundamental theorem we subsequently prove a number of additional
properties of the Fourier transform. One of the most famous is undoubtedly Parseval's identity
∞
−∞
| f (t) |2 dt =
∞
1
| F(ω) |2 dω,
2π −∞
which has an important interpretation in signal theory: if a signal has a 'finite
∞
energy-content' (meaning that −∞ | f (t) |2 dt < ∞), then the spectrum of the
signal also has a finite energy-content.
The fundamental theorem from section 7.1, together with its consequences from
section 7.2, conclude the Fourier analysis of non-periodic functions. The final section then treats Poisson's summation formula. Although not really a consequence
of the fundamental theorem, it forms an appropriate closing subject of this chapter,
since this formula gives an elegant relationship between the Fourier series and the
Fourier integral. Moreover, we will use Poisson's summation formula in chapter 9
to determine the Fourier transform of the so-called comb distribution.
164
7.1 The fundamental theorem
165
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know the Riemann–Lebesgue lemma and know its interpretation for non-periodic
functions
- know the concept of Cauchy principal value
- know and can apply the fundamental theorem of the Fourier integral
- know the uniqueness theorem for the Fourier transform
- know and can apply the reciprocity property
- know and can apply the convolution theorem in the frequency domain
- know and can apply Parseval's identities
- know the concepts of energy-content and of energy-signal
- can calculate definite integrals using the fundamental theorem and Parseval's identities
- know and can apply Poisson's summation formula∗ .
7.1
The fundamental theorem
In this section we give a precise meaning to the Fourier integral in (6.8). We will
prove that, under certain conditions on the function f in the time domain, the Fourier
integral converges and, just as in the case of Fourier series, will produce the original
function f . Hence, through the Fourier integral the function f (t) can be recovered
from its Fourier transform F(ω). The result is crucial for the remainder of the
Fourier theory and so we will present a proof of this result in this book. Before we
can give this proof, some preparatory results will be derived, although these are also
of interest in themselves. A first step is the so-called Riemann–Lebesgue lemma
on R. For the concepts absolutely integrable and piecewise continuous we refer to
definitions 6.2 and 2.3 respectively.
THEOREM 7.1
Riemann–Lebesgue lemma
Let f (t) be an absolutely integrable and piecewise continuous function on R. Then
lim
ω→±∞
F(ω) =
∞
lim
ω→±∞ −∞
f (t)e−iωt dt = 0.
(7.1)
Proof
Let > 0. Since f is absolutely integrable, there exist A, B ∈ R such that
∞
| f (t) | dt +
B
A
−∞
| f (t) | dt < /2.
On the remaining bounded interval [ A, B] we use the Riemann–Lebesgue lemma
from section 4.1 (theorem 4.2): there exists a G > 0 such that for | ω | > G one has
B
A
f (t)e−iωt dt < /2.
By applying the triangle inequality repeatedly, it follows that
∞
−∞
≤
≤
f (t)e−iωt dt
A
−∞
A
−∞
f (t)e−iωt dt +
| f (t) | dt +
∞
B
∞
B
f (t)e−iωt dt +
| f (t) | dt +
B
A
B
A
f (t)e−iωt dt
f (t)e−iωt dt ≤ /2 + /2 =
166
7 The fundamental theorem of the Fourier integral
for | ω | > G, where in the second step we used that
g(t) dt ≤
| g(t) | dt and
e−iωt = 1. This proves theorem 7.1.
Since e−iωt = cos ωt − i sin ωt, it follows immediately that (7.1) is equivalent to
∞
lim
ω→±∞ −∞
f (t) sin ωt dt = 0
∞
and
lim
ω→±∞ −∞
f (t) cos ωt dt = 0.
(7.2)
As a matter of fact, the Riemann–Lebesgue lemma is also valid if we only assume
that the function f is absolutely integrable on R. This more general theorem will not
be proven here, since we will only need this result for absolutely integrable functions
that are piecewise continuous as well; for these functions the proof is easier.
Theorem 7.1 has a nice intuitive interpretation: when we integrate the function f
against ever higher frequencies, so for increasing ω, then everything will eventually
cancel out. This is because the function f will change little relative to the strong
oscillations of the sine and cosine functions; the area of juxtapositioned oscillations
of f (t) sin ωt, for example, will cancel each other better and better for increasing ω.
Besides theorem 7.1 we will use the following identity in the proof of the fundamental theorem:
∞ sin t
t
0
dt =
π
.
2
(7.3)
This identity has been proven in section 4.4.1. Using theorem 7.1 and formula (7.3)
we can now prove the fundamental theorem. It will turn out, however, that the
Fourier integral will not necessarily exist as an improper Riemann integral. Hence,
to be able to formulate the fundamental theorem properly, we need the concept of a
Cauchy principal value of an integral.
DEFINITION 7.1
Cauchy principal value
A
The value of lim A→∞ −A f (t) dt is called the Cauchy principal value of the inte∞
gral −∞ f (t) dt, provided that this limit exists.
The difference between the Cauchy principal value and the improper Riemann
integral is the fact that here the limits tend to ∞ and −∞ at the same rate. We
recall that in the improper (Riemann) integral we are dealing with two independent
limits B → ∞ and A → −∞ (also see the end of section 6.1). When a function
has an improper integral, then the Cauchy principal value will certainly exist and it
will have the same value as the improper integral (just take 'A = B'). The converse
need not be true, as the next example shows.
EXAMPLE
∞
The improper integral −∞ t dt does not exist, but the Cauchy principal value is 0
A
since −A t dt = 0 for each A > 0 and hence also for A → ∞.
Since the Cauchy principal value of the Fourier integral is equal to
A
1
F(ω)eiωt dω,
A→∞ 2π −A
lim
it seems plausible to investigate for arbitrary A > 0 the integral
A
A
1
1
F(ω)eiωt dω =
2π −A
2π −A
∞
−∞
f (s)e−iωs ds eiωt dω
(7.4)
more thoroughly. If we may interchange the order of integration (we will return to
this in the proof of the fundamental theorem), then it would follow that
A
∞
1
1
F(ω)eiωt dω =
f (s)
2π −A
2π −∞
A
−A
eiω(t−s) dω ds.
7.1 The fundamental theorem
167
The inner integral can be calculated (as a matter of fact, it is precisely the Fourier
transform of the block function p2A at s − t; see section 6.3.1) and it then follows
that
A
1
1 ∞
sin A(t − s)
F(ω)eiωt dω =
f (s)
ds.
2π −A
π −∞
t −s
By changing to the variable τ = t − s we finally obtain the important formula
A
1
1 ∞
sin Aτ
F(ω)eiωt dω =
f (t − τ )
dτ.
2π −A
π −∞
τ
(7.5)
This result is important because it shows us why the Fourier integral will converge
to the value f (t) for A → ∞ (assuming for the moment that f is continuous at t).
To that end we take a closer look at the function D A (τ ) = sin Aτ/τ (for A > 0).
The value at τ = 0 is A, while the zeros of D A (τ ) are given by τ = kπ/A with
k ∈ Z (k = 0). For increasing A the zeros are thus getting closer and closer to each
other, which means that the function D A (τ ) has ever increasing oscillations outside
τ = 0. This is shown in figure 7.1. This makes it plausible that at a point t where
the function f is continuous we will have
1 ∞
sin Aτ
dτ = f (t),
f (t − τ )
τ
A→∞ π −∞
lim
(7.6)
under suitable conditions on f . The reason is that for large values of A the strong
oscillations of D A (τ ) outside τ = 0 will cancel everything out (compare this with
the interpretation of the Riemann–Lebesgue lemma), and so only the value of f at
A
DA(τ) = sinτAτ
0
π
A
FIGURE 7.1
The function D A (τ ) for a large value of A.
τ
168
7 The fundamental theorem of the Fourier integral
the point τ = 0 will contribute to the integral (of course, this is not a proof of (7.6)).
The constant π −1 in the left-hand side of (7.6) can easily be 'explained' since (7.3)
∞
implies that −∞ sin Aτ/τ dτ = π for every A > 0 and hence also for A → ∞;
but this is precisely (7.6) if we take for f the constant function 1.
It now becomes clear how we shall proceed with the proof of the fundamental
theorem. First we need to prove (7.6); combining this with (7.5) then immediately
gives the fundamental theorem of the Fourier integral. As a matter of fact, the
assumption that f should be continuous at t was only for our convenience: in the
following theorem we will drop this assumption and so we get a slightly different
version of (7.6).
THEOREM 7.2
Let f (t) be an absolutely integrable and piecewise smooth function on R. Then one
has for each t ∈ R that
1
sin Aτ
1 ∞
dτ = ( f (t+) + f (t−)) .
f (t − τ )
τ
2
A→∞ π −∞
lim
Step 1
Proof
In order to get a well-organized proof, we divide it into two steps.
By splitting the integral in the left-hand side at τ = 0, and changing from τ to
−τ in the resulting integral over (−∞, ∞), we obtain
1 ∞
sin Aτ
dτ
f (t − τ )
τ
A→∞ π −∞
sin Aτ
1 ∞
= lim
dτ.
( f (t − τ ) + f (t + τ ))
τ
A→∞ π 0
lim
(7.7)
Now note that by replacing Aτ by v, and so τ by v/A, and applying (7.3), it follows
that
lim
A→∞ 0
1 sin Aτ
τ
dτ = lim
A→∞ 0
A sin v
v
dv =
π
.
2
If we multiply this equality by ( f (t+) + f (t−)) /π we obtain
1 1 sin Aτ
1
( f (t+) + f (t−)) = lim
( f (t+) + f (t−)) dτ
2
τ
A→∞ π 0
(7.8)
(in fact, f (t+) + f (t−) does not depend on τ ). If we now look at the result to be
proven, it is quite natural to study the difference
1 ∞
1 1
sin Aτ
sin Aτ
dτ −
dτ
( f (t − τ ) + f (t + τ ))
( f (t+) + f (t−))
π 0
τ
π 0
τ
Step 2
of the right-hand sides of (7.7) and (7.8) for A → ∞. If we can show that this
difference tends to 0 for A → ∞, then the theorem is proven. We will do this in
step 2.
If we split the first integral in this difference at τ = 1, then we see that the
difference can be rewritten as I1 + I2 with
1 1 f (t − τ ) − f (t−) + f (t + τ ) − f (t+)
sin Aτ dτ,
π 0
τ
1 ∞ f (t − τ ) + f (t + τ )
I2 =
sin Aτ dτ.
π 1
τ
I1 =
170
7 The fundamental theorem of the Fourier integral
proving theorem 7.3: the Fourier integral converges as a Cauchy principal value to
( f (t+) + f (t−))/2.
The Fourier integral should in the first instance always be considered as a Cauchy
principal value. However, in many cases it is immediately clear that the integral
also exists as an improper Riemann integral, for example when F(ω) is absolutely
integrable. Nevertheless, one should not jump to conclusions too easily, and as a
warning we present a simple example showing that the Fourier integral can certainly
not always be considered as an improper integral.
EXAMPLE 7.1
Let f (t) be the function defined by f (t) = (t)e−t , where (t) is the unit step
function (see example 6.1). This function satisfies all the conditions of theorem 7.3
and F(ω) = 1/(1+iω) (see table 3). For t = 0 it then follows from the fundamental
theorem that
∞
1
−∞ 1 + iω
dω = π,
since ( f (t+) + f (t−))/2 = 1/2 for t = 0. This integral must be considered as
a Cauchy principal value since it does not exist as an improper Riemann integral.
This is because the imaginary part of 1/(1 + iω) is −ω/(1 + ω2 ) and
B
A
1
ω
dω =
ln(1 + B 2 ) − ln(1 + A2 ) ,
2
1 + ω2
which means that the limit does not exist for A → −∞ (or B → ∞).
For a continuous function f (t), the right-hand side of (7.9) equals f (t) since in
this case f (t+) = f (t−) for each t ∈ R. Let us give another example.
EXAMPLE 7.2
Take f (t) = e−a| t | , then F(ω) = 2a/(a 2 + ω2 ) (see table 3). The function f
satisfies all the conditions of the fundamental theorem and it is also continuous (and
even). For each t ∈ R we thus have
e−a| t | =
∞
1
2a
eiωt dω.
2π −∞ a 2 + ω2
(7.10)
Since F(ω) is even, this can also be written as a Fourier cosine transform (see section 6.4.6):
e−a| t | =
Inversion formula
Inverse Fourier transform
EXAMPLE 7.3
2 ∞
a
cos ωt dω.
π 0 a 2 + ω2
Formula (7.9) is called the inversion formula for the Fourier transform on R. The
name 'inversion formula' is clear: if the conditions of theorem 7.3 are met, then one
can recover the original function f (t) from its spectrum F(ω) using (7.9). Therefore, the function f (t) is called the inverse Fourier transform of F(ω).
Here we have proven the fundamental theorem under the condition that f is
piecewise smooth and absolutely integrable. There are many other conditions for
which the theorem remains valid. It is, however, remarkable that up till now there
are no known conditions for absolutely integrable functions that are both necessary
and sufficient (in other words, a minimal condition) for the inversion formula (7.9)
to be valid.
A well-known continuous function which doesn't satisfy the conditions of the fundamental theorem is f (t) = sin t/t. Although this function is piecewise smooth
(and even continuously differentiable), it is not absolutely integrable, as was mentioned in section 6.3.1.
7.1 The fundamental theorem
171
If f (t) is an even function, then F(ω) is also even (see section 6.4.6). Since one
then has for any A > 0 that
A
−A
A
F(ω) cos ωt dω = 2
0
F(ω) cos ωt dω
A
and
−A
F(ω) sin ωt dω = 0,
it follows that
∞
−∞
F(ω)eiωt dω = 2
∞
0
F(ω) cos ωt dω
as a Cauchy principal value. For even functions we thus obtain the following version
of the fundamental theorem:
Fundamental theorem for
even functions
1
2 ∞
Fc (ω) cos ωt dω = ( f (t+) + f (t−)) ,
π 0
2
(7.11)
where
∞
Fc (ω) =
f (t) cos ωt dt
0
is the Fourier cosine transform of f (see section 6.4.6). Note that the integral in
(7.11) is now an ordinary improper integral. When a function f (t) is only defined
for t > 0, then, just as in section 6.4.6, one can extend the function f to an even
function on R which will be denoted by f (t) again; formula (7.11) then holds for
this even extension.
For an odd function g(t) the function G(ω) is odd and one obtains in a similar
fashion a version of the fundamental theorem for odd functions:
Fundamental theorem for odd
functions
2 ∞
1
G s (ω) sin ωt dω = (g(t+) + g(t−)) ,
π 0
2
(7.12)
where
G s (ω) =
∞
0
g(t) sin ωt dt
is the Fourier sine transform of f (see section 6.4.6). When a function is only
defined for t > 0 and we extend it to an odd function on R, then (7.12) will hold for
this odd extension. Of course, the conditions of the fundamental theorem should be
satisfied in all of the preceding statements.
Now that we have done the hard work in proving the fundamental theorem, we
will reap the fruits of it in the next section.
EXERCISES
7.1
Verify for the spectra from the exercises in section 6.3 that the Riemann–Lebesgue
lemma holds.
7.2
Show that the Fourier transform of the block function p2A at the point ω = s − t
equals 2 sin(A(t − s))/(t − s) (see the derivation of (7.5) from (7.4)).
7.3
Show that for arbitrary C > 0 one has
lim
A→∞ 0
C sin Au
u
du =
π
2
(see step 1 in the proof of theorem 7.2).
172
7 The fundamental theorem of the Fourier integral
7.4
Calculate in a direct way (so in contrast to example 7.1) the Cauchy principal value
of
∞
1
dω.
1 + iω
−∞
7.5
a Check that the block function pa (t) satisfies the conditions of the fundamental
theorem and give the inversion formula.
b Use (7.3) to show that the integral in the fundamental theorem of part a exists as
improper integral.
7.6
Let the function f (t) be given by (see exercise 6.9b)
f (t) =
sin t
0
for | t | ≤ π,
elsewhere.
a Check that f (t) satisfies the conditions of the fundamental theorem and that
F(ω) exists as improper integral.
b Prove that
f (t) =
1 ∞ cos(π − t)ω − cos(π + t)ω
dω.
π 0
1 − ω2
7.7
Show that for an odd function the fundamental theorem can be written as in (7.12).
7.8
Consider the function
f (t) =
1
0
for 0 < t ≤ a,
for t > a.
a Use the Fourier sine transform of the function f (t) (see exercise 6.15b) to show
that for t > 0 (t = a) one has
f (t) =
2 ∞ 1 − cos aω
sin ωt dω.
π 0
ω
b To which value does the right-hand side of the identity in part a converge for
t = a?
7.9
The ordinary multiplication of real numbers has a unit, that is to say, there exists a
number e (namely the number 1) such that ex = x for all x ∈ R. Now take as multiplication of functions the convolution product f ∗ g. Use the Riemann–Lebesgue
lemma and the convolution theorem (theorem 6.13) to show that the convolution
product has no unit. In others words, there does not exist a function e such that
e ∗ f = f for all f . (All functions are assumed to be bounded, absolutely integrable
and piecewise smooth.)
7.2
Consequences of the fundamental theorem
In this section we use the fundamental theorem to derive a number of important
additional properties of the Fourier transform.
7.2.1
Uniqueness
From the fundamental theorem we immediately obtain the uniqueness of the Fourier
transform.
7.2 Consequences of the fundamental theorem
THEOREM 7.4
Uniqueness theorem
173
Let f (t) and g(t) be absolutely integrable and piecewise smooth functions on R
with spectra F(ω) and G(ω). If F(ω) = G(ω) on R, then f (t) = g(t) at all points
where f and g are continuous.
Proof
Let t ∈ R be a point where f and g are both continuous. Since F(ω) = G(ω), it
then follows from the fundamental theorem that
f (t) =
∞
∞
1
1
F(ω)eiωt dω =
G(ω)eiωt dω = g(t).
2π −∞
2π −∞
The uniqueness theorem is often applied (implicitly) when addressing the following frequently occurring problem: given F(ω), find a function f (t) with spectrum
F(ω). Let us assume that on the basis of a table, and perhaps in combination with
the properties of the Fourier transform, we know a function f (t) indeed having
F(ω) as its spectrum. (In most cases a direct calculation of the Fourier integral is
not a very clever method.) Theorem 7.4 then guarantees that the function we have
found is the only possibility within the set of absolutely integrable and piecewise
smooth functions. This is up to a finite number of points (on an arbitrary bounded
interval) where we can give the function f an arbitrary (finite) value (see figure 7.2).
This is because if f and g are two functions with the same spectrum, then we can
only conclude that ( f (t+) + f (t−))/2 = (g(t+) + g(t−))/2 at the points where
f and/or g are not continuous. These exceptional points are of little importance;
however, in order to formulate results like the fundamental theorem accurately, one
should keep them in mind. In some of the literature these exceptional points are
avoided by assuming that at a point t ∈ R where a piecewise smooth function is
not continuous, the function value is always defined as ( f (t+) + f (t−))/2. In that
case, theorem 7.4 is thus correct on R. This is the case, for example, for the function
h in figure 7.2.
EXAMPLE
We are looking for a function with spectrum F(ω) = 1/(1 + iω)2 . This function is
the product of 1/(1 + iω) with itself and from table 3 it follows that g(t) = (t)e−t
has as its spectrum precisely 1/(1 + iω). From the convolution theorem (theorem
6.13) it then follows that f (t) = (g ∗ g)(t) has F(ω) as its spectrum. It is easy to
calculate the convolution product (see exercise 6.27) and from this it follows that
f (t) = (t)te−t . According to theorem 7.4 this is the only absolutely integrable
and piecewise smooth function with spectrum F(ω).
The uniqueness theorem gives rise to a new notation. When f (t) is absolutely
integrable and piecewise smooth, and when F(ω) is the spectrum of f (t), then we
will write from now on
f (t) ↔ F(ω).
This expresses the fact that f and F determine each other uniquely according to
theorem 7.4. As before, one should adjust the value of f in the exceptional points,
where f is not continuous, as before. Often the following equivalent formulation of
theorem 7.4 is given (we keep the conditions of the theorem): when F(ω) = 0 on R,
then f (t) = 0 at all points t where f is continuous. This formulation is equivalent
because of the linearity property (see section 6.4.1).
7.2.2
Fourier pairs
In section 6.3 the Fourier transforms were calculated for a number of frequently used
functions. Using the fundamental theorem it will follow that Fourier transforms
174
7 The fundamental theorem of the Fourier integral
f (a+)
f
f (a –)
0
a
t
g
0
1
2
a
t
(h (a+) + h (a –))
h
0
a
t
FIGURE 7.2
Three functions f , g and h with the same spectrum.
usually occur in pairs, which immediately doubles the number of examples. Let
us assume that a function f (t) with spectrum F(ω) satisfies the conditions of the
fundamental theorem. For convenience we also assume that f (t) is continuous, and
according to theorem 7.3 we then have for each t ∈ R that
f (t) =
∞
1
F(ω)eiωt dω.
2π −∞
7.2 Consequences of the fundamental theorem
175
Let us moreover assume that this integral exists as an improper integral (which usually is the case). Both t and ω are just variables for which one may as well choose
another symbol. In particular we may interchange the role of t and ω, and it then
follows that
f (ω) =
∞
1
F(t)eiωt dt.
2π −∞
If we now change from the variable t to the variable −t we obtain
f (ω) =
∞
1
F(−t)e−iωt dt.
2π −∞
(7.13)
But this last integral is precisely the Fourier transform of the function F(−t)/2π .
Summarizing, we have proven the following theorem.
THEOREM 7.5
Let f (t) be an absolutely integrable and piecewise smooth function on R with spectrum F(ω). Assume that f is continuous and that the Fourier integral exists as an
improper Riemann integral. Then
F(−t) ↔ 2π f (ω).
Duality
Reciprocity
EXAMPLE 7.4
Hence, Fourier transforms almost always occur in pairs. This property is called
the duality or the reciprocity of the Fourier transform and is included as property
11 in table 4. Do not confuse 'duality' with the 'selfduality' from section 6.4.7. If
the function f (t) is not continuous, but merely piecewise continuous, then (7.13)
and the duality property will still hold, with the exception of the points where f is
not continuous (there the left-hand side of (7.13) has to be adjusted). A special case
arises when f is an even function. Then F is also an even function, from which it
follows that in this case F(t) ↔ 2π f (ω).
Take f (t) = e−a| t | as in example 7.2. This is an even and continuous function and
F(ω) is absolutely integrable. From theorem 7.5 we then obtain
a
a2 + t 2
↔ π e−a| ω | ,
(7.14)
and so we have found a new Fourier transform.
We called the mapping assigning Fourier transforms to functions the Fourier
transform. For the distribution theory in chapters 8 and 9 it is important to know
the image under the Fourier transform of the set S (S is the set of rapidly decreasing functions; see section 6.5). In theorem 6.12 it was already shown that F(ω) ∈ S
for f (t) ∈ S. In other words, the Fourier transform maps the space S into itself.
Moreover, functions in S certainly satisfy the conditions of the uniqueness theorem;
the Fourier transform is thus a one-to-one mapping in the space S. With the duality
property one can now easily show that the image of S is the whole of S, in other
words, the Fourier transform is also a mapping from S onto S.
THEOREM 7.6
The Fourier transform is a one-to-one mapping from S onto S.
Proof
For a given function f ∈ S we must find a function in the time domain having f as
its spectrum. Now let F(ω) be the spectrum of f (t). By theorem 6.12 the function
F belongs to S and so it is certainly absolutely integrable (see the remark preceding
theorem 6.12). All conditions of theorem 7.5 are thus satisfied and it follows that
the function F(−t)/2π ∈ S has the function f (ω) as its spectrum. We have thus
found the desired function and this proves theorem 7.6.
176
7 The fundamental theorem of the Fourier integral
Sometimes theorem 7.6 is also formulated as follows: for each F(ω) ∈ S there
exists a (unique) f (t) ∈ S such that F(ω) is the spectrum of f (t). We recall that
f (t) is called the inverse Fourier transform of F(ω) (see section 7.1). In particular
we see that the inverse Fourier transform of a function in S belongs to S again (the
'converse' of theorem 6.12). In the next example we apply theorem 7.6 to derive a
well-known result from probability theory.
EXAMPLE 7.5
Application of the reciprocity property to the Gauss function e−at will not result
in a new Fourier transform, since the spectrum of the Gauss function has a similar
form. Still, we will have a closer look at the Gauss function and the Fourier pair
2
π −ω2 /4a
e
a
e−at ↔
2
and
2
2
1
√ e−t /4a ↔ e−aω .
2 πa
(7.15)
From (6.15) it follows that
∞
2
1
√ e−t /4a dt = 1.
πa
−∞ 2
Probability distribution
Functions with integral over R equal to 1, which moreover are positive on R, are
called probability distributions in probability theory. If we now write
2
1
Wa (t) = √ e−t /4a ,
2 πa
Normal distribution
then Wa is a positive function with integral over R equal to 1 and so a probability
distribution. It is called the normal distribution. As an application of our results, we
can now prove a nice and important property of the normal distribution, namely
(Wa ∗ Wb )(t) = Wa+b (t).
(7.16)
(For those familiar with stochastic variables, this result can be reformulated as follows: if X and Y are two independent stochastic variables with normal distributions
Wa and Wb , then X + Y has normal distribution Wa+b .) To prove this result, we
first apply the convolution theorem from section 6.6 and then use (7.15):
F(Wa ∗ Wb )(ω) = (F Wa )(ω)(F Wb )(ω)
2
2
2
= e−aω e−bω = e−(a+b)ω = (F Wa+b )(ω).
Since F(Wa ∗ Wb ) ∈ S, the inverse Fourier transform Wa ∗ Wb will also belong to
S. Hence, both Wa ∗ Wb and Wa+b belong to S and (7.16) then follows from the
uniqueness of the Fourier transform on the space S (theorem 7.6).
7.2.3
Definite integrals
Using the fundamental theorem one can also calculate certain definite integrals. As
an example we take the block function pa (t) with spectrum 2 sin(aω/2)/ω (see
table 3). Applying the fundamental theorem for t = 0, it follows that
∞ sin(aω/2)
ω
−∞
dω = π.
Next we change to the variable t = aω/2 and use that the integrand is an even
function. We then obtain
∞ sin t
0
t
dt =
π
,
2
7.2 Consequences of the fundamental theorem
177
which proves (7.3) again. Note, however, that in proving the fundamental theorem
we have used (7.3), which means that we cannot claim to have found a new proof of
(7.3). For other choices for the function f (t), the fundamental theorem can indeed
provide new results that may be much harder to prove using other means. Let us
give some more examples.
EXAMPLE 7.6
Let qa (t) be the triangle function with spectrum 4 sin2 (aω/2)/(aω2 ) (see table 3 or
section 6.3.2). Applying theorem 7.3 for t = 0 we obtain
∞ 4 sin2 (aω/2)
1
dω = 1.
2π −∞
aω2
Now change to the variable t = aω/2 and use that the integrand is an even function.
We then find
∞ sin2 t
π
dt = .
2
t2
0
EXAMPLE 7.7
(7.17)
Take f (t) = e−a| t | . All preparations can be found in example 7.2: if we take t = 0
and a = 1 in (7.10), and change to the variable t = ω, we obtain
∞
1
−∞ 1 + t 2
dt = π.
(This result can easily be obtained in a direct way since a primitive of the integrand
is the well-known function arctan t.) Now write for a > 0
Pa (t) =
Cauchy distribution
a
1
.
π a2 + t 2
Then Pa is a positive function with integral over R equal to 1 and so Pa is a probability distribution (see example 7.5); it is called the Cauchy distribution.
7.2.4
Convolution in the frequency domain
In section 6.6 the convolution theorem (theorem 6.13) was proven: if f (t) ↔ F(ω)
and g(t) ↔ G(ω), then ( f ∗g)(t) ↔ F(ω)G(ω). The duality property suggests that
a similar result should hold for convolution in the frequency domain. Under certain
conditions on the functions f and g this is indeed the case. Since these conditions
will return more often, it will be convenient to introduce a new class of functions.
DEFINITION 7.2
Square integrable
∞
A function f : R → C is called square integrable on R if −∞ | f (t) |2 dt exists as
improper Riemann integral.
If a function f is absolutely integrable, then it need not be square integrable. For
example, the function f (t) = p2 (t) | t |−1/2 is absolutely integrable, but not square
integrable since | t |−1 is not integrable over the interval −1 ≤ t ≤ 1. One now has
the following convolution theorem in the frequency domain.
THEOREM 7.7
Convolution theorem in the
frequency domain
Let f (t) and g(t) be piecewise smooth functions which, in addition, are absolutely
integrable and square integrable on R. Then
f (t)g(t) ↔ (2π)−1 (F ∗ G)(ω).
Proof
We give the proof under the simplifying assumption that the spectra F(ω) and G(ω)
exist as improper Riemann integrals. Since (a + b)2 = a 2 + 2ab + b2 ≥ 0, we
178
7 The fundamental theorem of the Fourier integral
see that | f (t)g(t) | ≤ 1 (| f (t) |2 + | g(t) |2 ) for all t ∈ R. Since f and g are
2
square integrable, it now follows that f (t)g(t) is absolutely integrable. Hence, the
spectrum of f (t)g(t) will certainly exist. According to the fundamental theorem we
may replace f (t) in
F ( f (t)g(t))(ω) =
∞
−∞
f (t)g(t)e−iωt dt
by
∞
1
F(τ )eiτ t dτ,
2π −∞
where we may consider this integral as an improper Riemann integral (the exceptional points, where f (t) is not equal to this integral, are not relevant since we
integrate over the variable t). It then follows that
∞
∞
1
F(τ )g(t)e−i(ω−τ )t dτ dt
2π −∞ −∞
∞
∞
1
g(t)e−i(ω−τ )t dt F(τ ) dτ,
=
2π −∞ −∞
F ( f (t)g(t))(ω) =
where we have changed the order of integration. We state without proof that this
is allowed under the conditions of theorem 7.7. We recognize the inner integral as
G(ω − τ ). This proves theorem 7.7, since it follows that
F( f (t)g(t))(ω) =
∞
1
1
(F ∗ G)(ω).
F(τ )G(ω − τ ) dτ =
2π −∞
2π
For complicated functions one can now still obtain the spectrum, in the form of a
convolution product, by using theorem 7.7. In some cases this convolution product
can then be calculated explicitly.
EXAMPLE
Suppose we need to find the spectrum of the function (1 + t 2 )−2 . In example 7.4 we
showed that 1/(1+t 2 ) ↔ πe−| ω | and since all conditions of theorem 7.7 are met, it
follows that (1+t 2 )−2 ↔ (π/2)(e−| τ | ∗e−| τ | )(ω). By calculating the convolution
product (see exercise 6.26a), it then follows that (1+t 2 )−2 ↔ (π/2)(1+| ω |)e−| ω | .
7.2.5
Parseval's identities
In theorem 7.7 it was shown that
∞
−∞
f (t)g(t)e−iωt dt =
∞
1
F(τ )G(ω − τ ) dτ
2π −∞
and at the point ω = 0 this gives the identity
∞
−∞
f (t)g(t) dt =
∞
1
F(τ )G(−τ ) dτ.
2π −∞
Now replace g(t) by g(t) and use that g(t) ↔ G(−ω) (see table 4), then it follows
that
∞
−∞
f (t)g(t) dt =
∞
1
F(ω)G(ω) dω.
2π −∞
(7.18)
7.2 Consequences of the fundamental theorem
Parseval's identity
179
This result is known as the theorem of Parseval or as Parseval's identity and it thus
holds under the same conditions as theorem 7.7 (taking the complex conjugate has
no influence on the conditions). Since zz = | z |2 for z ∈ C, formula (7.18) reduces
for f (t) = g(t) to
∞
−∞
Plancherel's identity
Energy-signal
Energy-content
| f (t) |2 dt =
∞
1
| F(ω) |2 dω.
2π −∞
(7.19)
In order to distinguish between formulas (7.18) and (7.19) one often calls (7.19)
Plancherel's identity. We shall not make this distinction: both identities will be
called Parseval's identity. (Compare (7.18) and (7.19) with Parseval's identities for
Fourier series.) Note that in (7.19) it is quite natural to require f (t) to be square
integrable.
Identity (7.19) shows that square integrable functions have a Fourier transform
that is again square integrable. In signal theory a square integrable function is
also called a signal with finite energy-content or energy-signal for short. The value
∞
2
−∞ | f (t) | dt is then called the energy-content of the signal f (t) (also see chapter 1). Identity (7.19) shows that the spectrum of an energy-signal has again a finite
energy-content.
Parseval's identities can also be used to calculate certain definite integrals.
Again we consider the block function pa (t) from section 6.3.1 having spectrum
2 sin(aω/2)/ω. The function pa satisfies the conditions of theorem 7.7 and from
(7.19) it then follows that
EXAMPLE
∞
−∞
| pa (t) |2 dt =
∞ 4 sin2 (aω/2)
1
dω.
2π −∞
ω2
The left-hand side is easy to calculate and equals a. Changing to the variable t =
aω/2 in the right-hand side, we again obtain the result (7.17) from example 7.6.
In (7.18) we take f (t) = pa (t) and g(t) = pb (t) with 0 ≤ a ≤ b. It then follows
that
EXAMPLE
∞
−∞
pa (t) pb (t) dt =
∞ 4 sin(aω/2) sin(bω/2)
1
dω.
2π −∞
ω2
The left-hand side equals a (since a ≤ b), while the integrand in the right-hand side
is an even function. Hence,
∞ sin(aω/2) sin(bω/2)
π
dω = a.
4
ω2
0
Now change to the variable t = aω/2, then it follows for any c (= b/a) ≥ 1 that
∞ sin t sin ct
π
dt = .
2
t2
0
The previous example is the case a = b = 1.
EXERCISES
7.10
Show that theorem 7.4 is equivalent to the following statement: if F(ω) = 0 on R,
then f (t) = 0 at all points t where f is continuous.
7.11
Use the duality property to determine the Fourier transform of the function
sin(at/2)/t. Also see exercise 7.5, where we already verified the conditions.
7.12
Use the duality property to determine the Fourier transform of the function
sin2 (at/2)/t 2 .
180
7 The fundamental theorem of the Fourier integral
7.13
Use the convolution theorem in the frequency domain to determine the spectrum of
the function sin2 (at/2)/t 2 (see section 6.6 for pa ∗ pa ). Check your answer using
the result from exercise 7.12 and note that although sin(at/2)/t is not absolutely
integrable, theorem 7.7 still produces the correct result.
7.14
According to table 3 one has for a > 0 that (t)e−at ↔ 1/(a + iω). Can we now
use duality to conclude that 1/(a + it) ↔ 2π (−ω)eaω ?
7.15
Verify that the duality property applied to the relation
e−at ↔
2
π −ω2 /4a
e
a
leads to the result
2
2
1
√ e−t /4a ↔ e−aω .
2 πa
Then show that the second relation is a direct consequence of the first relation by
changing from a to 1/(4a) in the first relation. Hence, in this case we do not find a
new Fourier transform.
7.16
Determine the Fourier transform of the following functions:
a f (t) = 1/(t 2 − 2t + 2),
b f (t) = sin 2π(t − 3)/(t − 3),
c f (t) = sin 4t/(t 2 − 4t + 7),
d f (t) = sin2 3(t − 1)/(t 2 − 2t + 1).
7.17
Determine the function f (t) having the following function F(ω) as its spectrum:
a F(ω) = 1/(ω2 + 4),
b F(ω) = p2a (ω − ω0 ) + p2a (ω + ω0 ) for a > 0,
c F(ω) = e−3| ω−9 | .
7.18
Let f and g be two functions in S. Use the convolution theorem (theorem 6.13) and
theorem 7.6 to show that f ∗ g belongs to S.
7.19
Let Pa (t) be as in example 7.7. Show that Pa ∗ Pb = Pa+b . Here one may use that
Pa ∗ Pb is continuously differentiable (Pa is not an element of S; see exercise 6.20).
7.20
Consider the Fourier transform on the space S. Use the reciprocity property to show
that F 4 is the identity on S (up to a constant): F 4 ( f (t)) = 4π 2 f (t) for any f ∈ S.
7.21
For t > 0 we define the function g(t) by g(t) = e−t . Extend this function to an odd
function on R. The spectrum of g has been determined in exercise 6.14.
a Use the fundamental theorem to show that for t > 0 one has
∞ x sin xt
π
d x = e−t .
2
1 + x2
0
Why is this identity not correct for t = 0?
b Next use Parseval's identity to show that
∞
0
7.22
Use the function e−a| t | to show that
∞
0
7.23
x2
π
dx = .
4
(1 + x 2 )2
π
1
dx =
.
2ab(a + b)
(a 2 + x 2 )(b2 + x 2 )
a Determine the spectrum of the function sin4 t/t 4 using the convolution theorem
in the frequency domain.
b Calculate
∞ sin4 x
d x.
−∞ x 4
7.3 Poisson's summation formula∗
7.24
181
Find the function f (t) with spectrum 1/(1 + ω2 )2 and use this to give a new proof
of the identity from exercise 7.22 for the case a = b = 1.
7.3
Poisson's summation formula∗
The material in this section will only be used to determine the Fourier transform
of the so-called comb distribution in section 9.1.3 and also to prove the sampling
theorem in chapter 15. Sections 7.3 and 9.1.3 and the proof of the sampling theorem
can be omitted without any consequences for the remainder of the material.
With the conclusion of section 7.2 one could state that we have finished the theory of the Fourier integral for non-periodic functions. In the next two chapters we
extend the Fourier analysis to objects which are no longer functions, but so-called
distributions. Before we start with distribution theory, the present section will first
examine Poisson's summation formula. It provides an elegant connection between
the Fourier series and the Fourier integral. Moreover, we will use Poisson's summation formula in chapter 9 to determine the Fourier transform of the so-called comb
distribution, and in chapter 15 to prove the sampling theorem. We note, by the way,
that in the proof of Poisson's summation formula we will not use the fundamental
theorem of the Fourier integral.
In order to make a connection between the Fourier series and the Fourier integral,
we will try to associate a periodic function with period T with an absolutely integrable function f (t). We will do this in two separate ways. First of all we define the
periodic function f p (t) in the following obvious way:
f p (t) =
∞
f (t + nT ).
(7.20)
n=−∞
Replacing t by t + T in (7.20), it follows from a renumbering of the sum that
f p (t + T ) =
∞
f (t + (n + 1)T ) =
n=−∞
∞
f (t + nT ) = f p (t).
n=−∞
Hence, the function f p (t) is indeed periodic with period T . There is, however,
yet another way to associate a periodic function with f (t). First take the Fourier
transform F(ω) of f (t) and form a sort of Fourier series associated with f (note
again that f is non-periodic):
1 ∞
F(2πn/T )e2πint/T .
T n=−∞
(7.21)
(We will see that this is in fact the Fourier series of f p (t).) If we replace t by
t + T , then (7.21) remains unchanged and (7.21) is thus, as a function of t, also periodic with period T . (We have taken F(2πn/T )/T instead of F(n) since a similar
connection between Fourier coefficients and the Fourier integral has already been
derived in (6.9).) Poisson's summation formula now states that the two methods to
obtain a periodic function from a non-periodic function f (t) lead to the same result.
Of course we have to require that the resulting series converge, preferably absolutely. In order to give a correct statement of the theorem, we also need to impose
some extra conditions on the function f (t).
THEOREM 7.8
Poisson's summation formula
Let f (t) be an absolutely integrable and continuous function on R with spectrum
F(ω). Let T > 0 be a constant. Assume furthermore that there exist constants
p > 1, A > 0 and M > 0 such that | f (t) | < M | t |− p for | t | > A. Also assume
182
7 The fundamental theorem of the Fourier integral
∞
n=−∞ | F(2πn/T ) | converges. Then
that
∞
f (t + nT ) =
n=−∞
1 ∞
F(2π n/T )e2πint/T
T n=−∞
(7.22)
(with absolutely convergent series). In particular
∞
f (nT ) =
n=−∞
1 ∞
F(2π n/T ).
T n=−∞
(7.23)
Proof
Define f p (t) as in (7.20). Without proof we mention that, with the conditions on the
function f (t), the function f p (t) exists for every t ∈ R and that it is a continuous
function. Furthermore, we have already seen that f p (t) is a periodic function with
period T . The proof now consists of the determination of the Fourier series of f p (t)
and subsequently applying some of the results from the theory of Fourier series.
For the nth Fourier coefficient cn of f p (t) one has
cn =
1 T
1 T ∞
f p (t)e−2πint/T dt =
f (t + kT )e−2πint/T dt.
T 0
T 0 k=−∞
Integrating term-by-term we obtain
cn =
T
1 ∞
f (t + kT )e−2πint/T dt.
T k=−∞ 0
From the conditions on the function f (t) it follows that this termwise integration is
allowed, but again this will not be proven. Changing to the variable τ = t + kT in
the integral, it then follows that
cn =
(k+1)T
1 ∞
f (τ )e−2πinτ/T e2πink dτ.
T k=−∞ kT
For k, n ∈ Z one has e2πink = 1. Furthermore, the intervals [kT, (k+1)T ] precisely
fill up all of R when k runs through the set Z, and so
cn =
1 ∞
1
f (τ )e−2πinτ/T dτ = F(2πn/T ).
T −∞
T
(7.24)
This determines the Fourier coefficients cn of f p (t) and because of the assumption
on the convergence of the series ∞
n=−∞ | F(2πn/T ) | we now have that
∞
| cn |
converges.
n=−∞
Since cn e2πint/T = | cn |, it then also follows that the Fourier series of f p (t)
converges absolutely (see theorem 4.5). For the moment we call the sum of this
series g(t), then g(t) is a continuous function with Fourier coefficients cn . The
two continuous functions f p (t) and g(t) thus have the same Fourier coefficients and
according to the uniqueness theorem 4.4 it then follows that f p (t) = g(t). Hence,
∞
n=−∞
f (t + nT ) = f p (t) = g(t) =
1 ∞
F(2π n/T )e2πint/T ,
T n=−∞
7.3 Poisson's summation formula∗
183
which proves (7.22). For t = 0 we obtain (7.23).
In the proof of theorem 7.8 a number of results were used without proof. All
of these results rely on a property of series – the so-called uniform convergence –
which is not assumed as a prerequisite in this book. The reader familiar with the
properties of uniform convergent series can find a more elaborate proof of Poisson's
summation formula in, for example, The theory of Fourier series and integral by
P.L. Walker, Theorem 5.30.
We will call both (7.22) and (7.23) Poisson's summation formula. From the proof
we see that the right-hand side of (7.22) is the Fourier series of f p (t), that is, of the
function f (t) made periodic according to (7.20). The occurring Fourier coefficients
are obtained from the spectrum F(ω) using (7.24). In this manner we have linked
the Fourier series to the Fourier integral. It is even possible to derive the fundamental
theorem of the Fourier integral from the fundamental theorem of Fourier series using
Poisson's summation formula. This gives a new proof of the fundamental theorem
of the Fourier integral. We will not go into this any further. In conclusion we present
the following two examples.
EXAMPLE 7.8
Take f (t) = a/(a 2 + t 2 ) with a > 0, then F(ω) = πe−a| ω | (see table 3). We want
to apply (7.23) with T = 1 and so we have to check the conditions. The assumption
about the convergence of the series ∞
n=−∞ | F(2πn) | is easy since
∞
e−2πa| n | = 1 + 2
n=−∞
∞
e−2πan ,
n=1
which is a geometric series with ratio r = e−2πa . Since a > 0 it follows that
| r | < 1, and so the geometric series converges (see section 2.4.1). In this case we
even know the sum:
∞
e−2πa| n | = 1 + 2
n=−∞
e−2πa
1 + e−2πa
=
.
−2πa
1−e
1 − e−2πa
The condition on the function f (t) is also easy to verify: for t = 0 one has | f (t) | ≤
a/t 2 and so the condition in theorem 7.8 is met if we take p = 2, M = a and A > 0
arbitrary. Poisson's summation formula can thus be applied. The right-hand side of
(7.23) has just been calculated and hence we obtain
∞
π 1 + e−2πa
1
=
.
2
2
a 1 − e−2πa
n=−∞ a + n
(7.25)
By rewriting (7.25) somewhat, it then follows for any a > 0 that
∞
π 1 + e−2πa
1
1
=
− 2.
2 + n2
2a 1 − e−2πa
a
2a
n=1
Now take the limit a ↓ 0, then the left-hand side tends to
calculation will show that
1
π 1 + e−2πa
− 2
a↓0 2a 1 − e−2πa
2a
lim
∞
2
n=1 1/n , while a little
π2
πa(1 + e−2πa ) − (1 − e−2πa )
=
2 (1 − e−2πa )
6
a↓0
2a
= lim
(apply, for example, De l'Hˆ pital's rule three times). This gives a new proof of the
o
famous identity (also see exercise 4.8)
∞
1
π2
=
.
2
6
n
n=1
CHAPTER 8
Distributions
INTRODUCTION
Many new concepts and theories in mathematics arise from the fact that one is confronted with problems that existing theories cannot solve. These problems may
originate from mathematics itself, but often they arise elsewhere, such as in physics.
Especially fundamental problems, sometimes remaining unsolved for years, decades
or even centuries, have a very stimulating effect on the development of mathematics (and science in general). The Greeks, for example, tried to find a construction
of a square having the same area as the unit circle. This problem is known as the
'quadrature of the circle' and remained unsolved for some two thousand years. Not
until 1882 it was found that such a construction was impossible, and it was discovered that the area of the unit circle, hence the number π, was indeed a very special
real number.
Many of the concepts which one day solved a very fundamental problem are now
considered obvious. Even the concept of 'function' has one day been heavily debated, in particular relating to questions on the convergence of Fourier series. Problems arising in the context of the solutions of quadratic and cubic equations were
solved by introducing the now so familiar complex numbers. As is well-known, the
complex numbers form an extension of the set of real numbers.
In this chapter we will introduce new objects, the so-called 'distributions', which
form an extension of the concept of function. For twenty years, these distributions
were used successfully in physics, prior to the development, in 1946, of a mathematical theory which could handle these problematic objects. It will turn out that
these distributions are an important tool, just as the complex numbers. They are
indispensable when describing, for example, linear systems in chapter 10.
In section 8.1 we will show how distributions arise in the Fourier analysis of
non-periodic functions. We will first concentrate on the so-called delta function – a
misleading term by the way, since it is by no means a function. In section 8.2 we
then present a mathematically rigorous introduction of distributions, and we treat our
first important examples. We will show, among other things, that most functions can
be considered as distributions; hence, distributions form an extension of functions
(although not every function can be considered as a distribution).
It is remarkable that distributions can always be differentiated, as will be established in section 8.3. In this way, one can obtain new distributions by differentiation.
In particular one can start with an ordinary function, consider it as a distribution, and
then differentiate it (as a distribution). In this manner one can obtain distributions
from ordinary functions, which themselves can then no longer be considered as
functions. For example, the delta function mentioned above arises as the derivative
of the unit step function. In the final section of this chapter two more properties
will be developed, which will be useful later on: multiplication and scaling. Fourier
analysis will not return until chapter 9.
188
8.1 The problem of the delta function
189
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know and can apply the definition of distributions
- know how to consider functions as distributions
- know the definition of a number of specific distributions: the delta function, the
principal value 1/t
- know how to differentiate distributions
- can add distributions and multiply them by a constant
- can multiply distributions by polynomials and some more general functions
- can scale distributions and know the concept of time reversal for distributions
- know the concepts even and odd distributions.
8.1
The problem of the delta function
Without any doubt, the most famous distribution is the 'delta function'. Although
the name suggests otherwise, this object is not a function. This is because, as we
shall see in a moment, a function cannot have the prescribed properties of the delta
function. A precise definition of the delta function will be given in section 8.2. First
we will show that the notion of the delta function arises naturally in the Fourier
analysis of non-periodic functions.
In section 6.2 we already noted that the constant function f (t) = 1 has no Fourier
transform. However, a good approximation of f is the block function p2a (t) for
very large values of a and in fact we would like to take the limit a → ∞. Since the
spectrum of p2a (t) is the function 2 sin aω/ω, we will be inclined to believe that
from this we should get the spectrum of the function f (t) = 1 as a → ∞. But what
precisely is lima→∞ 2 sin aω/ω? When this is considered as a pointwise limit, that
is, for each ω ∈ R fixed, then there is no value of ω such that the limit exists, since
limx→∞ sin x does not exist. If we want to obtain a meaningful result, we need to
attach a different meaning to the limit lima→∞ 2 sin aω/ω. Now in theorem 7.2 we
have shown that for an absolutely integrable and piecewise smooth function one has
1 ∞ sin aτ
f (t − τ ) dτ = f (t),
a→∞ π −∞
τ
lim
(8.1)
where we assume for convenience that f is continuous at t. By substituting t = 0
and changing from the variable τ to −ω, we obtain that
lim
1
a→∞ π
∞ sin aω
−∞
ω
f (ω) dω = f (0)
(8.2)
when f is continuous at t = 0. This enables us to give a new interpretation for
lima→∞ 2 sin aω/ω: for any absolutely integrable and continuously differentiable
function f (ω) formula (8.2) is valid. Only within this context will the limit have a
meaning. There is no point in asking whether we may interchange the limit and the
integral in (8.2). We have come up with this new interpretation precisely because
the original limit lima→∞ 2 sin aω/ω had no meaning. Still, one often defines the
symbol δ(ω) by
1
2 sin aω
lim
,
(8.3)
2π a→∞
ω
and then the limit and the integral are interchanged in (8.2). For the new object δ(ω)
one then obtains
δ(ω) =
∞
−∞
δ(ω) f (ω) dω = f (0).
(8.4)
190
8 Distributions
Formulas (8.3) and (8.4) should not be taken literally; the limit in (8.3) does not
exist and (8.4) should only be considered as a symbolic way of writing (8.2). Yet, in
section 8.2 it will turn out that the object δ(ω) can be given a rigorous mathematical
meaning that is very close to (8.4). And although δ(ω) is called the delta function,
it will no longer be a function, but a so-called 'distribution'. The general theory of
distributions, of which the delta function is one the most important examples, will
be treated in a mathematically correct way in the next section.
We recall that studying the limit lima→∞ 2 sin aω/ω was motivated by the search
for the spectrum of the constant function f (t) = 1. Since we can write 2πδ(ω) =
lima→∞ 2 sin aω/ω, taking (8.3) as starting point, it is now plausible that 2πδ(ω)
will be the spectrum of the constant function f (t) = 1.
Conversely, it can be made plausible in a similar way that the spectrum of the
delta function δ(t) is the constant function F(ω) = 1. To do so, we will take a closer
look at (8.4) (which, by the way, is often taken as the defining property in much
of the engineering literature). For example, (8.4) should be valid for absolutely
integrable and piecewise smooth functions which, moreover, are continuous at t =
0. The function
f (t) =
1
0
for a < t < b,
elsewhere,
satisfies all these conditions as long as a = 0 and b = 0. From (8.4) it then follows
b
that a δ(ω) dω = f (0) for all a, b ∈ R with a = 0, b = 0 and a < b. If we now
b
take a < 0 and b > 0, then f (0) = 1 and so a δ(ω) dω = 1. This suggests that
the integral of the delta function over R equals 1, that is,
∞
−∞
δ(ω) dω = 1.
(8.5)
If, on the other hand, we take a < b < 0 or 0 < a < b, then f (0) = 0 and so
b
a
δ(ω) dω = 0
for all a, b ∈ R with a < b < 0 or 0 < a < b.
(8.6)
A function satisfying both (8.5) and (8.6) must have a very extraordinary behaviour!
(Based on (8.5) and (8.6) one can, for instance, conclude that there is not a single
point in R where δ(ω) is continuous.) Sometimes this is solved by describing the
delta function as a function being 0 everywhere, except at the point ω = 0 (in order
for (8.6) to hold) and in addition having integral over R equal to 1 (in order for (8.5)
to hold). However, such a function satisfying (8.5) and (8.6) cannot exist since an
integral does not change its value if the value of the integrand is changed at one
point. Hence, the value of δ(ω) at the point ω = 0 is not relevant for the integral as
a whole, which means that the integral will be 0 since δ(ω) = 0 outside the point
ω = 0.
The above description of the delta function still has some useful interpretations.
Let us consider the block function a −1 pa (t) of height a −1 and duration a for ever
decreasing a (see figure 8.1). (For small values of a we can interpret a −1 pa (t)
physically as an impulse: a big force applied during a short time.) For a ↓ 0 we
obtain an object equal to 0 everywhere except at the point t = 0 where the limit
will be ∞; moreover, the integral over R of a −1 pa (t) will equal 1 for all a > 0,
and so in the limit a ↓ 0 the integral over R will equal 1 as well. We thus obtain
an object satisfying precisely the description of the delta function given above. It is
then plausible that
∞
lim
a↓0 −∞
a −1 pa (t) f (t) dt = f (0),
(8.7)
8.1 The problem of the delta function
191
p 1/4(t )
4
–
p 1/2(t )
2
p 1(t )
1
1
2
p 2(t )
1
4
p4(t )
–2
–1
–
1
2
–1
4
1
4
1
2
1
2
t
FIGURE 8.1
Block functions approximating the delta function.
and looking at (8.4) we are thus led to believe that
lim a −1 pa (t) = δ(t).
a↓0
(8.8)
As a matter of fact, one can prove that (8.7) is indeed correct, under certain conditions on the function f (t). Formulas (8.7) and (8.8) now present us with a situation which is entirely analogous to the situation in (8.2) and (8.3). We can use
this to make it plausible that the spectrum of the delta function δ(t) is the constant function F(ω) = 1. This is because the spectrum of a −1 pa (t) is the function
(2 sin aω/2)/aω and for arbitrary ω ∈ R one has lima↓0 sin aω/aω = 1. We thus
indeed find that the spectrum of δ(t) = lima↓0 a −1 pa (t) will equal the constant
function F(ω) = 1 (this also follows if we interchange limit and integral in (8.7)
and take e−iωt for the function f (t)). Also note that the duality or reciprocity property of the Fourier transform seems to hold for the delta function as well: δ(t) ↔ 1
and 1 ↔ 2πδ(ω).
Of course, all the conclusions in this section rest upon intuitive derivations. In the
next section a mathematically rigorous definition of distributions, and in particular
of the delta function, will be given. In chapter 9 all of the results on the Fourier
analysis of the delta function described above will be proven and the more general
theory of the Fourier transform of distributions will be treated.
192
8 Distributions
EXERCISE
8.1
There are many ways to obtain the delta function as a symbolic limit of functions.
We have already seen two of these limits (in (8.3) and (8.8)). Now consider the
function Pa (t) from example 7.7:
Pa (t) =
a
.
π(a 2 + t 2 )
a Sketch π Pa (t) for a = 1 and a = 1/4.
b Show that lima↓0 Pa (t) = 0 for t = 0, while the limit equals ∞ for t = 0.
Since the integral of Pa (t) over R is 1 (see example 7.7), it is plausible that δ(t) =
lima↓0 Pa (t).
c Determine the limit of the spectrum of Pa (t) for a ↓ 0 and conclude that it is
plausible that the constant function 1 is the spectrum of δ(t).
8.2
Definition and examples of distributions
8.2.1
Definition of distributions
Loosely speaking, a complex-valued function f is a prescription assigning a value
f (t) ∈ C to every t ∈ R. Complex-valued functions are thus mappings from R
to C. In section 8.1 we have seen that the expression lima→∞ 2 sin aω/ω only
makes sense in the context of the integral in (8.2). In fact, (8.2) assigns the value
2π f (0) to every absolutely integrable and, say, continuously differentiable function f (ω). Here we have discovered an important new principle: the expression
lima→∞ 2 sin aω/ω (written symbolically as 2πδ(ω)) can be considered as a mapping assigning to every absolutely integrable and continuously differentiable function f (ω) a certain complex number (namely 2π f (0)). This new principle will be
used to give a mathematically rigorous definition of distributions.
Keeping in mind the Fourier analysis of distributions, it turns out that it is not
very convenient to work with continuously differentiable functions. In order to get
a nice theory it is necessary to use a set of functions which is mapped into itself
by the Fourier transform. We already know such a set of functions, namely the set
S of rapidly decreasing functions (see sections 6.5 and 7.2.2). Now distributions
will be mappings assigning a complex number to every f ∈ S. The choice of S is
determined by its usefulness in Fourier analysis. It is quite possible to define certain
distributions as mappings from other sets of functions to C, for example from the set
of all continuous functions, or the set of all continuously differentiable functions, to
C. However, we will mainly confine ourselves to mappings from S to C.
One additional condition is imposed on these mappings: linearity. We will illustrate this using our example from section 8.1. If we replace f in (8.2) by c f , where
c is an arbitrary complex constant, then c can be taken outside the integral as well
as outside the limit. Hence, we assign to the function c · f the complex number
c · 2π f (0). So, if we multiply f by c, then the complex number 2π f (0) assigned to
f is also multiplied by c. Next we replace f in (8.2) by a sum g +h of two functions
g and h. Then
lim
∞ 2 sin aω
a→∞ −∞
= lim
ω
(g + h)(ω) dω
∞ 2 sin aω
ω
= 2πg(0) + 2π h(0).
a→∞ −∞
g(ω) dω + lim
∞ 2 sin aω
a→∞ −∞
ω
h(ω) dω
8.2 Definition and examples of distributions
193
Hence, we assign to the sum g + h the sum 2π(g + h)(0) of the complex numbers
2πg(0) and 2π h(0). Together, these two properties show that the mapping is linear (see also chapter 1 on linear systems). This finally brings us to the following
definition of the concept of distribution.
DEFINITION 8.1
Distribution
A distribution T is a linear mapping assigning a complex number to every rapidly
decreasing function φ.
We denote the image of a φ ∈ S under the mapping T by T, φ ; note that
T, φ ∈ C. A distribution is thus a mapping T : S → C satisfying
T, cφ = c T, φ ,
T, φ1 + φ2 = T, φ1 + T, φ2 ,
where φ, φ1 and φ2 are elements of S and c ∈ C.
One uses the notation T, φ to prevent confusion with functions. For the same
reason it is customary in distribution theory to denote elements in S with the Greek
symbols φ, ψ, etc. In section 8.2.3 we will see that many functions can be considered as distributions; it would then be very confusing to use the symbols f , g, etc.
for elements in S as well. Although a distribution T is a linear mapping on S, we
will nevertheless often write T (t) to express the fact that T acts on functions that
depend on the variable t.
8.2.2
The delta function
Of course, our first example of a distribution should be the delta function δ(t). In
section 8.1 we have argued that (8.4) is the crucial 'property' of δ(t): to a function
φ(t) the value φ(0) is assigned. This will be taken as the definition of the delta
function.
DEFINITION 8.2
Delta function
The delta function δ(t) (or δ for short) is the distribution defined by
δ(t), φ = φ(0)
for φ ∈ S.
Let us verify that δ is indeed a distribution. It is clear that δ is a mapping from S
to C since φ(0) ∈ C. One also has
δ, cφ = (cφ)(0) = c · φ(0) = c δ, φ
and
δ, φ1 + φ2 = (φ1 + φ2 )(0) = φ1 (0) + φ2 (0) = δ, φ1 + δ, φ2 ,
which proves the linearity of δ. The delta function is thus indeed a distribution. In
many books δ, φ = φ(0) is written as
∞
−∞
Sifting property
δ(t)φ(t) dt = φ(0),
(8.9)
just as we have done in (8.4) in connection with (8.2). Relation (8.9) is often called
the sifting property of the delta function; the value of φ at the point 0 is 'sifted
out' by δ. The graphical representation of the delta function in figure 8.2 is also
based on this property: we draw an arrow at the point 0 of height 1. If we agree
that the integral in (8.9) is a symbolic representation, then there is no objection.
Of course, one cannot prove results about the delta function by applying properties
from integral calculus to this integral. This is because it is only a symbolic way
of writing. However, calculating in an informal way with the integral (8.9) may
provide conjectures about possible results for the delta function. As an example one
194
Dirac delta function
8 Distributions
can get an impression of the derivative of the delta function by using integration by
parts (see example 8.8).
The delta function was introduced explicitly for the first time by the English
physicist P.A.M. Dirac in 1926. He was certainly not the first, nor the only one,
to have some notion of a delta function. The classical result (8.2) for example, is
already very close to how the delta function operates. Dirac was the first, however,
to give an explicit meaning to the delta function and to introduce a separate notation
for it. For this reason the delta function is often called the Dirac function or Dirac
delta function. In the years following the introduction of the delta function, its use
produced many results, which in physical practice turned out to be correct. Not until
1946 was a rigorous distribution theory developed by the French mathematician
L. Schwartz.
A slightly more general delta function is obtained by assigning to a function
φ ∈ S not the value φ(0), but the value φ(a), where a ∈ R is fixed. This distribution
is denoted by δ(t − a), so
δ(t − a), φ = φ(a).
Delta function at a
(8.10)
We will call δ(t − a) the delta function at the point a. If we choose 0 for the point
a, then we simply call this the delta function. Symbolically (8.10) is sometimes
written as
∞
−∞
δ(t − a)φ(t) dt = φ(a).
(8.11)
We represent the distribution δ(t − a) graphically by an arrow of height 1 at the
point a. See figure 8.2.
1
δ(t )
δ(t – a )
a
0
t
FIGURE 8.2
The delta function at 0 and at a.
At the start of this section it was noted that in order to define a distribution it is
not always necessary to confine ourselves to the space S. The definition of δ(t − a)
for example, and in particular of δ, is also meaningful for any continuous function
φ. Hence, definition 8.2 and (8.10) are often given for all continuous functions φ.
8.2.3
Generalized functions
Function as distribution
Functions as distributions
Distributions are often called generalized functions because they form an extension
of the concept of 'function'. Just as any real number can be considered as a complex
number, a function apparently can be considered as a distribution. This comparison
with R as a subset of C is not entirely correct, since not all functions can be considered as distributions. If, however, the function f (t) is absolutely integrable, then it
8.2 Definition and examples of distributions
195
can certainly be considered as a distribution T f by defining
Tf ,φ =
∞
−∞
f (t)φ(t) dt
for φ ∈ S.
(8.12)
First, it has to be shown that the integral in (8.12) exists. But for φ ∈ S one has in
particular that φ(t) is bounded on R, say | φ(t) | ≤ M. It then follows that
∞
−∞
f (t)φ(t) dt ≤
∞
−∞
| f (t)φ(t) | dt ≤ M
∞
−∞
| f (t) | dt,
and since f is absolutely integrable, the integral exists. Next we have to show that
T f is indeed a distribution. For each φ ∈ S the integral in (8.12) gives a complex
number. Hence, T f is a mapping from S to C. The linearity of T f follows immediately from the linearity of integration (see also, for example, section 6.4.1) and so
T f is indeed a distribution. In this way one can consider any absolutely integrable
function f as a distribution. But now a problem arises. How do we know for sure
that two different functions f and g also lead to two different distributions T f and
Tg ? This ought to be true if we consider distributions as an extension of functions.
(Two real numbers that are unequal will also be unequal when considered as complex numbers.) But what do we actually mean by 'unequal' or 'equal' distributions?
DEFINITION 8.3
Equality of distributions
Two distributions T1 and T2 are called equal if T1 , φ = T2 , φ for all φ ∈ S. In
this case we write T1 = T2 .
We now mention without proof that T f = Tg implies that f = g at the points
where both f and g are continuous. When no confusion is possible, the distribution
T f is simply denoted by f . It is then customary to use the phrase ' f as distribution'.
We close this section with some examples of distributions that will often return.
Among other things, these examples will show that many functions which are not
absolutely integrable still define a distribution T f through (8.12).
EXAMPLE 8.1
The function f (t) = 1
The constant function f (t) = 1 is not absolutely integrable over R. Still it defines
precisely as in (8.12) a distribution, simply denoted by 1:
1, φ =
∞
−∞
φ(t) dt
for φ ∈ S.
(8.13)
Since φ ∈ S, there exists a constant M such that (1 + t 2 ) | φ(t) | ≤ M. Then the
integral in (8.13) exists since
∞
−∞
φ(t) dt ≤
∞
−∞
| φ(t) | dt ≤ M
∞
1
dt < ∞.
−∞ 1 + t 2
It is now rather easy to show that 1 is indeed a distribution (that is, a linear mapping
from S to C). In chapter 9 it will turn out that 1 is the spectrum of the delta function.
This has been already been made plausible in section 8.1.
EXAMPLE 8.2
Unit step function
The unit step function (t) (see example 6.1) defines a distribution, again denoted
by (t), or for short, by
,φ =
∞
−∞
(t)φ(t) dt =
∞
0
φ(t) dt
for φ ∈ S.
Note that (8.12) is again applied here. Further details are almost the same as in
example 8.1.
196
8 Distributions
EXAMPLE 8.3
Sign function
The sign function sgn t is defined by
for t > 0,
1
0
for t = 0,
sgn t =
−1 for t < 0.
This function defines a distribution sgn t by
sgn t, φ =
∞
−∞
sgn tφ(t) dt =
∞
0
(φ(t) − φ(−t)) dt
for φ ∈ S.
Here (8.12) is again applied. Further details are the same as in the previous two
examples.
EXAMPLE 8.4
The function f (t) = | t |
The function f (t) = | t | also defines a distribution using (8.12), and again it will
simply be denoted by | t |:
|t |,φ =
∞
−∞
| t | φ(t) dt =
∞
0
t (φ(t) + φ(−t)) dt
for φ ∈ S.
In order to show that the integral exists, we use in this case that there exists a constant
M > 0 such that t (1 + t 2 )φ(t) ≤ M on R. From here on, the proof is exactly the
same as in example 8.1.
EXAMPLE 8.5
Principal value 1/t
The function 1/t is not absolutely integrable on R and even φ(t)/t with φ ∈ S
may not be absolutely integrable on R since the point t = 0 may cause a problem.
Hence, for 1/t we cannot use (8.12) to define a distribution. This problem can be
solved by resorting to a variant of the Cauchy principal value from definition 7.1.
Loosely speaking, in this case we let the limits at t = 0 tend to zero at the same
rate; the distribution arising in this way is denoted by pv(1/t) (pv from 'principal
value'). The precise definition of the distribution pv(1/t) is as follows:
1
φ(t)
pv , φ = lim
dt = lim
t
α↓0 | t |≥α t
α↓0
∞ φ(t)
α
t
dt +
−α φ(t)
−∞
t
dt
(8.14)
for φ ∈ S. The existence of the right-hand side is again the most difficult step
in proving that pv(1/t) is a distribution. To this end we split the integral in the
right-hand side of (8.14) as follows:
α≤| t |≤1
φ(t)
φ(t)
dt +
dt.
t
| t |≥1 t
(8.15)
First look at the second integral in (8.15). Since | 1/t | ≤ 1 for | t | ≥ 1, it follows
that
∞
φ(t)
| φ(t) | dt ≤
| φ(t) | dt,
dt ≤
| t |≥1 t
| t |≥1
−∞
and in example 8.1 it was shown that the last integral exists for φ ∈ S. Hence, the
second integral in (8.15) exists. For the first integral in (8.15) we note that for any
α > 0 one has
1 φ(0)
−α φ(0)
φ(0)
dt =
dt +
dt = 0,
t
t
α
−1
α≤| t |≤1 t
since 1/t is an odd function. Hence,
φ(t)
φ(t) − φ(0)
dt =
dt.
t
t
α≤| t |≤1
α≤| t |≤1
8.3 Derivatives of distributions
197
Now define ψ(t) = (φ(t) − φ(0))/t for t = 0 and ψ(0) = φ (0). Then ψ is
continuous at the point t = 0 since
lim ψ(t) = lim
t→0
t→0
φ(t) − φ(0)
= φ (0) = ψ(0).
t −0
It then follows that
lim
α↓0 α≤| t |≤1
φ(t)
φ(t) − φ(0)
ψ(t) dt.
dt = lim
dt = lim
t
t
α↓0 α≤| t |≤1
α↓0 α≤| t |≤1
Since ψ is continuous at t = 0, the limit α → 0 exists and it follows that
1
φ(t)
dt =
ψ(t) dt.
t
α↓0 α≤| t |≤1
−1
lim
Since ψ is a continuous function on the closed and bounded interval [−1, 1], it then
follows that this final integral, and so the right-hand side of (8.14), exists.
EXERCISES
8.2
Let δ(t − a) be defined as in (8.10).
a Show that δ(t − a) is a distribution, that is, a linear mapping from S to C.
b Derive the symbolic notation (8.11) by interchanging the limit and the integral
in (8.1) (hint: also use that δ(−t) = δ(t), which is quite plausible on the basis of
(8.3) or (8.8), and which will be proven in section 8.4).
8.3
Show that T f as defined in (8.12) is a linear mapping.
8.4
Show that 1 as defined in (8.13) is indeed a distribution.
8.5
Prove that the integral in example 8.2 exists and check that is a distribution.
8.6
Do the same as in exercise 8.5 for the distribution sgn t from example 8.3.
8.7
Prove that | t | from example 8.4 defines a distribution.
8.8
For fixed a ∈ R we define T by T, φ = φ (a) for φ ∈ S. Show that T is a
distribution.
8.9
Let the function f (t) = | t |−1/2 be given.
a Show that f is integrable on the interval [−1, 1]. Is f integrable on R?
b Show that f defines a distribution by means of (8.12). In particular it has to be
shown that the defining integral exists (hint: for | t | ≥ 1 one has | t |−1/2 ≤ 1).
8.10
Prove for the following functions that (8.12) defines a distribution:
a f (t) = t,
b f (t) = t 2 .
8.3
Derivatives of distributions
Switching on a (direct current) apparatus at time t = 0 can be described using the
unit step function (t). This, however, is an ideal description which will not occur
in reality of course. More likely there will be a very strong increase in a very short
time interval. Let u(t) be the function describing the switching on in a realistic way.
The ideal function (t) and a typical 'realistic' function u(t) are drawn in figure 8.3.
We assume for the moment that u(t) is differentiable and that u(t) increases from
the value 0 to the value 1 in the time interval 0 ≤ t ≤ a. The derivative u (t) of
u(t) equals 0 for t < 0 and t > a, while between t = 0 and t = a the function
198
8 Distributions
⑀ (t )
1
a
0
t
u (t )
1
a
0
t
u'(t )
1
a
0
t
FIGURE 8.3
The ideal (t), the realistic u(t) and the derivative u (t).
u (t) quickly reaches its maximum and then decreases to 0 rapidly. The graph of a
typical u (t) is also drawn in figure 8.3. If we now take the limit a ↓ 0, then u(t)
will transform into the function (t), while u (t) seems to tend towards the delta
function. This is because u(t) will have to increase faster and faster over an ever
smaller interval; the derivative will then attain ever increasing values in the vicinity
of t = 0. In the limit a ↓ 0 an object will emerge which is 0 everywhere, except at
the point t = 0, where the value becomes infinitely large. Since, moreover,
∞
−∞
u (t) dt =
a
0
u (t) dt = [u(t)]a = 1
0
for every a > 0, (8.5) is valid as well. We thus obtain an object fitting precisely
the description of the delta function from section 8.1. Hence, it is plausible that the
derivative of (t) will be the delta function.
8.3 Derivatives of distributions
199
In order to justify these conclusions mathematically, we should first find a definition for the derivative of a distribution. Of course, this definition should agree
with the usual derivative of a function, since distributions are an extension of functions. Now let f be an absolutely integrable function with continuous derivative f
being absolutely integrable as well. Then f defines a distribution T f and from an
integration by parts it then follows for φ ∈ S that
Tf ,φ =
∞
−∞
f (t)φ(t) dt = [ f (t)φ(t)]∞ −
−∞
∞
−∞
f (t)φ (t) dt.
Since φ ∈ S, one has limt→±∞ φ(t) = 0 (for | φ(t) | ≤ M/(1 + t 2 )). Moreover,
the final integral can be considered as the distribution T f applied to the function
φ (t). We thus have
Tf ,φ = − Tf ,φ .
(8.16)
There is now only one possible definition for the derivative of a distribution.
DEFINITION 8.4
Derivative of distributions
The derivative T of a distribution T is defined by
T , φ = − T, φ
for φ ∈ S.
Note that T, φ will certainly make sense because T is a distribution and φ ∈ S
whenever φ ∈ S. From the linearity of T and of differentiation the linearity of T
immediately follows; hence, T is indeed a distribution. This means in particular that
T has a derivative as well. As for functions this is called the second derivative of
T and it is denoted by T . Applying definition 8.4 twice, it follows that T , φ =
T, φ . This process can be repeated over and over again, so that we reach the
remarkable conclusion that a distribution can be differentiated an arbitrary number
of times. Applying definition 8.4 k times, it follows that the kth derivative T (k) of a
distribution T is given by
T (k) , φ = (−1)k T, φ (k)
Derivative of (t)
for φ ∈ S.
(8.17)
In particular it follows that any absolutely integrable function, considered as a distribution, is arbitrarily often differentiable. This gives us an obvious way to find
new distributions. Start with a function f and consider it as a distribution T f (if
possible). Differentiating several times if necessary, one will in general obtain a
distribution which no longer corresponds to a function. In the introduction to this
section we have in fact already seen a crucial example of this process. For we have
argued there that the derivative of the unit step function (t) should be the delta
function. This can now be proven using definition 8.4. For according to definition
8.4 the distribution is given by
,φ = − ,φ = −
∞
0
φ (t) dt = −[φ(t)]∞ ,
0
where in the second step we used the definition of
one has limt→±∞ φ(t) = 0 and so it follows that
, φ = φ(0) = δ, φ
for φ ∈ S.
(see example 8.2). For φ ∈ S
(8.18)
= δ. The informal derivation in the
According to definition 8.3 we then have
introduction to this section has now been made mathematically sound.
In order to handle distributions more easily, it is convenient to be able to multiply
them by a constant and to add them. The definitions are as follows.
200
8 Distributions
DEFINITION 8.5
Let S and T be distributions. Then cT (c ∈ C) and S + T are defined by
cT, φ = c T, φ
for φ ∈ S,
S + T, φ = S, φ + T, φ
for φ ∈ S.
EXAMPLE 8.6
The distribution cδ(t − a) is given by cδ(t − a), φ = cφ(a). For c ∈ R
this is graphically represented by an arrow of height c at the point t = a. See
figure 8.4.
c δ(t – a )
c
t
a
0
FIGURE 8.4
The distribution cδ(t − a).
EXAMPLE 8.7
The distribution 2 (t) + 3iδ(t) is given by
2 (t) + 3iδ(t), φ = 2
∞
0
φ(t) dt + 3iφ(0)
for φ ∈ S.
We close this section by determining some derivatives of distributions.
EXAMPLE 8.8
Derivative of δ(t)
We have just shown that = δ, considered as distribution. Differentiating again,
we obtain that = δ . Just as for δ, there is a simple description for the distribution
δ:
δ , φ = − δ, φ = −φ (0)
for φ ∈ S.
(8.19)
Hence, the distribution δ assigns the complex number −φ (0) to a function φ. Note
also that δ is still well-defined when applied to continuously differentiable functions
φ. Symbolically one writes (8.19) as
∞
−∞
δ (t)φ(t) dt = −φ (0).
(8.20)
This expression can be derived symbolically from (8.9) by taking the function φ
instead of φ in (8.9) and performing a formal integration by parts. This example can
be extended even further by repeated differentiation. Then the distributions δ , δ (3) ,
etc. will arise.
Let f be a function with continuous derivative f and assume that both f and
f can be considered as a distribution through (8.12). The distribution T f then has
the distribution (T f ) as derivative; if our definitions have been put together well,
then (T f ) = T f should be true. For then the two concepts of 'derivative' coincide.
Using (8.16) the proof reads as follows: T f , φ = − T f , φ = (T f ) , φ for all
φ ∈ S. When no confusion is possible, one often simply writes f , when actually
T f is meant. Most often we then use the phrase f as distribution.
202
8 Distributions
1
⑀(t ) cost
0
π
2
π
t
1
δ(t ) – ⑀ (t ) sint
0
π
2
π
t
FIGURE 8.5
The function (t) cos t and its derivative, considered as distribution.
If we take for f the function (t), then a = 0 and (0+) − (0−) = 1. Moreover,
(t) = 0 for t = 0, so T f = 0. It then follows from (8.21) that = δ, considered
as distribution, in accordance with previous results. In a similar way one obtains
examples 8.9 and 8.10 from (8.21).
EXERCISES
8.11
Let T be defined as in definition 8.4. Show that T is a linear mapping from S to
C.
8.12
a√ Which complex number is assigned to φ ∈ S by the distribution 2δ(t) −
i 3δ (t) + (1 + i)sgn t?
b Show that the function f (t) = at 2 + bt + c with a, b and c ∈ C defines a
distribution through (8.12).
8.13
a Show that for distributions S and T one has (S+T ) = S +T and (cT ) = cT .
Hence, differentiation of distributions is linear.
b Show that for the constant function f (t) = c one has f = 0 as distribution.
8.14
a
b
8.15
a Calculate the derivative of the distribution sgn t in a direct way, using the definition of the derivative of a distribution.
Which complex number is assigned to φ ∈ S by the distribution δ (3) ?
To which set of functions can one extend the definition of δ (3) ?
8.4 Multiplication and scaling of distributions
203
b Verify that sgn t = 2 (t) − 1 for all t = 0 and then use exercise 8.13 to determine the derivative of sgn t again.
c Determine the second derivative of | t |.
8.16
Show how examples 8.9 and 8.10 arise as special cases of the jump-formula (8.21).
8.17
Determine the derivative of the following distributions:
a pa (t),
b
(t) sin t.
8.18
The (discontinuous) function f (t) is given by
t +1
π
f (t) =
2
t − 2t + 5
for t < 1,
for t = 1,
for t > 1.
a Verify that f (t) defines a distribution by means of (8.21) (also see exercises 8.10
and 8.12b).
b Determine the derivative of f (t) as distribution.
8.19
Let a ≤ 0 be fixed and consider f (t) = (t)eat . Prove that f (t) − a f (t) = δ(t)
(considered as distributions).
8.20
Define for fixed a ∈ R (a = 0) the function g(t) by g(t) = (t)(sin at)/a. Prove
that g (t) + a 2 g(t) = δ(t) (considered as distributions).
8.4
Multiplication and scaling of distributions
In the previous section it was shown that distributions can be added and multiplied
by a complex constant. We start this section with a treatment of the multiplication of
distributions. Multiplication is important in connection with convolution theorems
for the Fourier transform. This is because the convolution product changes into an
ordinary product under the Fourier transform. If we want to formulate similar results
for distributions, then we ought to be able to multiply distributions. However, in
general this is not possible (in contrast to functions). The function f (t) = | t |−1/2 ,
for example, is integrable on, say, the interval [−1, 1] and thus it defines a distribution through (8.12) (see exercise 8.9). But f 2 (t) = 1/ | t | is not integrable on
an interval containing 0; hence, one cannot define a distribution using (8.12). Still,
multiplication is possible in a very limited way: distributions can be multiplied by
polynomials. As a preparation we will first prove the following theorem.
THEOREM 8.1
Let φ ∈ S and p be a polynomial. Then pφ ∈ S.
Proof
A polynomial p(t) is of the form an t n + an−1 t n−1 + · · · + a1 t + a0 with ai ∈ C. If
φ ∈ S, then certainly cφ ∈ S for c ∈ C. The sum of two elements in S also belongs
to S. Hence, it is sufficient to show that t k φ(t) ∈ S for φ ∈ S and k ∈ N. But this
has already been observed in section 6.5 (following theorem 6.11).
We can now define the product of a distribution and a polynomial.
DEFINITION 8.6
Product of distribution and
polynomial
Let T be a distribution and p a polynomial. The distribution pT is defined by
pT, φ = T, pφ
for φ ∈ S.
(8.22)
204
8 Distributions
Theorem 8.1 shows that the right-hand side of (8.22) is meaningful, since T is a
distribution and pφ ∈ S. Since T is linear, it immediately follows that pT is linear
as well:
pT, cφ = T, p(cφ) = T, c( pφ) = c T, pφ = c pT, φ
and
pT, φ1 + φ2 = T, p(φ1 + φ2 ) = T, pφ1 + pφ2
= T, pφ1 + T, pφ2 = pT, φ1 + pT, φ2 .
This proves that pT is indeed a distribution.
EXAMPLE 8.11
For a polynomial p one has
p(t)δ(t) = p(0)δ(t).
This is because according to definitions 8.6 and 8.2 we have pδ, φ = δ, pφ =
( pφ)(0) = p(0)φ(0) for any φ ∈ S. But p(0)φ(0) = p(0) δ, φ = p(0)δ, φ
and according to definition 8.3 the distributions pδ and p(0)δ are thus equal. In
particular we have for p(t) = t:
tδ(t) = 0.
Similarly one has for the delta function δ(t − a) that
p(t)δ(t − a) = p(a)δ(t − a).
Product of δ and a
continuous function
Often a distribution can be multiplied by many more functions than just the polynomials. As an example we again look at the delta function δ(t), which can be
defined on the set of all continuous functions (see section 8.2.2). Now if f is a
continuous function, then precisely as in (8.22) one can define the product f δ by
f δ, φ = δ, f φ ,
where φ is an arbitrary continuous function. From the definition of δ(t) it follows
that f δ, φ = f (0)φ(0) = f (0) δ, φ and so one has
f (t)δ(t) = f (0)δ(t)
for any continuous function f (t). For the general delta function δ(t − a) it follows
analogously for any continuous function f (t) that
f (t)δ(t − a) = f (a)δ(t − a)
for a ∈ R.
(8.23)
Similarly one can, for example, multiply the distribution δ by continuously differentiable functions.
The reason why we are constantly working with the space S lies in the fact that
S is very suitable for Fourier analysis. Moreover, it can be quite tedious to find
out exactly for which set of functions the definition of a distribution still makes
sense. And finally, it would be very annoying to keep track of all these different
sets of functions (the continuous functions for δ, the continuously differentiable
functions for δ , etc.). We have made an exception for (8.23) since it is widely used
in practical applications and also because in much of the literature the delta function
is introduced using continuous functions.
We close this section with a treatment of the scaling of distributions. As in the
case of the definition of the derivative of a distribution, we first take a look at the
situation for an absolutely integrable function f (t). For a ∈ R with a = 0 one has
8.4 Multiplication and scaling of distributions
205
for the scaled function f (at) that
∞
−∞
f (at)φ(t) dt = | a |−1
∞
−∞
f (τ )φ(a −1 τ ) dτ
for φ ∈ S.
This follows by changing to the variable τ = at, where for a < 0 we should pay
attention to the fact that the limits of integration are interchanged; this explains the
factor | a |−1 . If we consider this result as an identity for the distributions associated
with the functions, then it is clear how scaling of distributions should be defined.
DEFINITION 8.7
Scaling of distributions
Let T be a distribution and a ∈ R with a = 0. Then the scaled distribution T (at) is
defined by
T (at), φ(t) = | a |−1 T (t), φ(a −1 t)
for φ ∈ S.
(8.24)
According to definitions 8.7 and 8.2 one has for the scaled delta distribution δ(at):
EXAMPLE 8.12
δ(at), φ(t) = | a |−1 δ(t), φ(a −1 t) = | a |−1 φ(0) = | a |−1 δ(t), φ(t)
for any φ ∈ S. Hence,
δ(at) = | a |−1 δ(t).
Time reversal of distribution
Even and odd distribution
A special case of scaling occurs for a = −1 and is called time reversal. For
the delta function one has δ(−t) = δ(t), which means that the delta function remains unchanged under time reversal. We recall that a function is called even when
f (−t) = f (t) and odd when f (−t) = − f (t). Even and odd distributions are
defined in the same way. A distribution T is called even when T (−t) = T (t) and
odd when T (−t) = −T (t). The delta function δ(t) is thus an example of an even
distribution.
EXERCISES
8.21
In example 8.10 it was shown that ( (t) cos t) = δ(t) − (t) sin t. Derive this result
again by formally applying the product rule for differentiation to the product of (t)
and cos t. (Of course, a product rule for differentiation of distributions cannot exist,
since in general the product of distributions does not exist.)
8.22
Show that for the delta function δ(t − a) one has p(t)δ(t − a) = p(a)δ(t − a),
where p(t) is a polynomial and a ∈ R.
8.23
The derivative δ of the delta function can be defined by δ , φ = −φ (0) for the set
of all continuously differentiable functions φ. Let f (t) be a continuously differentiable function.
a Give the definition of the product f (t)δ (t).
b Show that f (t)δ (t) = f (0)δ (t) − f (0)δ(t).
c Prove that tδ (t) = −δ(t) and that t 2 δ (t) = 0.
8.24
Show that for the scaled derivative of the delta function one has δ (at) =
a −1 | a |−1 δ (t) for a = 0.
8.25
Show that the product t · pv(1/t) is equal to the distribution 1.
8.26
Show that a distribution is even if and only if one has T, φ(t) = T, φ(−t) for
all φ ∈ S, while T is odd if and only if T, φ(t) = − T, φ(−t) .
8.27
a
b
Show that the distributions sgn t and pv(1/t) are odd.
Show that the distribution | t | is even.
206
8 Distributions
SUMMARY
Distributions are linear mappings from the space of rapidly decreasing functions S
to C. The delta function δ(t − a), for example, assigns the number φ(a) to any
φ ∈ S (a ∈ R). Many ordinary functions f can be considered as distribution by
φ→
∞
−∞
f (t)φ(t) dt
for φ ∈ S.
Examples of this are the constant function 1, the unit step function (t), the sign
function sgn t and any absolutely integrable function. Like the delta function, the
distribution pv(1/t) is not of this form.
Distributions are arbitrarily often differentiable. The delta function is the derivative of the unit step function. More generally, one has the following. Let f be a
function with a jump of magnitude c at the point t = a. Then the derivative of f (as
distribution) contains the distribution cδ(t − a) at the point t = a.
Distributions can simply be added and multiplied by a constant. In general they
cannot be multiplied together. It is possible to multiply a distribution by a polynomial. Sometimes a distribution can also be multiplied by more general functions.
For example, for the delta function one has f (t)δ(t − a) = f (a)δ(t − a) (a ∈ R)
for any continuous function f (t). Finally, one can scale distributions with a real
constant a = 0. For a = −1 this is called time reversal. This also gives rise to the
notions even and odd distributions.
SELFTEST
8.28
Given is the function f (t) = ln | t | for t = 0.
a Show that 0 ≤ ln t ≤ t for t ≥ 1 and conclude that 0 ≤ ln | t | ≤ | t | for
| t | ≥ 1.
b Show that f is integrable over [−1, 1]. Use part a to show that f defines a
distribution by means of (8.12).
c Prove that f defines an even distribution.
8.29
The continuous function f (t) is given by
f (t) =
a
b
c
8.30
t2
2t
for t ≥ 0,
for t < 0.
Prove that f can be considered as a distribution.
Find the derivative of f as distribution.
Determine the second derivative of f as distribution.
Consider the second derivative δ (t) of the delta function.
a For which set of functions f (t) can one define the product f (t)δ (t)?
b Prove that f (t)δ (t) = f (0)δ(t) − 2 f (0)δ (t) + f (0)δ (t).
c Show that t 2 δ (t) = 2δ(t) and that t 3 δ (t) = 0.
d Show that for a = 0 one has for the scaled distribution δ (at) that δ (at) =
a −2 | a |−1 δ (t).
CHAPTER 9
The Fourier transform of distributions
INTRODUCTION
In the previous chapter we have seen that distributions form an extension of the familiar functions. Moreover, in most cases it is not very hard to imagine a distribution
intuitively as a limit of a sequence of functions. Especially when introducing new
operations for distributions (such as differentiation), such an intuitive representation
can be very useful. In section 8.1 we applied this method to make it plausible that
the Fourier transform of the delta function is the constant function 1, and also that
the reciprocity property holds in this case.
The purpose of the present chapter is to develop a rigorous Fourier theory for
distributions. Of course, the theory has to be set up in such a way that for functions
we recover our previous results; this is because distributions are an extension of
functions. This is why we will derive the definition of the Fourier transform of
a distribution from a property of the Fourier transform of functions in section 9.1.
Subsequently, we will determine the spectrum of a number of standard distributions.
Of course, the delta function will be treated first.
In section 9.2 we concentrate on the properties of the Fourier transform of distributions. The reciprocity property for distributions is proven. We also treat the
correspondence between differentiation and multiplication. Finally, we show that
the shift properties also remain valid for distributions.
It is quite problematic to give a rigorous definition of the convolution product
or to state (let alone prove) a convolution theorem. Let us recall that the Fourier
transform turns the convolution product into an ordinary multiplication (see section
6.6). But in general one cannot multiply two distributions, and so the convolution
product of two distributions will not exist in general. In order to study this problem,
we start in section 9.3 with an intuitive version of the convolution product of the
delta function (and derivatives of the delta function) with an arbitrary distribution.
We then look at the case where the distributions are defined by functions. This
will tell us in which situations the convolution product of two distributions exists.
Finally, the convolution theorem for distributions is formulated. The proof of this
theorem will not be given; it would lead us too far into the theory of distributions.
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know and can apply the definition of the Fourier transform of a distribution
- know the Fourier transform of the delta function and the principal value 1/t
- can determine the Fourier transform of periodic signals and periodic signals that
are switched on
- know and can apply the properties of the Fourier transform of distributions
- know and can apply the convolution product and the convolution theorem for distributions in simple cases
- know the comb distribution and its Fourier transform∗ .
208
9.1 Definition and examples
9.1
209
The Fourier transform of distributions: definition and examples
It should come as no surprise that we will start section 9.1.1 with the definition of the
Fourier transform of a distribution. In section 9.1.2 we then determine the Fourier
transform of a number of standard distributions. We also discuss how the Fourier
transform of periodic signals (considered as distributions) can be determined. Finally, we treat the so-called comb distribution and its Fourier transform in section
9.1.3.
9.1.1
Definition of the Fourier transform of distributions
In section 8.1 we sketched an intuitive method to obtain the Fourier transform of a
distribution. First, a distribution is considered symbolically as a limit of functions
f a whose Fourier transforms Fa are known. Next, the limit of these functions Fa
is determined. So when the distribution T is given by T = lim f a (for a → ∞ for
example) and f a ↔ Fa , then the Fourier transform of T is given by F T = lim Fa
(for a → ∞). Mathematically, however, this method has some serious problems.
What is meant by the symbolic limits T = lim f a and F T = lim Fa ? Is the choice
of the sequence of functions f a (and hence Fa ) uniquely determined? And if not, is
F T uniquely determined then? One can solve all of these problems at once using
the rigorous definition of distributions given in the previous chapter. The only thing
still missing is the rigorous definition of the Fourier transform – or spectrum – of a
distribution. Of course, such a definition has to be in agreement with our earlier definition of the Fourier transform of a function, since distributions form an extension
of functions. Precisely as in the case of the definition of the derivative of a distribution, we therefore start with a function f that can be considered as a distribution T f
according to (8.12). Is it then possible, using the properties of the Fourier transform
of ordinary functions, to come up with a definition of the Fourier transform of the
distribution T f ? Well, according to the selfduality property in section 6.4.7 one has
for any φ ∈ S
∞
−∞
F(t)φ(t) dt =
∞
−∞
f (t) (t) dt,
where is the spectrum of φ and F is the spectrum of f . From theorem 6.12 it
follows that ∈ S and so the identity above can also be considered as an identity
for distributions: TF , φ = T f , . From this, it is obvious how one should define
the Fourier transform (or spectrum) of an arbitrary distribution.
DEFINITION 9.1
Fourier transform or
spectrum of distributions
For a distribution T the Fourier transform or spectrum F T is defined by
F T, φ = T,
where
,
(9.1)
is the Fourier transform of φ ∈ S.
We recall once again theorem 6.12, which established that
∈ S for φ ∈ S.
Hence, the right-hand side of (9.1) is well-defined and from the linearity of T it
follows immediately that F T is indeed a distribution. The mapping assigning the
spectrum F T to a distribution T is again called the Fourier transform. In section
9.1.2 we determine the Fourier transform of a number of distributions.
210
9 The Fourier transform of distributions
9.1.2
Examples of Fourier transforms of distributions
First of all, we will use definition 9.1 to determine the Fourier transform of the delta
function. We have
F δ, φ = δ,
Spectrum of δ(t)
=
∞
(0) =
−∞
φ(t) dt
for φ ∈ S,
where in the last step we used the definition of the ordinary Fourier transform. From
example 8.1 it then follows that Fδ, φ = 1, φ , which shows that the spectrum of
δ is indeed the function 1. A short symbolic proof is obtained by taking e−iωt for φ
in (8.9). It then follows that
∞
−∞
δ(t)e−iωt dt = 1,
(9.2)
which states precisely (but now symbolically) that the spectrum of δ(t) is the function 1. Figure 9.1 shows δ(t) and its spectrum.
1
δ(t )
t
0
F δ(ω)
1
ω
0
FIGURE 9.1
The delta function δ(t) and its spectrum.
Conversely, the spectrum of the function 1 is determined as follows:
F 1, φ = 1,
=
∞
−∞
(ω) dω.
For φ ∈ S the inversion formula (7.9) certainly holds. Applying it for t = 0, it
follows that
F 1, φ =
Spectrum of 1
∞
−∞
(ω) dω = 2πφ(0) = 2π δ, φ = 2πδ, φ
for φ ∈ S.
Hence, the spectrum of the function 1 (as distribution) is given by the distribution
2πδ(ω). Symbolically, this is sometimes written as
∞
−∞
e−iωt dt = 2πδ(ω).
(9.3)
9.1 Definition and examples
211
(Note that the integral in the left-hand side does not exist as an improper integral;
see exercise 6.1.) The results that have been derived intuitively in section 8.1 have
now all been proven. Note that δ(t) ↔ 1 and 1 ↔ 2πδ(ω) constitute a Fourier
pair in the sense of section 7.2.2. In section 9.2.3 we will prove that indeed the
reciprocity property for distributions remains valid. In the next example the delta
function at the point a is treated (see (8.10) for its definition).
EXAMPLE 9.1
Spectrum of δ(t − a)
The spectrum of δ(t −a) is the function e−iaω (considered as distribution of course).
The proof is similar to the case a = 0:
F δ(t − a), φ = δ(t − a),
=
(a) =
∞
−∞
φ(t)e−iat dt
for φ ∈ S.
If we now call the variable of integration ω instead of t, then the result follows:
F δ(t − a), φ = e−iaω , φ . Conversely, one has that 2πδ(ω − a) is the spectrum
Spectrum of eiat
of the function eiat :
F eiat , φ = eiat ,
=
∞
−∞
eiat (t) dt
and according to the inversion formula (7.9) it then follows that Feiat , φ =
2πφ(a) = 2πδ(ω − a), φ (φ ∈ S), which proves the result. We thus again have a
Fourier pair δ(t − a) ↔ e−iaω and eiat ↔ 2πδ(ω − a). Also note that the function
eiat is a periodic function with period 2π/a.
Just as for the Fourier transform of functions, the Fourier transform of distributions is a linear mapping. For distributions S and T and a, b ∈ C one thus has
F (aS + bT ) = aF S + bF T .
EXAMPLE 9.2
Since the Fourier transform is linear, it follows from example 9.1 that the spectrum
of cos at = (eiat + e−iat )/2 is the distribution π(δ(ω − a) + δ(ω + a)). This is
represented graphically in figure 9.2.
For the Fourier transform of distributions one has the following result.
THEOREM 9.1
The Fourier transform is a one-to-one mapping on the space of distributions.
Proof
Because of the linearity of the Fourier transform, it is sufficient to show that F T = 0
implies that T = 0. So let us assume that F T = 0, then F T, φ = 0 for all φ ∈ S.
From definition 9.1 it then follows that T,
= 0 for all φ ∈ S. But according to
theorem 7.6 one can write any ψ ∈ S as the spectrum of some φ ∈ S. Hence,
T, ψ = 0 for all ψ ∈ S, which means that T = 0 (definition 8.3).
Theorem 9.1 is often used (implicitly) in the following situation. Let a distribution U be given. Suppose that by using some table, perhaps in combination with the
properties of the Fourier transform, we have found a distribution whose spectrum
is the given distribution U . We may then conclude on the basis of theorem 9.1 that
we have found the only possibility. For a given distribution T there is thus only one
distribution U which is the spectrum of T . As for functions, T ↔ U will mean
that U is the spectrum of T and that the distributions T and U determine each other
uniquely.
EXAMPLE 9.3
Suppose that we are looking for a distribution T whose spectrum is the distribution
U = 4δ(ω − 3) − 2δ(ω + 2). In example 9.1 it was shown that eiat ↔ 2πδ(ω − a).
212
9 The Fourier transform of distributions
1
cos at
π
2a
0
π
a
t
π
πδ(ω – a ) + πδ(ω + a )
–a
ω
a
0
FIGURE 9.2
The function cos at and its spectrum.
From the linearity of the Fourier transform it then follows that T = (2/π)e3it −
(1/π)e−2it is a distribution with spectrum U . Theorem 9.1 guarantees that it is the
only one.
In the next example the spectrum of the distribution pv(1/t) will be determined.
EXAMPLE 9.4
Spectrum of pv(1/t)
The spectrum of the distribution pv(1/t) from example 8.5 is the distribution
−πisgn ω. A mathematically rigorous proof of this result would lead us too far
into the theory of distributions. Instead we will only give the following formal proof
(using a certain assumption, one can give a rigorous proof; see exercise 9.23).
F pv(1/t), φ = pv(1/t),
= lim
α→0 | t |≥α
Now apply the definition of spectrum for
1
α→0 | t |≥α t
Fpv(1/t), φ = lim
∞
−∞
(t)
dt
t
for φ ∈ S.
(t), then
φ(ω)e−iωt dω dt.
214
9 The Fourier transform of distributions
with a converging Fourier series:
∞
f (t) =
∞
ck e2πikt/T =
k=−∞
ck eikω0 t ,
k=−∞
where ω0 = 2π/T . Assume, moreover, that f defines a distribution by means of
(8.12). The Fourier series then also defines a distribution and the spectrum of this
distribution can be calculated as follows:
∞
F
ck eikω0 t
∞
=
k=−∞
∞
ck F eikω0 t = 2π
k=−∞
ck δ (ω − kω0 ) .
(9.5)
k=−∞
Here we assumed that the Fourier transform F and the summation may be interchanged. We will not prove that this is indeed allowed.
In fact, (9.5) shows once again that a periodic function f (t) has a line spectrum:
the spectrum consists of delta functions at the points ω = kω0 (k ∈ Z) with 'weight'
equal to the kth Fourier coefficient ck (also see section 3.3).
The comb distribution and its spectrum∗
9.1.3
Comb or shah distribution
The material in this section can be omitted without any consequences for the remainder of the book. Furthermore, we note that Poisson's summation formula from
section 7.3∗ will be used in an essential way.
The main reason that we treat the comb distribution is the fact that it is widely
used in the technical literature to represent the sampling of a continuous-time signal
(see later on in this section).
The comb or shah distribution is defined by
,φ =
∞
φ(k)
for φ ∈ S.
(9.6)
k=−∞
First we will have to show that is well-defined, that is, the series in the right-hand
side of (9.6) converges. But for φ ∈ S there exists a constant M > 0 such that
(1 + t 2 ) | φ(t) | ≤ M. Hence, it follows that (compare with example 8.1)
∞
| φ(k) | ≤ M
k=−∞
∞
∞
1
1
= M + 2M
.
1 + k2
1 + k2
k=−∞
k=1
But 1 + k 2 > k 2 for k > 0 and
∞
φ(k) ≤
k=−∞
∞
∞
−2 converges. Hence,
k=1 k
| φ(k) | < M 1 + 2
k=−∞
∞
k −2
< ∞.
k=1
It now immediately follows that defines a distribution. Since one has δ(t − k),
φ = φ(k) (see (8.10)), we obtain from (9.6) that
,φ =
∞
δ(t − k), φ
k=−∞
for every φ ∈ S. Because of this, one often writes
(t) =
∞
k=−∞
δ(t − k)
=
(t) as
9.1 Definition and examples
Train of delta functions
215
and calls a train of delta functions. The distribution is then graphically represented as in figure 9.4a. Often, is used to represent the sampling of a continuousa
(t )
1
–5 –4 –3 –2 –1 0
1
2
3
4
5
6
t
7
b
(ω/2π)
2π
–2π
0
2π
4π
ω
FIGURE 9.4
The impulse train (a) and its spectrum (b).
time signal (for example in Van den Enden and Verhoeckx (1987), chapter 3 (in
Dutch)). For when f (t) is a bounded continuous-time signal, then the product
f (t) (t) exists (we state this without proof) and one has (as in definition 8.6)
f (t) (t), φ =
(t), f φ =
∞
f (k)φ(k) =
k=−∞
∞
f (k)δ(t − k), φ
k=−∞
for all φ ∈ S. Hence we obtain that
f (t) (t) =
∞
f (k)δ(t − k).
k=−∞
Sampling
The right-hand side of this can indeed be interpreted as the sampling of the function
f (t) at times t = k (k ∈ Z); see figure 9.5.
After this short intermezzo on sampling, we now determine the spectrum of the
comb distribution . From the definition of it follows that
F
,φ =
,
=
∞
(k)
for φ ∈ S.
k=−∞
From example 7.9 it follows that Poisson's summation formula (7.23) is valid for
functions in S. If we apply it with T = 2π, then it follows that
F
,φ =
∞
k=−∞
(k) = 2π
∞
φ(2πk) =
(ω/2π), φ
k=−∞
for every φ ∈ S (in the last step we used the scaling property for distributions
with scaling factor 1/2π; see definition 8.7). We have thus proven that F
=
(ω/2π), where (ω/2π) is the scaled comb distribution ; the spectrum of the
9.2 Properties of the Fourier transform
217
9.2
Let S and T be distributions with spectra U and V respectively. Show that for
a, b ∈ C the spectrum of aS + bT equals aU + bV . The Fourier transform of
distributions is thus a linear mapping.
9.3
Determine the spectrum of the following distributions:
a δ(t − 4),
b e3it ,
c sin at,
d 4 cos 2t + 2ipv(1/t).
9.4
Determine the distribution T whose spectrum is the following distributions:
a δ(ω + 5),
b 2πδ(ω + 2) + 2πδ(ω − 2),
c sgn ω + 2 cos ω.
9.5
Show that the spectrum of an even (odd) distribution is again an even (odd) distribution.
9.6∗
a
b
Verify that defines a distribution.
ikω by
Define the distribution ∞
k=−∞ e
∞
eikω , φ =
k=−∞
eikω , φ
for φ ∈ S.
k=−∞
Show that F
9.2
∞
=
∞
ikω .
k=−∞ e
Properties of the Fourier transform
Most of the properties of the Fourier transform of functions, as derived in section
6.4, can easily be carried over to distributions. In this section we examine shifting,
differentiation and reciprocity (in exercise 9.9 scaling is considered as well).
9.2.1
Shift in time and frequency domains
First we treat the shifting property in the time domain. We will show that for a
distribution T with spectrum U one has T (t − a) ↔ e−iaω U (ω) (a ∈ R), just
as for functions (see section 6.4.3). Apart from the proof of this property, there
are two problems that we have to address. Does the product of U (ω) and e−iaω
exist? And what do we mean by the shifted distribution T (t − a)? We start with
the first problem. The product e−iaω U (ω) can be defined precisely as in (8.22) by
e−iaω U (ω), φ = U (ω), e−iaω φ (φ ∈ S). This definition makes sense since it
follows immediately from the product rule for differentiation and from e−iaω = 1
that e−iaω φ ∈ S. This solves the first problem. We now handle the second problem
and define the shifted distribution T (t − a).
DEFINITION 9.2
Shifted distribution
For a distribution T (t) the distribution T (t − a) shifted over a ∈ R is defined by
T (t − a), φ(t) = T (t), φ(t + a) .
As before (for example, for differentiation, scaling and the Fourier transform),
this definition is a direct generalization of the situation that occurs if we take T
equal to a distribution T f , where f is a function (see (8.12)).
218
9 The Fourier transform of distributions
EXAMPLE
As is suggested by the notation, the delta function δ(t − a) at the point a is indeed the shifted delta function δ(t). This is proven as follows: δ(t − a), φ(t) =
δ(t), φ(t + a) = φ(a), which is in agreement with the earlier definition of δ(t −a)
in (8.10).
We can now prove the shifting property for distributions. When T (t) is a distribution with spectrum U (ω), then
F(T (t − a)), φ(t) = T (t − a), (t) = T (t), (t + a)
for φ ∈ S.
From the shifting property in theorem 6.4 it follows that
T (t), (t + a) = T (t), F(e−iat φ(t)) = U (ω), e−iaω φ(ω) .
Hence, F(T (t − a)), φ(t) = e−iaω U (ω), φ(ω) , proving that
Shift property
T (t − a) ↔ e−iaω U (ω).
EXAMPLE
Since δ(t) ↔ 1, it follows from the shifting property that δ(t − a) ↔ e−iaω , in
accordance with example 9.1.
(9.9)
In a similar way one can prove the shifting property in the frequency domain (see
exercise 9.15):
eiat T ↔ U (ω − a)
9.2.2
Spectrum of δ (t)
(9.10)
Differentiation in time and frequency domains
For the ordinary Fourier transform, differentiation in one domain corresponded to
multiplication (by −it or iω) in the other domain (see sections 6.4.8 and 6.4.9). This
is also the case for the Fourier transform of distributions. We start by determining
the spectrum of the derivative δ (t) of the delta function. Successively applying
definitions 9.1, 8.4 and 8.2, we obtain that
Fδ , φ = δ ,
= − δ,
=−
According to theorem 6.8 one has
F δ , φ = F(itφ(t))(0) =
∞
−∞
(0)
for φ ∈ S.
(ω) = F(−itφ(t))(ω) and so
itφ(t) dt = it, φ
for φ ∈ S.
The variable t in the integral – and in the distribution – is irrelevant; by changing to
the variable ω we have thus shown that δ ↔ iω. In general one has for an arbitrary
distribution T with spectrum U that T ↔ iωU . In fact, as for the spectrum of the
derivative of the delta function, it follows that
FT , φ = T ,
= − T,
= T, F(itφ(t)) = U, itφ(t) .
In the last step we use that T ↔ U . Again, the variable t is irrelevant and if we
consider the distribution U as acting on functions in the variable ω, then we may
also write F T , φ = U, iωφ(ω) . From definition 8.6 of the multiplication of
distributions by polynomials, we then finally obtain that F T , φ = iωU, φ(ω) ,
which proves the result. More generally, one has (see exercise 9.8c):
Differentiation in time
domain
T (k) ↔ (iω)k U
when
T ↔ U.
(9.11)
9.2 Properties of the Fourier transform
219
Similarly, one can prove the differentiation rule in the frequency domain (see
exercise 9.10c):
Differentiation in frequency
domain
(−it)k T ↔ U (k)
when
T ↔U
(9.12)
(compare with section 6.4.9). This rule is mainly used to determine the spectrum of
t k T when the spectrum of T is known.
EXAMPLE
We know from section 9.1.2 that 1 ↔ 2πδ(ω). From (9.12) (for k = 1 and k = 2)
it then follows that t ↔ 2πiδ (ω) and t 2 ↔ −2πδ (ω).
9.2.3
Reciprocity
The Fourier pair δ ↔ 1 and 1 ↔ 2πδ suggests that the reciprocity property also
holds for distributions. This is indeed the case: if U is the spectrum of T , then
2π T (ω) is the spectrum of U (−t), or
Reciprocity
U (−t) ↔ 2π T (ω)
(9.13)
(where U (−t) is the scaled distribution from definition 8.7 with a = −1). For
its proof we first recall the reciprocity property for functions: if φ ↔ , then
(−t) ↔ 2πφ(ω). Hence, it follows that
2π T (ω), φ(ω) = T (ω), 2πφ(ω) = T (ω), F( (−t)) = U (t), (−t) .
Formula (9.13) is then proven by applying definition 8.7 of scaling. For then it
follows for every φ ∈ S that
2π T (ω), φ(ω) = U (−t), (t) = FU (−t), φ(ω) .
EXAMPLE
For T we take the delta function, so U is the function 1. Since 1 is an even distribution, it follows from (9.13) that 1 ↔ 2πδ, in accordance with our previous results
from section 9.1.2.
EXAMPLE 9.6
Spectrum of sgn t
The distribution sgn t is odd (see exercise 8.27a). Applying (9.13) to the result of
example 9.4, we obtain that the spectrum of πisgn t is the distribution 2π pv(1/ω),
that is,
sgn t ↔ −2ipv(1/ω).
Spectrum of (t)
(9.14)
Since (t) = (1 + sgn t)/2, where (t) is the unit step function, it follows that
(t) ↔ πδ(ω) − ipv(1/ω),
(9.15)
where we used the linearity of the Fourier transform. The function (t) and its
spectrum are shown in figure 9.6; imaginary values are represented by dashed curves
in this figure.
EXAMPLE 9.7
We know from (8.18) that = δ. From (9.11) it then follows that Fδ = F =
iωF . Apparently, iωF = 1. Now we have shown in example 9.6 that F =
πδ(ω) − ipv(1/ω) and indeed we have iω(π δ(ω) − ipv(1/ω)) = πiωδ(ω) +
ωpv(1/ω) = 0 + 1 = 1 (see example 8.11 and exercise 8.25).
At the end of section 9.1.2 we determined the spectrum of periodic functions. One
can use (9.15) to determine the spectrum of periodic functions that are 'switched on'.
220
9 The Fourier transform of distributions
⑀ (t )
1
t
0
π
( )
πδ(ω) – i pv 1
ω
ω
0
FIGURE 9.6
The unit step and its spectrum.
Let f (t) be a periodic function with period T and converging Fourier series
f (t) =
∞
ck e2πikt/T .
k=−∞
Switched-on periodic signal
We will call (t) f (t) a switched-on periodic signal. In (9.15) the spectrum of (t)
has been determined, and from the shifting property (9.10) it then follows that
(t)eiktω0 ↔ πδ(ω − kω0 ) − ipv(1/(ω − kω0 )),
Spectrum of switched-on
periodic signal
where ω0 = 2π/T . If we now assume, as in section 9.1.2, that the Fourier transform
and the summation may be interchanged, then it follows that the spectrum of a
switched-on periodic signal (t) f (t) is given by
∞
k=−∞
ck πδ(ω − kω0 ) − ipv
1
ω − kω0
.
(9.16)
9.3 Convolution
221
EXERCISES
9.7
The spectrum of sgn t is −2ipv(1/ω) (see example 9.6). Verify that (t) = (1 +
sgn t)/2 (t = 0) and that the spectrum of (t) is given by πδ(ω) − ipv(1/ω) (as was
stated in example 9.6).
9.8
a
b
c
9.9
a Let T be a distribution with spectrum U . Show that for a = 0 the scaled distribution T (at) has | a |−1 U (ω/a) as its spectrum.
b Determine the spectrum of δ(4t + 3).
9.10
a Show that −it T ↔ U when T ↔ U .
b Use part a to determine the spectrum of tδ(t) and tpv(1/t); check your answers
using example 8.11 and exercise 8.25.
c Show that (−it)k T ↔ U (k) when T ↔ U .
9.11
Determine the spectrum of tδ (t) and tδ (t) using (9.12); check your answers using
exercises 8.23 and 8.30.
9.12
From the identity = δ it follows that F = Fδ = 1. The differentiation rule
(9.11) then leads to iωF = 1. Hence, F = 1/iω, where we have to consider
the right-hand side as a distribution, in other words, as pv(1/ω). Thus we obtain
F = −ipv(1/ω). Considering the result from example 9.6 (or exercise 9.7), this
cannot be true since the term πδ(ω) is missing. Find the error in the reasoning given
above.
9.13
Let T be a distribution. Define for a ∈ R the product eiat T by eiat T, φ =
Determine in a direct way (without using (9.11)) the spectrum of δ .
Let T be a distribution with spectrum U . Show that T ↔ −ω2 U .
Prove (9.11): T (k) ↔ (iω)k U when T ↔ U .
T, eiat φ . Show that eiat T is a distribution.
9.14
Let T f be a distribution defined by a function f through (8.12). Show that for this
case, definition 9.2 of a shifted distribution reduces to a simple change of variables
in an integral.
9.15
Prove (9.10): eiat T ↔ U (ω − a) when T ↔ U .
9.16
Determine the spectra of the following distributions:
a
(t − 1),
b
(t)eiat ,
c
(t) cos at,
d δ (t − 4) + 3i,
e πit 3 + (t)sgn t.
9.17
Determine the distribution T whose spectrum is the following distributions:
a pv(1/(ω − 1)),
b (sin 3ω)pv(1/ω),
c
(ω),
d δ(3ω − 2) + δ (ω).
9.3
Convolution
Turning the convolution product into an ordinary multiplication is an important
property of the Fourier transform of functions (see the convolution theorem in section 6.6). In this section we will see to what extent this result remains valid for
222
9 The Fourier transform of distributions
distributions. We are immediately confronted with a fundamental problem: in general we cannot multiply two distributions S and T . Hence, the desired property
F (S ∗ T ) = F S · F T will not hold for arbitrary distributions S and T . This is
because the right-hand side of this expression will in general not lead to a meaningful expression. If we wish to stick to the important convolution property, then we
must conclude that the convolution product of two distributions is not always welldefined. Giving a correct definition of convolution is not a simple matter. Hence,
in section 9.3.1 we first present an intuitive derivation of the most important results
on convolution. For the remainder of the book it is sufficient to accept these intuitive results as being correct. For completeness (and because of the importance of
convolution) we present in section 9.3.2 a mathematically rigorous definition of the
convolution of distributions as well as proofs of the intuitive results from section
9.3.1. Section 9.3.2 can be omitted without any consequences for the remainder of
the book.
9.3.1
Intuitive derivation of the convolution of distributions
As we often did, we first concentrate ourselves on the delta function δ(t). Let us
return to the intuitive definition of the delta function from section 8.1. If we interchange limit and integral in (8.1), then we obtain the symbolic expression
∞
−∞
δ(τ ) f (t − τ ) dτ = f (t).
(9.17)
Changing from the variable t − τ to τ in the integral leads to
∞
−∞
f (τ )δ(t − τ ) dτ = f (t)
(9.18)
(also see the similar formula (8.11) and exercise 8.2, although there we also used that
δ(−t) = δ(t)). If we now pretend that the delta function is an ordinary absolutely
integrable function, then we recognize in the left-hand sides of (9.17) and (9.18)
precisely the convolution of δ and f (see definition 6.4) and so apparently (δ ∗
f )(t) = f (t) and ( f ∗ δ)(t) = f (t). This is written as δ ∗ f = f ∗ δ = f for
short. If we now consider δ and f as distributions again, then one should have for
an arbitrary distribution T that
δ ∗ T = T ∗ δ = T.
(9.19)
In a similar way one can derive an intuitive result for the convolution of δ with a
distribution T . For the delta function δ (t − a) one has δ (t − a), φ = −φ (a) for
each φ ∈ S. This can again be symbolically written as
∞
−∞
δ (τ − a) f (τ ) dτ = − f (a),
where, as in (9.17) and (9.18), we now write f instead of φ (for a = 0 this is (8.20)).
Now δ (τ − a) is an odd distribution (see exercise 9.18), so δ (τ − a) = −δ (a − τ ),
and if we also write t instead of a, then we see that
∞
−∞
δ (t − τ ) f (τ ) dτ = f (t).
The left-hand side can again be interpreted as the convolution of δ with f and
apparently one has (δ ∗ f )(t) = f (t). By changing from the variable t − τ to τ it
also follows that ( f ∗ δ )(t) = f (t). If we consider δ and f as distributions, then
9.3 Convolution
223
one should have for an arbitrary distribution T that
δ ∗T = T ∗δ = T .
(9.20)
Analogously, one can derive for the higher order derivatives δ (k) of the delta function
that
δ (k) ∗ T = T ∗ δ (k) = T (k) .
(9.21)
In section 9.3.2 we will define convolution of distributions in a mathematically rigorous way and we will prove formulas (9.19) – (9.21).
We will close this subsection by showing that the convolution theorem (theorem
6.13) is valid in the context of formulas (9.19) and (9.20), that is to say,
F (δ ∗ T ) = Fδ · F T
and
F(δ ∗ T ) = Fδ · F T.
(9.22)
First of all, the first identity in (9.22) follows immediately from δ ∗T = T and Fδ =
1 since F(δ ∗ T ) = F T = Fδ · F T . To prove the second identity in (9.22) we first
note that (9.11) implies that F T = iωF T . But iω = Fδ (see section 9.2.2) and
hence it follows from (9.20) that indeed F(δ ∗ T ) = F T = iωF T = Fδ · F T .
Similarly, one can show that the convolution theorem also holds for higher order
derivatives of the delta function. Finally, in this manner one can also verify that
the convolution theorem in the frequency domain, that is, S · T ↔ (U ∗ V )/2π
when S ↔ U and T ↔ V , is valid whenever the delta function or a (higher order)
derivative of the delta function occurs in the convolution product. In section 9.3.2
we will discuss in some more detail the convolution theorems for distributions and
we will formulate a general convolution theorem (theorem 9.2).
9.3.2
Mathematical treatment of the convolution of distributions∗
In this section we present a mathematically correct treatment of the convolution of
distributions. In particular we will prove the results that were derived intuitively in
section 9.3.1. As mentioned at the beginning of section 9.3, section 9.3.2 can be
omitted without any consequences for the remainder of the book.
In order to find a possible definition for S ∗ T , we start, as usual by now, with two
functions f and g for which we assume that the convolution product f ∗ g exists
and, moreover, defines a distribution by means of (8.12). It then follows for φ ∈ S
that
T f ∗g , φ =
∞
−∞
( f ∗ g)(t)φ(t) dt =
∞
∞
−∞
−∞
f (τ )g(t − τ ) dτ φ(t) dt.
Since we are only looking for the right definition, we might as well assume that we
may interchange the order of integration. We then see that
T f ∗g , φ =
=
∞
−∞
∞
−∞
f (τ )
f (τ )
∞
−∞
∞
−∞
g(t − τ )φ(t) dt dτ
g(t)φ(t + τ ) dt dτ
(9.23)
(change the variable from t − τ to t in the last step). For each fixed τ ∈ R the
latter inner integral can be considered as the distribution Tg applied to the function
φ(t + τ ). Here φ(t + τ ) should be considered as a function of t with τ kept fixed.
In order to show this explicitly, we denote this by Tg (t), φ(t + τ ) ; the distribution
Tg (t) acts on the variable t in the function φ(t + τ ). Now Tg (t), φ(t + τ ) is
a complex number for each τ ∈ R and we will assume that the mapping τ →
224
9 The Fourier transform of distributions
Tg (t), φ(t + τ ) from R into C is again a function in S; let us denote this function
by ψ(τ ). The right-hand side of (9.23) can then be written as the distribution T f
applied to this function ψ(τ ), which means that (9.23) can now be written as
T f ∗g , φ = T f (τ ), ψ(τ ) = T f (τ ), Tg (t), φ(t + τ ) .
This finally gives us the somewhat complicated definition of the convolution of two
distributions.
DEFINITION 9.3
Convolution of distributions
Let S and T be two distributions and define for τ ∈ R the function ψ(τ ) by
ψ(τ ) = T (t), φ(t + τ ) . If for each φ ∈ S the function ψ(τ ) belongs to S, then
the convolution product S ∗ T is defined by
S ∗ T, φ = S(τ ), ψ(τ ) = S(τ ), T (t), φ(t + τ ) .
(9.24)
The condition that ψ(τ ) should belong to S is the reason that often S ∗ T cannot
be defined.
EXAMPLE 9.8
For T take the delta function δ(t), then the function ψ(τ ) is given by ψ(τ ) =
T (t), φ(t + τ ) = δ(t), φ(t + τ ) = φ(τ ), because of the definition of δ(t).
Hence, we have ψ(τ ) = φ(τ ) in this case and so we certainly have that ψ ∈ S. The
convolution S ∗ δ thus exists for all distributions S and S ∗ δ, φ = S(τ ), φ(τ )
(φ ∈ S). This then proves the identity S ∗ δ = S in (9.19).
EXAMPLE 9.9
For each distribution T we have that T ∗ δ exists and T ∗ δ = T . In fact,
T ∗ δ , φ = T (τ ), δ (t), φ(t + τ ) = T (τ ), −φ (τ ) , because of the action of
δ (see (8.19)). Hence T ∗ δ exists and T ∗ δ , φ = T, −φ = T , φ for all
φ ∈ S (in the last step we used definition 8.4). This proves the identity T ∗ δ = T
in (9.20).
Convolution of distributions is not easy. For example, for functions f and g we
know that f ∗ g = g ∗ f , in other words, the convolution product is commutative
(see section 6.6). This result also holds for distributions. But even this simplest
of properties of convolution will not be proven in this book; it would lead us too
far into the theory of distributions. Without proof we state the following result: if
S ∗ T exists, then T ∗ S also exists and S ∗ T = T ∗ S. In particular it follows from
examples 9.8 and 9.9 that for an arbitrary distribution T one has T ∗ δ = δ ∗ T = T
and T ∗ δ = δ ∗ T = T . In exactly the same way one obtains more generally that
for any distribution T and any k ∈ Z+ one has
T ∗ δ (k) = δ (k) ∗ T = T (k) .
(9.25)
Convolution of distributions is also quite subtle and one should take great care using
it. For functions the convolution product is associative: ( f ∗ g) ∗ h = f ∗ (g ∗ h) for
functions f , g and h. For distributions this is no longer true. Both the convolutions
(R ∗ S) ∗ T and R ∗ (S ∗ T ) may well exist without being equal. An example: one
has (1 ∗ δ ) ∗ = 1 ∗ = 0 ∗ = 0, while 1 ∗ (δ ∗ ) = 1 ∗ = 1 ∗ δ = 1!
Finally, we treat the convolution theorems for distributions. However, even the
formulation of these convolution theorems is a problem, since the convolution of
two distributions may not exist. Hence, we first have to find a set of distributions for
which the convolution exists. To do so, we need a slight extension of the distribution
theory. We have already noted a couple of times that distributions can often be
defined for more than just the functions in S. The delta function δ, for example,
is well-defined for all continuous functions, δ for all continuously differentiable
functions, etc. Now let E be the space consisting of all functions that are infinitely
many times differentiable. All polynomials, for example, belong to the space E. Any
9.3 Convolution
Distribution on E
225
function in S belongs to E as well, since functions in S are in particular infinitely
many times differentiable (see definition 6.3).
Now let T be a distribution, so T is a linear mapping from S to C. Then it could
be that T is also well-defined for all functions in E, which means that T, φ is also
meaningful for all φ ∈ E (note that more functions belong to E than to S: S is a
subset of E). If this is the case, then we will say that T is a distribution that can be
defined on E. Such distributions play an important role in our convolution theorems.
EXAMPLE
The delta function and all the derivatives of the delta function belong to the distributions that can be defined on E.
EXAMPLE
(This example uses the comb distribution from section 9.1.3∗ .) The comb distribution cannot be defined on the space E. This is because the function φ(t) = 1
is an element of E (the function 1 is infinitely many times differentiable with all
∞
derivatives equal to 0) and , φ = ∞
k=−∞ φ(k) =
k=−∞ 1 diverges.
Distributions that can be defined on E form a suitable class of distributions for
the formulation of the convolution theorem.
THEOREM 9.2
Convolution theorem for
distributions
Let S and T be distributions with spectra U and V respectively. Assume that S is
a distribution that can be defined on the space E. Then U is a function in E, both
S ∗ T and U · V are well-defined, and S ∗ T ↔ U · V .
We will not prove this theorem. It would lead us too far into the distribution
theory. However, we can illustrate the theorem using some frequently occurring
convolution products.
EXAMPLE 9.10
Take for S in theorem 9.2 the delta function δ. The delta function can certainly be
defined on the space E. The spectrum U of δ is the function 1 and this is indeed a
function in E. Furthermore, F (δ ∗ T ) = F δ · F T according to theorem 9.2. But this
is obvious since we know that δ ∗ T = T and Fδ = 1 (also see section 9.3.1).
EXAMPLE 9.11
One can define δ on the space E as well. Furthermore, we showed in section 9.2.2
that δ ↔ iω. Note that the spectrum U of δ is thus indeed a function in E. Applying theorem 9.2 establishes that δ ∗ T ↔ iωV when T ↔ V . This result can again
be proven in a direct way since we know that δ ∗ T = T and T ↔ iωV when
T ↔ V (also see section 9.3.1).
Often, the conditions of theorem 9.2 are not satisfied in applications. If one
doesn't want to compromise on mathematical rigour, then one has to check case
by case whether or not the operations used (multiplication, convolution, Fourier
transform) are well-defined. The result S ∗ T ↔ U · V from theorem 9.2 will then
turn out to be valid in many more cases.
Finally, we expect on the basis of the reciprocity property that a convolution
theorem in the frequency domain exists as well (compare with section 7.2.4). We
will content ourselves with the statement that this is indeed the case: S · T ↔
(U ∗ V )/2π, if all operations occurring here are well-defined. This is the case
when, for example, U is a distribution that can be defined on the space E.
EXAMPLE 9.12
Take for S the function eiat , considered as a distribution. The spectrum U of S is
2πδ(ω − a) (see example 9.1) and this distribution can indeed be defined on the
space E. For an arbitrary distribution T with spectrum V it then follows from the
convolution theorem in the frequency domain that the spectrum of eiat T is equal to
(2πδ(ω − a) ∗ V )/2π . Hence, eiat T ↔ δ(ω − a) ∗ V when T ↔ V . Note that
according to the shift property in the frequency domain one has eiat T ↔ V (ω − a)
(see exercise 9.15). Apparently, δ(ω − a) ∗ V = V (ω − a).
226
9 The Fourier transform of distributions
EXERCISES
9.18
Show that δ (t − a) is an odd distribution.
9.19
Show that δ ∗ | t | = sgn t.
9.20∗
Let T (t) be a distribution and a ∈ R. Show that T (t) ∗ δ(t − a) exists and that
T (t) ∗ δ(t − a) = T (t − a) (see example 9.12).
9.21∗
Prove that δ(t − b) ∗ δ(t − a) = δ(t − (a + b)) (a, b ∈ R). Look what happens when
you apply the convolution theorem to this identity!
9.22∗
Let the piecewise smooth function f be equal to 0 outside a bounded interval; say
f (t) = 0 for | t | ≥ A (where A > 0 is a constant).
a Prove that the spectrum F of f exists (in the ordinary sense). Hint (which also
applies to parts b and c): the function f is bounded.
b Show that f defines a distribution by means of (8.12).
c Show that the distribution defined in part b is also well-defined on E.
d Let T be an arbitrary distribution with spectrum V . We simply write f and F
for the distributions defined by the functions f and F. Conclude from part c that
f ∗ T exists and that f ∗ T ↔ F · V .
e Determine the spectrum of p2 ∗ ( p2 is the block function, the unit step
function).
t
f Determine the spectrum of −∞ f (τ ) dτ = ( f ∗ )(t).
9.23∗
For functions the only solution to f (t) = 0 (for all t ∈ R) is the constant function
f (t) = c. In this exercise we assume that the same result is true for distributions:
the only distribution T with T = 0 is the distribution T = c (c a constant). Using
this assumption one can determine the spectrum U of the distribution pv(1/t) in a
mathematically rigorous way.
a Use the result t · pv(1/t) = 1 from exercise 8.25 to show that U satisfies the
equation U = −2πiδ(ω).
b Show that all solutions of the equation S = δ are necessarily of the form S =
+ c, where c is a constant. (Hint: when both S1 and S2 are solutions to S = δ,
what then will S1 − S2 satisfy?) Conclude that U (ω) = −2πi( (ω) + c).
c Conclude from exercise 8.27a that U is odd. Use this to determine the constant
c from part b and finally conclude that U (ω) = −πisgn ω.
SUMMARY
The Fourier transform F T of a distribution T is defined by F T, φ = T, ,
where is the Fourier transform of φ ∈ S. In this way the Fourier transform is
extended from functions to distributions. The most well-known Fourier pair for distributions is δ(t) ↔ 1 and 1 ↔ 2πδ(ω). More generally one has δ(t − a) ↔ e−iaω
and eiat ↔ 2πδ(ω − a). The latter result can be used to determine the spectrum, in the sense of distributions, of a periodic signal which defines a distribution
and, moreover, has a convergent Fourier series. Another important Fourier pair is
pv(1/t) ↔ −πisgn ω and sgn t ↔ −2ipv(1/ω), from which it follows in particular
that (t) ↔ πδ(ω) − ipv(1/ω). Finally, we mention the comb distribution or impulse train (t), whose spectrum is again a comb distribution, but scaled by 1/2π:
∞
(t) ↔ (ω/2π), or ∞
k=−∞ δ(t − k) ↔ 2π
k=−∞ δ(ω − 2πk).
Quite a few of the properties of the ordinary Fourier transform remain valid for
distributions. When U is the spectrum of T , then 2π T (ω) is the spectrum of U (−t)
(reciprocity). Furthermore, one has the differentiation properties T (k) ↔ (iω)k U
and (−it)k T ↔ U (k) , as well as the shift properties T (t − a) ↔ e−iaω U (ω) and
9.3 Convolution
227
eiat T (t) ↔ U (ω − a). Since the spectrum of (t) is known, the latter property can
be used to determine the spectrum of switched-on periodic signals.
Convolution theorems do not hold for arbitrary distributions. In general, multiplication as well as convolution of two distributions are not well-defined. When
S and T are distributions with spectra U and V , then the convolution theorems
S ∗ T ↔ U · V and S · T ↔ (U ∗ V )/2π are correct whenever all the operations occurring are well-defined. One has, for example, that δ (k) ∗ T (k ∈ Z+ )
is well-defined for each distribution T and that δ (k) ∗ T = T ∗ δ (k) = T (k) and
δ (k) ∗ T ↔ (iω)k U when T ↔ U . A list of some standard Fourier transforms
of distributions and a list of properties of the Fourier transform of distributions are
given in tables 5 and 6.
SELFTEST
9.24
Determine the spectra of the following distributions:
a δ(t − 3),
b cos t δ(t + 4),
c t 2 (t),
d (2 (t) cos t) ,
e (δ(7t − 1)) ,
2
−2 (2k+1)it .
f π −π ∞
k=−∞ (2k + 1) e
2
9.25
Find the distributions whose spectrum is given by the following distributions:
a δ(ω − 1) − δ(ω + 1),
b ω2 ,
c eiω/2 /4,
d ω3 sin ω,
e cos(ω − 4).
9.26
Consider the distribution U = (ω − 1)2 in the frequency domain.
a Use the shift property to determine a distribution whose spectrum is U .
b Use the differentiation property to determine a distribution whose spectrum is
the distribution ω2 − 2ω + 1.
c Note that U = ω2 − 2ω + 1; the answers in parts a and b should thus be the
same. Give a direct proof of this fact.
9.27
Let T be a distribution with spectrum U .
a Use the convolution theorem to determine the spectrum of T ∗ δ in terms of U .
Check the result by applying the differentiation property to the distribution T .
b The function | t | defines a distribution. Let V be the spectrum of | t |. Show that
δ ∗ | t | = 2δ and prove that V satisfies ω2 V = −2.
CHAPTER 10
Applications of the Fourier integral
INTRODUCTION
The Fourier transform is one of the most important tools in the study of the transfer
of signals in control and communication systems. In chapter 1 we have already discussed signals and systems in general terms. Now that we have the Fourier integral
available, and are familiar with the delta function and other distributions, we are
able to get a better understanding of the transfer of signals in linear time-invariant
systems. The Fourier integral plays an important role in continuous-time systems
which, moreover, are linear and time-invariant. These have been introduced in chapter 1 and will be denoted here by LTC-systems for short, just as in chapter 5.
Systems can be described by giving the relation between the input u(t) and the
corresponding output or response y(t). This can be done in several ways. For
example, by a description in the time domain (in such a description the variable
t occurs), or by a description in the frequency domain. The latter means that a
relation is given between the spectra (the Fourier transforms) U (ω) and Y (ω) of,
respectively, the input u(t) and the response y(t).
In section 10.1 we will see that for LTC-systems the relation between u(t) and
y(t) can be expressed in the time domain by means of a convolution product. Here
the response h(t) to the unit pulse, or delta function, δ(t) plays a central role. In fact,
the response y(t) to an input u(t) is equal to the convolution product of h(t), the socalled impulse response, and u(t). Hence, if the impulse response is known, then
the system is known in the sense that the response to an arbitrary input can be determined. Properties of LTC-systems, such as stability and causality, can then immediately be derived from the impulse response. Moreover, applying the convolution
theorem is now obvious and so the Fourier transform is going to play an important
role.
In section 10.2 we will see that the frequency response H (ω) of a system, which
we introduced in chapter 1, is nothing else but the Fourier transform of the impulse
response, and that a description of an LTC-system in the frequency domain is simply
given by Y (ω) = U (ω)H (ω), where U (ω) and Y (ω) are the spectra of, respectively,
the input u(t) and the corresponding output y(t). Properties of a continuous-time
system can then also be derived from the frequency response. For an all-pass system
or a phase shifter, for example, which is an LTC-system with the property that the
energy-content of the output is equal to the energy-content of the corresponding
input, the modulus of the frequency response is constant. Another example is the
ideal low-pass filter, which is characterized by a frequency response being 0 outside
a certain frequency band.
Continuous-time systems occurring in practice usually have a rational function
of ω as frequency response. Important examples are the electrical RCL-networks
having resistors, capacitors and coils as their components. The reason is that for
these systems the relation between an input and the corresponding response can
229
230
10 Applications of the Fourier integral
be described in the time domain by an ordinary differential equation with constant
coefficients. These will be considered in section 10.3.
Applications of the Fourier transform are certainly not limited to just the transfer of signals in systems. The Fourier integral can also be successfully applied to
all kinds of physical phenomena, such as heat conduction, which can be described
mathematically by a partial differential equation. These kinds of applications are
examined in section 10.4.
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know the concepts of impulse response and step response of an LTC-system and
can determine these in simple cases
- know the relation between an input and the corresponding output and can apply it
to calculate outputs
- can determine the stability and causality of an LTC-system using the impulse response
- know the concept of frequency response and can determine it in simple cases
- know the relation between the input and the output in the frequency domain and
can apply it to calculate outputs
- know what is meant by an all-pass system and an ideal low-pass filter
- can determine the impulse response of a stable and causal LTC-system described
by a linear differential equation with constant coefficients
- can apply the Fourier transform in solving partial differential equations with initial
and boundary conditions.
10.1
The impulse response
In order to introduce the impulse response, we start, as in chapter 1, with an example
of a simple continuous-time system, namely the electrical network from figure 1.1
in chapter 1, consisting of a series connection of a voltage source, a resistor and a
coil. The relation between the voltage and the current in this network is described by
(1.1), which we will recall here. We denote the voltage by u(t), however, since we
will consider it as an input, and the current by y(t), being the corresponding output.
Relation (1.1) mentioned above then reads as follows:
y(t) =
LTC-system
1 t
e−(t−τ )R/L u(τ ) dτ.
L −∞
(10.1)
In chapter 1 we have seen that this relation allows us to consider the network as
a continuous-time system which is linear and time-invariant. We recall that we
denote this by LTC-system for short. First we will show that relation (10.1) between
the input u(t) and the corresponding output y(t) can be written as a convolution
product. For this we utilize the unit step function (t). Since (t − τ ) = 0 for τ > t,
it follows that
t
−∞
e−(t−τ )R/L u(τ ) dτ =
∞
−∞
e−(t−τ )R/L (t − τ )u(τ ) dτ.
From definition 6.4 of the convolution product we then obtain that
y(t) = (h ∗ u)(t),
where h(t) is the signal
h(t) =
1 −t R/L
(t).
e
L
10.1 The impulse response
231
If we now take as input the unit pulse or delta function δ(t), then it follows from
(9.19) that h(t) = (h ∗ δ)(t) and so we can view h(t) as the output corresponding
to the unit pulse. We will show that for LTC-systems in general, the response to the
unit pulse plays the same important role. To this end we first use property (9.19) to
write an input u(t) in the form
u(t) = (u ∗ δ)(t) =
Superposition
Superposition rule
−∞
u(τ )δ(t − τ ) dτ.
We say that the signal u(t) is now written as a (continuous) superposition of shifted
unit pulses δ(t − τ ). Now if h(t) is the response to δ(t), then, by the time-invariance
of the system, the response to δ(t − τ ) is h(t − τ ). Next, we tacitly assume that for
LTC-systems the linearity property can be extended to the so-called superposition
rule. By this we mean that the linearity property not only holds for finite sums of
inputs, but also for infinite sums of inputs, and even for 'continuous sums', that is,
for integrals. Applying this superposition rule gives for a system L:
y(t) = Lu(t) = L
=
Impulse response
∞
∞
−∞
∞
−∞
u(τ )δ(t − τ ) dτ =
∞
−∞
u(τ )Lδ(t − τ ) dτ
u(τ )h(t − τ ) dτ.
The signal h(t) is called the impulse response of the system L. Hence,
δ(t) → h(t).
(10.2)
We have now established the following important property for LTC-systems.
Let u(t) be an input of an LTC-system L with impulse response h(t). Then
Lu(t) = (h ∗ u)(t).
EXAMPLE 10.1
Distortion free system
(10.3)
A continuous-time system is called distortion free when the response y(t) to an
input u(t) is given by
y(t) = K u(t − t0 ),
where K and t0 are constants with K ≥ 0. Compared to the input, the response is
of the same form and is shifted over a time t0 . This system is an LTC-system. The
linearity can easily be verified. The time-invariance follows from
u(t − t1 ) → K u(t − t1 − t0 ) = y(t − t1 )
for all t1 .
Also note that the given system is causal if t0 ≥ 0. The impulse response is the
response to δ(t) and thus equal to h(t) = K δ(t − t0 ). According to (10.3) the
response y(t) to an input u(t) is then given by
y(t) = (h ∗ u)(t) = K
∞
−∞
u(τ )δ(t − t0 − τ ) dτ = K u(t − t0 ).
By (10.3), an LTC-system is completely determined by the impulse response.
Hence, all properties of an LTC-system can be derived from the impulse response.
For instance, an LTC-system is real if and only if the impulse response is real (see
exercise 10.1b) and an LTC-system is causal if and only if the impulse response is
causal (see exercise 10.1a). Another property, important for the physically realizable
systems, is stability as defined in definition 1.3. Using the impulse response one can
verify this property in the following way.
232
10 Applications of the Fourier integral
THEOREM 10.1
An LTC-system with impulse response h(t) (h(t) being an ordinary function) is sta∞
ble if and only if −∞ | h(t) | dt is convergent, in other words, if h(t) is absolutely
integrable.
Proof
∞
Let −∞ | h(t) | dt be convergent and let the input u(t) be bounded, that is to say,
there exists a number M such that | u(t) | ≤ M for all t. For the corresponding
response it then follows from (10.3) that
∞
| y(t) | =
≤ M
u(τ )h(t − τ ) dτ ≤
−∞
∞
−∞
∞
−∞
| u(τ ) | | h(t − τ ) | dτ
| h(τ ) | dτ.
The output is thus bounded and so the system is stable.
∞
Now let −∞ | h(t) | dt be divergent. Take as input
h(−t)
for h(−t) = 0,
| h(−t) |
u(t) =
0
for h(−t) = 0.
This signal is bounded: | u(t) | ≤ 1. Using (10.3) we find for the corresponding
response y(t) at t = 0:
y(0) =
∞
−∞
u(τ )h(−τ ) dτ =
∞
−∞
| h(τ ) | dτ = ∞.
The response is not bounded and so the system isn't stable.
Strictly speaking, we cannot apply this theorem when instead of an ordinary function h(t) is a distribution, so for example when h(t) = δ(t). In fact, for a distribution
the notion 'absolute integrability' has no meaning. Consider for example the distortion free system from example 10.1. The impulse response is the distribution
K δ(t − t0 ). Still, this system is stable, as follows immediately from definition 1.3
of stability. For if | u(t) | ≤ M for all t and a certain M, then one has for the corresponding response that | y(t) | = K | u(t − t0 ) | ≤ K M, so the response is bounded
as well.
For systems occurring in practice the impulse response will usually consist of
an ordinary function with possibly a finite number of shifted delta functions added
to this. In order to verify the stability for these cases, one only needs to check
the absolute integrability of the ordinary function. We will demonstrate this in the
following example.
EXAMPLE 10.2
A system L is given for which the relation between an input u(t) and the response
y(t) is given by
y(t) = u(t − 1) +
t
−∞
e−2(t−τ ) u(τ ) dτ.
In exercise 10.2 the reader is asked to show that L is an LTC-system. The impulse
response can be found by substituting the unit pulse δ(t) for u(t), resulting in h(t) =
δ(t − 1) + r (t), where r (t) = e−2t (t). The response y(t) to an input u(t) can be
written as y(t) = u(t − 1) + y1 (t), where y1 (t) is the response to u(t) of an LTCsystem with impulse response h 1 (t) = r (t). Since
∞
−∞
| h 1 (t) | dt =
∞
0
e−2t dt = 1 < ∞,
2
10.1 The impulse response
233
the system is stable according to theorem 10.1. So if | u(t) | ≤ M, then | y1 (t) | ≤ L
for some L and we thus have
| y(t) | ≤ | u(t − 1) | + L ≤ M + L .
This establishes the stability of the LTC-system.
EXAMPLE 10.3
Integrator
An LTC-system for which the relation between an input u(t) and the corresponding
output y(t) is given by
y(t) =
t
−∞
u(τ ) dτ
is called an integrator. Since
t
−∞
δ(τ ) dτ = (t),
the impulse response of the integrator is equal to the unit step function (t). According to property (10.3) we thus have y(t) = ( ∗ u)(t). The unit step function (t) is
not absolutely integrable. Hence, it follows from theorem 10.1 that the integrator is
unstable.
We close this section with the introduction of the so-called step response a(t) of
an LTC-system. The step response is defined as the response to the unit step function
(t), so
Step response
(t) → a(t).
(10.4)
From property (10.3) it follows that a(t) = ( ∗ h)(t), where h(t) is the impulse
response of the system. Convolution with the unit step function is the same as
integration (see example 10.3 of the integrator), so
a(t) =
t
−∞
h(τ ) dτ.
(10.5)
This relation implies that h(t) is the derivative of a(t), but not in the ordinary sense.
Thanks to the introduction of the distributional derivative, we can say that h(t) is the
distributional derivative of a(t): a (t) = h(t). Apparently, the impulse response follows easily once the step response is known and using (10.3) one can then determine
the response to an arbitrary input.
EXERCISES
10.1
Given is an LTC-system L.
a Show that the system L is causal if and only if the impulse response h(t) of L is
a causal signal.
b Show that the system L is real if and only if the impulse response h(t) of L is a
real signal.
10.2
The relation between an input u(t) and the corresponding response y(t) of a system
L is given by
y(t) = u(t − 1) +
a
b
t
−∞
e−2(t−τ ) u(τ ) dτ.
Show that L is a real and causal LTC-system.
Determine the step response of the system.
234
10 Applications of the Fourier integral
10.3
Given is an LTC-system L with step response a(t). Determine the response to the
rectangular pulse p2 (t).
10.4
Given is a stable LTC-system with impulse response h(t).
a Show that the response to the constant input u(t) = 1 is given by the constant
output y(t) = H (0) for all t, where
H (0) =
∞
−∞
h(τ ) dτ.
b Let u(t) be a differentiable input with absolutely integrable derivative u (t) and
for which u(−∞), defined by limt→−∞ u(t), exists. Show that u(t) = u(−∞) +
(u ∗ )(t).
c Show that for the response y(t) to u(t) one has y(t) = H (0)u(−∞)+(u ∗a)(t).
10.5
For an LTC-system the step response a(t) is given by a(t) = cos(2t)e−3t (t).
a Determine the impulse response h(t).
b Show that the system is stable.
10.6
Given are the LTC-systems L1 and L2 in a series connection. We denote this socalled cascade system by L2 L1 . The response y(t) to an input u(t) is obtained as
indicated in figure 10.1.
u (t)
y (t)
L1
L2
FIGURE 10.1
Cascade system of exercise 10.6.
a Give an expression for the impulse response h(t) of L2 L1 in terms of the impulse
responses h 1 (t) and h 2 (t) of, respectively, L1 and L2 (t) .
b Show that if both L1 and L2 are stable, then L2 L1 is also stable.
10.2
The frequency response
In the previous section we saw that for an LTC-system the relation between an input and the corresponding output is given in the time domain by the convolution
product (10.3). In this section we study the relation in the frequency domain. It
is quite natural to apply the convolution theorem to (10.3). Since we have to take
into account that delta functions may occur in u(t) as well as in h(t), we will need
the convolution theorem that is also valid for distributions. We will assume that for
the LTC-systems and inputs under consideration, the convolution theorem may always be applied in distributional sense as well. As a result we obtain the following
important theorem.
THEOREM 10.2
Let U (ω), Y (ω) and H (ω) denote the Fourier transforms of, respectively, an input
u(t), the corresponding output y(t), and the impulse response h(t) of an LTC-system.
Then
Y (ω) = H (ω)U (ω).
(10.6)
Property (10.6) describes how an LTC-system operates in the frequency domain.
The spectrum of an input is multiplied by the function H (ω), resulting in the spectrum of the output.
10.2 The frequency response
EXAMPLE 10.4
235
The impulse response of the integrator (see example 10.3) is the unit step function
(t): h(t) = (t). The Fourier transform of h(t) is 1/(iω) + πδ(ω) (see table 5;
−ipv(1/ω) is written as 1/(iω) for short). Hence, according to (10.6) the integrator
is described in the frequency domain by
Y (ω) = (1/(iω) + πδ(ω)) U (ω) = U (ω)/(iω) + πU (ω)δ(ω)
= U (ω)/(iω) + πU (0)δ(ω).
In the last equality we assumed that U (ω) is continuous at ω = 0.
In this example of the integrator we see that substituting ω = 0 is meaningless.
The reason for this is the instability of the integrator. For stable systems the impulse
response h(t) is absolutely integrable according to theorem 10.1 and H (ω) will then
be an ordinary function, and even a continuous one, defined for all ω (see section
6.4.11).
Now consider in particular the time-harmonic signal u(t) = eiωt with frequency
ω as input for an LTC-system. According to property (10.3) the response is equal to
y(t) = (h ∗ u)(t) =
∞
−∞
h(τ )eiω(t−τ ) dτ = H (ω)eiωt .
Hence,
eiωt → H (ω)eiωt .
(10.7)
This is nothing new. In chapter 1 we have derived that the response of an LTC-system
to a time-harmonic signal is again a time-harmonic signal with the same frequency.
In this chapter the frequency response H (ω) has been introduced through property
(10.7). We conclude that the following theorem holds.
THEOREM 10.3
Frequency response
The frequency response H (ω) of an LTC-system is the spectrum of the impulse response:
h(t) → H (ω).
EXAMPLE 10.5
(10.8)
For an LTC-system the impulse response is given by h(t) = e−t (t). The system is
stable since
∞
−∞
| h(t) | dt =
∞
0
e−t dt = 1 < ∞.
The Fourier transform of h(t) can be found in table 3: H (ω) = 1/(1 + iω). The
response to the input eiωt is thus equal to eiωt /(1 + iω).
Transfer function
System function
From chapter 1 we know that the frequency response H (ω) is also called the
transfer function or the system function of an LTC-system. Apparently, an LTCsystem can be described in the frequency domain by the frequency response H (ω).
EXAMPLE 10.6
Ideal low-pass filter
As an example we consider the so-called ideal low-pass filter with frequency response H (ω) given by
H (ω) =
e−iωt0
0
for | ω | ≤ ωc ,
for | ω | > ωc .
Hence, only frequencies below the cut-off frequency ωc can pass. By an inverse
Fourier transform we find the impulse response of the filter:
h(t) =
∞
ωc
1
1
sin(ωc (t − t0 ))
.
H (ω)eiωt dω =
eiω(t−t0 ) dω =
2π −∞
2π −ωc
π(t − t0 )
236
10 Applications of the Fourier integral
The function h(t) is shown in figure 10.2. The impulse response h(t) has a maximum value ωc /π at t = t0 ; the main pulse of the response is concentrated at t = t0
and has duration 2π/ωc . Note that h(t) is not causal, which means that the system
is not causal. The step response of the filter follows from integration of h(t):
a(t) =
t
−∞
h(τ ) dτ =
1 ωc (t−t0 ) sin x
sin(ωc (τ − t0 ))
dτ =
d x,
π(τ − t0 )
π −∞
x
−∞
t
where we used the substitution ωc (τ − t0 ) = x. Using the sine integral (see chapter
4), the step response can be written as
a(t) =
1
1
+ Si(ωc (t − t0 )).
2 π
Note that a(t0 ) = 1 and that the initial and final values a(−∞) = 0 and a(∞) = 1
2
are approached in an oscillating way (see figure 10.2b). The maximal overshoot
occurs at t = t0 + π/ωc and amounts to 9% (compare this with Gibbs' phenomenon
in chapter 4). In this example we will also determine the response to a periodic
a
b
ωc
π
1.09
1
h (t)
t0
0
a (t)
0.5
t
0
t0
t
π π
ωc ωc
FIGURE 10.2
Impulse response (a) and step response (b) of ideal low-pass filter.
signal u(t) given by the Fourier series
u(t) =
∞
cn einω0 t ,
n=−∞
where ω0 = 2π/T . According to property (10.7), the response to einω0 t equals
H (nω0 )einω0 t =
einω0 (t−t0 )
0
for | nω0 | ≤ ωc ,
for | nω0 | > ωc .
Here we assume that ωc is not an integer multiple of ω0 . Let N be the integer
uniquely determined by N ω0 < ωc < (N + 1)ω0 . For the response y(t) to the
periodic input u(t) it then follows that
N
y(t) =
cn einω0 (t−t0 ) = u N (t − t0 ),
n=−N
where u N (t) denotes the N th partial sum of the Fourier series of u(t).
The frequency response can also show us how the energy-content (see section
1.2.3) of an input is effected by an LTC-system. When an energy-signal u(t) is
10.2 The frequency response
237
applied to an LTC-system, then by Parseval's identity (see (7.19)) one has for the
energy-content of the corresponding output y(t) that
∞
−∞
| y(t) |2 dt =
∞
∞
1
1
| Y (ω) |2 dω =
| H (ω) |2 | U (ω) |2 dω.
2π −∞
2π −∞
All-pass system
Note that for LTC-systems whose amplitude response | H (ω) | is identical to 1, the
energy-content of the output equals the energy-content of the input. Such a system
is called an all-pass system.
EXAMPLE 10.7
For an LTC-system the frequency response H (ω) is given by
H (ω) =
ω − i −iωt0
e
.
1 + iω
Here t0 is real. Since
ω2 + 1
= 1,
1 + ω2
ω−i
ω − i −iωt0
e
=
=
1 + iω
1 + iω
the system is an all-pass system.
In this section we have established the importance of the frequency response for
LTC-systems. An important class of LTC-systems in practical applications has the
property that the frequency response is a rational function of ω. Examples are the
electrical networks. In the next section we will examine these in more detail.
EXERCISES
10.7
For an LTC-system L the impulse response is given by h(t) = δ(t) + te−t (t).
a Determine the frequency response of the system L.
b Determine for all real ω the response of the LTC-system to the input u(t) = eiωt .
10.8
For an LTC-system L the frequency response is given by
H (ω) =
a
b
10.9
cos ω
.
ω2 + 1
Determine the impulse response h(t) of the system L.
Determine the response to the input u(t) = δ(t − 1).
For a low-pass filter the frequency response H (ω) is given by the graph of
figure 10.3.
1
H (ω)
–ω c =
3
2
0
ωc =
FIGURE 10.3
Frequency response of the low-pass filter of exercise 10.9.
3
2
ω
238
10 Applications of the Fourier integral
u (t)
1
0
π/2
π
3π/2
2π
t
2ω c
ω
FIGURE 10.4
Periodic signal u(t) of exercise 10.9.
H (ω)
1
–2ωc
–ω c
0
ωc
FIGURE 10.5
Band-pass filter of exercise 10.10.
a Determine the impulse response of the filter.
b To the filter we apply a periodic signal u(t) with period 2π which, on the time
interval [0, 2π), is given by the graph of figure 10.4. Find the response to this
periodic signal u(t).
10.10
For an ideal band-pass filter the frequency response is given by the graph of
figure 10.5.
a Determine the impulse response of the filter.
b Use the sine integral to determine the step response of the filter.
10.11
For an LTC-system the frequency response H (ω) is given by
Band-pass filter
H (ω) =
iω + 1 iω − 2
·
.
iω − 1 iω + 2
a Determine the impulse response of the system.
b Is the system causal? Justify your answer.
c To the system we apply a signal u(t) with a finite energy-content. Show that
the energy-content of the response y(t) to u(t) is equal to the energy-content of the
input u(t).
10.12
Band-limited signal
An LTC-system with frequency response H (ω) is given. To the system we apply
a so-called band-limited signal u(t). This means that the spectrum U (ω) satisfies
U (ω) = 0 for | ω | ≥ ωc for some ωc > 0.
a Show that the output y(t) is also band-limited.
10.3 Causal stable systems
239
b For the band-limited signal the values u(nT ) for n ∈ Z are given, where T is
such that ωs = 2π/T > 2ωc . In chapter 15 we will derive that this signal can then
be written as
∞
u(t) =
u(nT )
2 sin( 1 ωs (t − nT ))
2
n=−∞
ωs (t − nT )
.
Show that for the response y(t) one has
∞
y(t) =
u(nT )h a (t − nT ),
n=−∞
where h a (t) is the signal given by
h a (t) =
10.3
ωs /2
1
H (ω)eiωt dω.
ωs −ωs /2
Causal stable systems and differential equations
An example of an LTC-system, occurring quite often in electronics, is an electric
network consisting of linear elements: resistors, capacitors and inductors, whose
properties should not vary in time (time-invariance). For these systems one can
often derive a differential equation, from which the frequency response can then be
determined quite easily (see theorem 5.2). From this, the impulse response can be
determined by using the inverse Fourier transform, and subsequently one obtains the
response to any input by a convolution. Let us start with an example.
EXAMPLE 10.8
In figure 10.6 a series connection is drawn, consisting of a voltage source, a resistor
with resistance R and a capacitor with capacitance C. This circuit or network can
be considered as a causal LTC-system with input the voltage u(t) drop across the
voltage source and with output the voltage drop y(t) across the capacitor. We will
i (t)
+
R
+
u (t)
C
y (t)
–
–
FIGURE 10.6
An RC-network.
now determine the impulse response. The frequency response can be obtained as
follows. Let i(t) be the current in the network, then it follows from Kirchhoff's
voltage law and Ohm's law that
u(t) = Ri(t) + y(t).
The voltage–current relation for the capacitor is as follows:
y(t) =
1 t
i(τ ) dτ.
C −∞
240
10 Applications of the Fourier integral
By differentiation it follows that the relation between u(t) and y(t) can be described
by the following differential equation:
RC
dy
+ y(t) = u(t).
dt
From theorem 5.2 it then follows that the frequency response H (ω) is equal to
H (ω) =
1
.
1 + iω RC
Finally, the impulse response is obtained by an inverse Fourier transform. From
table 3 we conclude that
h(t) =
1 −t/RC
e
(t).
RC
By (10.3), the response to an arbitrary input u(t) then equals
y(t) = (u ∗ h)(t) =
t
1
u(τ )e−(t−τ )/RC dτ.
RC −∞
In this section we will consider causal and stable LTC-systems described in the
time domain by an ordinary differential equation with constant coefficients. Examples are the electrical networks with resistors, capacitors and inductors, the so-called
RLC-networks, having one source (a voltage or current source); see also chapter 5.
Using Kirchhoff's voltage law and the voltage–current relations for the resistor, capacitor and inductor, one can derive that the relation between the input u(t) and
the response y(t) in such networks can always be described by a linear differential
equation with constant coefficients of the form
am
dm y
d m−1 y
dy
+ a0 y
+ am−1 m−1 + · · · + a1
dt m
dt
dt
dnu
d n−1 u
du
= bn n + bn−1 n−1 + · · · + b1
+ b0 u
dt
dt
dt
(10.9)
with n ≤ m. Causal and stable systems described by (10.9) are of practical importance because they can physically be realized, for example as an electric network.
If we have a complete description of a system, then the response y(t) to a given
input u(t) must follow uniquely from the given description. For periodic signals
this problem has already been discussed in chapter 5, section 5.1. However, if u(t)
is known in (10.9) (and so the right-hand side is known), then we know from the
theory of ordinary differential equations that a solution y(t) still contains m (the
order of the differential equation) unconstrained parameters. Apparently, more data
from the output or the system are required to determine y(t) uniquely for a given
u(t). If, for example, y(t) and all the derivatives of y(t) up to order m −1 are known
at t = 0, that is to say, if the initial values are known, then it follows from the theory
of ordinary differential equations that y(t) is uniquely determined for all t.
In this chapter we will assume that the differential equation (10.9) describes a
causal and stable LTC-system. The impulse response is then causal and, after removing delta functions that might occur (see example 10.2), absolutely integrable.
Using this, one is able to find a unique solution for the impulse response by substituting u(t) = δ(t) in (10.9). It is easier, though, to first calculate the frequency
response (see theorem 5.2). According to theorem 5.2 one can use the characteristic polynomial A(s) = am s m + am−1 s m−1 + · · · + a1 s + a0 and the polynomial
B(s) = bn s n + bn−1 s n−1 + · · · + b1 s + b0 to write the frequency response as
A(iω)H (ω)eiωt = B(iω)eiωt .
10.3 Causal stable systems
241
We now impose the condition that A(iω) = 0 for all real ω, which means that the
polynomial A(s) has no zeros on the imaginary axis. Dividing by A(iω)eiωt is then
permitted, which leads to the result
H (ω) =
bn (iω)n + bn−1 (iω)n−1 + · · · + b1 (iω) + b0
B(iω)
=
.
m +a
m−1 + · · · + a (iω) + a
A(iω)
am (iω)
m−1 (iω)
1
0
(10.10)
The frequency response is thus a rational function of ω. Because we assumed that
n ≤ m, the degree of the denominator is at most equal to the degree of the numerator.
The impulse response follows from the inverse Fourier transform of the frequency
response. To that end we apply the partial fraction expansion technique, explained
in chapter 2, to the rational function B(s)/A(s). Let s1 , s2 , . . . , sm be the zeros of
the polynomial A(s) and assume, for convenience, that these zeros are simple. Since
n ≤ m, the partial fraction expansion leads to the representation
m
H (ω) = c0 +
ck
,
iω − sk
k=1
where c0 , c1 , . . . , cm are certain constants. Inverse transformation of c0 gives the
signal c0 δ(t). Inverse transformation for Re sk < 0 gives (see table 3)
esk t (t) ↔
1
,
iω − sk
while for Re sk > 0 it gives (use time reversal)
−esk t (−t) ↔
1
.
iω − sk
Finally, when Re sk = 0, so sk = iω0 for some ω0 , then
1
1 iω0 t
sgn t ↔
e
.
2
i(ω − ω0 )
We have assumed that the system is causal and stable. This then implies that for
k = 1, 2, . . . , m the zeros sk must satisfy Re sk < 0. Apparently, the zeros of A(s)
lie in the left-half plane Re s < 0 of the complex s-plane. The impulse response
h(t) of the LTC-system then looks as follows:
m
h(t) = c0 δ(t) +
ck esk t (t).
k=1
We conclude that for a description of a causal and stable system by means of a
differential equation of type (10.9), the zeros of A(s) must lie in the left-half plane
of the complex s-plane. Hence, there should also be no zeros on the imaginary axis.
We formulate this result in the following theorem.
THEOREM 10.4
When an LTC-system is described by an ordinary differential equation of type (10.9),
then the system is causal and stable if and only if the zeros s of the characteristic
polynomial satisfy Re s < 0.
EXAMPLE 10.9
Consider the RC-network from figure 10.7. The input is the voltage drop u(t) across
the voltage source and the output is the voltage drop y(t) between nodes A and B
of the network. The network is considered as an LTC-system. The relation between
u(t) and y(t) is described by the differential equation
RC
dy
du
+ y = −RC
+ u.
dt
dt
10.4 Partial differential equations
243
+
R
+
R
u (t)
y (t)
–
C
–
FIGURE 10.9
RC-network of exercise 10.14.
a
b
c
10.14
Determine the frequency response H (ω).
Determine the response to u(t) = cos(ω0 t).
Determine the response to u(t) = cos(ω0 t) (t).
The RC-network from figure 10.9 is considered as an LTC-system with input the
voltage drop u(t) across the voltage source and output y(t) the voltage drop across
the resistor and the capacitor. The relation between u(t) and y(t) is given by the
differential equation
2
dy
1
du
1
+
y=
+ 1+
dt
RC
dt
RC
u.
a Determine the frequency response and the impulse response of the system.
b Consider the so-called inverse system, which takes y(t) as input and u(t) as
response. Determine the transfer function H1 (ω) and the impulse response h 1 (t) of
the inverse system.
c Determine (h ∗ h 1 )(t).
10.15
For an LTC-system the impulse response h(t) is given by h(t) = t n e−at (t). Here
n is a non-negative integer and a a complex number with Re a > 0. Show that the
system is stable.
10.4
Boundary and initial value problems for partial differential
equations
In the previous sections it has been shown that the Fourier transform is an important
tool in the study of the transfer of continuous-time signals in LTC-systems. We
applied the Fourier transform to signals f (t) being functions of the time t, and
the Fourier transform F(ω) could then be interpreted as a function defined in the
frequency domain. However, applications of the Fourier transform are not restricted
to continuous-time signals only. For instance, one can sometimes apply the Fourier
transform successfully in order to solve boundary and initial value problems for
partial differential equations with constant coefficients. In this section an example
will be presented. The techniques that we will use are the same as in section 5.2 of
chapter 5. By a separation of variables one first determines a class of functions that
satisfy the differential equation as well as the linear homogeneous conditions. From
this set of functions one subsequently determines by superposition a solution which
also satisfies the inhomogeneous conditions.
244
10 Applications of the Fourier integral
Again we take the heat equation from chapter 5 as an example, and we will also
use the notations introduced in that chapter:
u t = ku x x
for x ∈ R and t > 0.
(10.11)
The function u(x, t) describes the heat distribution in a cylinder shaped rod at the
point x in the longitudinal direction and at time t. Now consider the conditions
u(x, 0) = f (x) for x ∈ R and t > 0,
u(x, t) is bounded.
This means that we assume that the rod has infinite length, that the heat distribution
at time t = 0 is known, and that we are only interested in a bounded solution. A
linear homogeneous condition is characterized by the fact that a linear combination
of functions that satisfy the condition, also satisfies that condition. In our case this
is the boundedness condition. Verify this for yourself.
Separation of variables gives u(x, t) = X (x)T (t), where X is a function of x
only and T is a function of t only. In section 5.2 of chapter 5 we derived that
this u(x, t) satisfies the given heat equation if for some arbitrary constant c (the
separation constant) one has
X − cX = 0,
T − ckT = 0.
In order to satisfy the linear homogeneous condition as well, X (x)T (t) has to be
bounded, and this implies that both X (x) and T (t) have to be bounded functions.
From the differential equation for T (t) it follows that T (t) = αeckt for some α.
And since T (t) has to be bounded for t > 0, the constant c has to satisfy c ≤ 0.
(Unless α = 0, but then we obtain the trivial solution T (t) = 0, which is of no
interest to us.) We therefore put c = −ω2 , where ω is a real number.
For ω = 0 the differential equation for X (x) has as general solution X (x) =
αx + β. The boundedness of X (x) then implies that α = 0. For ω = 0 the
differential equation has as general solution X (x) = αeiωx +βe−iωx . This function
is bounded for all α and β since
| X (x) | ≤ | α | eiωx + | β | e−iωx = | α | + | β | .
From the above it follows that the class of functions we are looking for, that is,
satisfying the differential equation and being bounded, can be described by
2
X (x)T (t) = eiωx e−kω t ,
where ω ∈ R.
Now the final step is to construct, by superposition of functions from this class,
a solution satisfying the inhomogeneous condition as well. This means that for a
certain function F(ω) we will try a solution u(x, t) of the form
u(x, t) =
∞
−∞
2
F(ω)e−kω t eiωx dω.
If we substitute t = 0 in this integral representation, then we obtain that
f (x) =
∞
−∞
F(ω)eiωx dω.
We can now apply the theory of the Fourier transform. If we interpret ω as a frequency, then this integral shows that, up to a factor 2π , f (x) equals the inverse
10.4 Partial differential equations
245
Fourier transform of F(ω). Hence, F(ω) is the Fourier transform of f (x), up to a
factor 2π:
F(ω) =
∞
1
f (x)e−iωx d x.
2π −∞
We have thus found a solution of the heat conduction problem for the infinite rod,
however, without worrying about convergence problems. In fact one should verify
afterwards that the u(x, t) we have found is indeed a solution. We will omit this
and express this by saying that we have determined a formal solution. When, for
example, f (x) = 1/(1 + x 2 ), then it follows from table 3 that F(ω) = 1 e−| ω | and
2
so a formal solution is given by
Formal solution
u(x, t) = 1
2
∞
−∞
2
e−| ω | e−kω t eiωx dω.
EXERCISES
10.16
Let f (x) = 1/(1 + x 2 ). Determine formally the bounded solution of the following
problem from potential theory:
u x x + u yy = 0
u(x, 0) = f (x)
10.17
for −∞ < x < ∞ and y > 0,
for −∞ < x < ∞.
Determine formally the bounded solution T (x, t) of the heat conduction equation
Tx x = Tt
for −∞ < x < ∞ and t > 0
with initial condition
T (x, 0) =
T1
T2
for x ≥ 0,
for x < 0.
SUMMARY
The Fourier transform is an important tool in the study of linear time-invariant
continuous-time systems (LTC-systems). These systems possess the important property that the relation between an input u(t) and the corresponding output y(t) is
given in the time domain by means of the convolution product
y(t) = (h ∗ u)(t),
where h(t) is the response of the LTC-system to the delta function or unit pulse
δ(t). The signal h(t) is called the impulse response. An LTC-system is completely
determined when the impulse response is known. Properties of an LTC-system can
be derived from the impulse response. For example, an LTC-system is stable if the
impulse response, ignoring possible delta functions, is absolutely integrable.
The step response a(t) is defined as the response to the unit step function (t).
The derivative of the step response as distribution is equal to the impulse response.
The frequency response H (ω), introduced in chapter 1, turned out to be equal to
the Fourier transform of the impulse response h(t). The frequency response has the
special property that the LTC-system can be described in the frequency domain by
Y (ω) = H (ω)U (ω).
246
10 Applications of the Fourier integral
Here U (ω) and Y (ω) are the spectra of, respectively, the input u(t) and the corresponding output y(t). Hence, an LTC-system is known when H (ω) is known: in
principle the response y(t) to any input u(t) can then be determined.
An ideal low-pass filter is characterized as an LTC-system for which H (ω) = 0
outside the pass-band −ωc < ω < ωc . An LTC-system for which H (ω) = 1 for all
ω is called an all-pass system. An all-pass system has the property that the energycontent of the output equals the energy-content of the corresponding input.
For practical applications the causal and stable systems are important, for which
the relationship between u(t) and y(t) can be described by an ordinary differential
equation= bn n + bn−1 n−1 + · · · + b1
+ b0 u.
dt
dt
dt
Examples are the electrical networks consisting of resistors, capacitors and inductors, whose physical properties should be time-independent. The frequency response is a rational function. Stability and causality can be established by looking
at the location of the zeros of the denominator of the frequency response. These
should lie in the left-half plane of the complex plane.
The Fourier transform can also successfully be applied to functions depending on
a position variable. Particularly for boundary and initial value problems for linear
partial differential equations, the Fourier transform can be a valuable tool.
SELFTEST
10.18
A system is described by
y(t) =
a
b
c
d
t
t−1
e−(t−τ ) u(τ ) dτ.
Determine the frequency response and the impulse response.
Is the system causal? Justify your answer.
Is the system stable? Justify your answer.
Determine the response to the block function p2 (t).
10.19
For an LTC-system the step response a(t) is given by a(t) = e−t (t).
a Determine the impulse response.
b Determine the frequency response.
c Determine the response to the input u(t) = e−t (t).
10.20
The frequency response of an ideal low-pass filter is given by
H (ω) =
1
0
for | ω | ≤ ωc ,
for | ω | > ωc .
Determine the response to the periodic input u(t) with period T = 5π/ωc given by
u(t) = t for 0 ≤ t < T .
10.21
For an LTC-system the frequency response H (ω) is given by
1 − | ω | for | ω | ≤ ω ,
c
ωc
H (ω) =
0
for | ω | > ωc .
Part 4
Laplace transforms
INTRODUCTION TO PART 4
In the previous two parts we considered various forms of Fourier analysis: for periodic functions in part 2 and for non-periodic functions and distributions in part 3. In
this part we examine the so-called Laplace transform. On the one hand it is closely
related to the Fourier transform of non-periodic functions, but on the other hand it
is more suitable in certain applied fields, in particular in signal theory. In physical
reality we usually study signals that have been switched on at a certain moment in
time. One then chooses this switch-on time as the origin of the time-scale. Hence,
in such a situation we are dealing with functions on R which are zero for t < 0,
the so-called causal functions (see section 1.2.4). The Fourier transform of such a
function f is then given by
F(ω) =
∞
0
f (t)e−iωt dt,
where ω ∈ R. A disadvantage of this integral is the fact that, even for very simple
functions, it often does not exist. For the unit step function (t) for example, the
integral does not exist and in order to determine the spectrum of (t) we had to
resort to distribution theory. If we multiply (t) by a 'damping factor' e−σ t for
an arbitrary σ > 0, then the spectrum will exist (see section 6.3.3). It turns out
that this is true more generally: when f (t) is a function that is zero for t < 0
and whose spectrum does not exist, then there is a fair chance that the spectrum of
g(t) = e−σ t f (t) does exist (under certain conditions on σ ∈ R). Determining the
spectrum of g(t) boils down to the calculation of the integral
∞
0
f (t)e−(σ +iω)t dt
for arbitrary real σ and ω. The result will be a new function, denoted by F again,
which no longer depends on ω ∈ R, but on σ + iω ∈ C. Hence, if we write
s = σ +iω, then this assigns to any causal function f (t) a function F(s) by defining
F(s) =
∞
0
f (t)e−st dt.
The function F(s) is called the Laplace transform of the causal function f (t) and
the mapping assigning the function F(s) to f (t) is called the Laplace transform.
When studying phenomena where one has to deal with switched-on signals, the
Laplace transform is often given preference over the Fourier transform. In fact, the
Laplace transform has a better way 'to deal with the switch-on time t = 0'. Another
advantage of the Laplace transform is the fact that we do not need distributions very
often, since the Laplace transform of 'most' functions exists as an ordinary integral.
For most applications it therefore suffices to use only a very limited part of the
distribution theory.
250
Another noticeable difference with the Fourier analysis from parts 2 and 3 is the
role of the fundamental theorem. Although the fundamental theorem of the Laplace
transform can easily be derived from the one for the Fourier integral, it will play
an insignificant role in part 4. In order to recover a function f (t) from its Laplace
transform F(s) we will instead use a table, the properties of the Laplace transform
and partial fraction expansions.
To really understand the fundamental theorem of the Laplace transform would
require an extensive treatment of the theory of functions from C to C. These functions are called complex functions and the Laplace transform is indeed a complex
function: to s ∈ C the number F(s) ∈ C is assigned (we recall that, in contrast,
the Fourier transform is a function from R to C). For a rigorous treatment of the
Laplace transform at least some knowledge of complex functions is certainly necessary. We therefore start part 4 with a short introduction to this subject in chapter 11.
A thorough study of complex functions, necessary in order to use the fundamental
theorem of the Laplace transform in its full strength, lies beyond the scope of this
book.
Following the brief introduction to complex functions in chapter 11, we continue
in chapter 12 with the definition of the Laplace transform of a causal function. A
number of standard Laplace transforms are calculated and some properties, most of
which will be familiar by now, are treated.
Chapter 13 starts with a familiar subject as well, namely convolution. However,
we subsequently treat a number of properties not seen before in Fourier analysis,
such as the so-called initial value and final value theorems. We also consider the
Laplace transform of distributions in chapter 13. Finally, the fundamental theorem
of the Laplace transform is proven and a method is treated to recover a function
f (t) from its Laplace transform F(s) by means of a partial fraction expansion. As
in parts 2 and 3, we apply the Laplace transform to the theory of linear systems and
(partial) differential equations in the final chapter 14.
Pierre Simon Laplace (1749 – 1827) lived and worked at the end of an epoch that
started with Newton, in which the study of the movement of the planets formed an
important stimulus for the development of mathematics. Theories developed in this
period were recorded by Laplace in the five-part opus M´ canique Celeste, which
e
he wrote during the years 1799 to 1825. The shape of the earth, the movements of
the planets, and the distortions in their orbits were described in it. Another major
work by Laplace, Th´ orie analytique des probabilit´ s, deals with the calculus of
e
e
probabilities. Both standard works do not only contain his own material, but also
that of his predecessors. Laplace, however, made all of this material coherent and
moreover wrote extensive introductions in non-technical terms.
Laplace's activities took place in a time where mathematicians were no longer
mainly employees of monarchs at courts, but instead were employed by universities
and technical institutes. Previously, mathematicians were given the opportunity to
work at courts, since enlightened monarchs were on the one hand pleased to have
famous scientists associated with their courts, and on the other hand because they
realized how useful mathematics and the natural sciences were for the improvement
of production processes and warfare. Mathematicians employed by universities and
institutes were also given significant teaching tasks. Laplace himself was professor
of mathematics at the Paris military school and was also a minister in Napoleon's
cabinet for some time. He considered mathematics mainly as a beautiful toolbox
which could benefit the progress of the natural sciences.
In Laplace's epoch the idea prevailed that mathematics was developed to such
an extent that all could be explained. Based on Newton's laws, numerous different
phenomena could be understood. This vision arose from the tendency to identify
251
mathematics mainly with astronomy and mechanics. It led Laplace to the following famous statement: "An intelligence which could oversee all forces acting in
nature at a specific moment and, moreover, all relative positions of all parts present
in nature, and which would also be sufficiently comprehensive to subject all these
data to a mathematical analysis, could in one and the same formula encompass the
movements of the largest bodies in the universe as well as that of the lightest atom:
nothing would remain uncertain for her, and the future as well as the past would
be open to her." Hence, any newly developed mathematics would at best be more
of the same. However, in the first decades of the nineteenth century mathematicians, such as Fourier, adopted a new course. In the twentieth century, the view
that mathematics could explain everything was thoroughly upset. First by quantum
mechanics, which proved that the observer always influences the observed object,
and subsequently by chaos theory, which proved that it is impossible to determine
the initial state of complex systems, such as the weather, sufficiently accurately to
be able to predict all future developments. Notwithstanding, the Laplace transforms
remain a very important tool for the analysis and further development of systems
and electronic networks.
CHAPTER 11
Complex functions
INTRODUCTION
In this chapter we give a brief introduction to the theory of complex functions. In
section 11.1 some well-known examples of complex functions are treated, in particular functions that play a role in the Laplace transform. In sections 11.2 and 11.3
continuity and differentiability of complex functions are examined. It will turn out
that both the definition and the rules for continuity and differentiability are almost
exactly the same as for real functions. Still, complex differentiability is surprisingly different from real differentiability. In the final section we will briefly go into
this matter and treat the so-called Cauchy–Riemann equations. The more profound
properties of complex functions cannot be treated in the context of this book.
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know the definition of a complex function and know the standard functions z n , e z ,
sin z and cos z
- can split complex functions into a real and an imaginary part
- know the concepts of continuity and differentiability for complex functions
- know the concept of analytic function
- can determine the derivative of a complex function.
11.1
Definition and examples
The previous parts of this book dealt almost exclusively with functions that were defined on R and could have values in C. In this part we will be considering functions
that are defined on C (and can have values in C).
DEFINITION 11.1
Complex function
Domain
Range
EXAMPLE 11.1
A function f is called a complex function when f is defined on a subset of C and
has values in a subset of C.
Note that in particular C itself is a subset of C. It is customary to denote the
variable of a complex function by the symbol z. The set of all z ∈ C for which a
complex function is well-defined is called the domain of the function. The range of
a complex function f is the set of values f (z), where z runs through the domain of
f.
The function f (z) = z is a complex function with domain C and range C. The
function assigning the complex conjugate z to each z ∈ C, has domain C and range
C as well. In fact, since z = z, it follows that z ∈ C is the complex conjugate
of z ∈ C. In figure 11.1 the function f (z) = z is represented: for a point z the
image-point z is drawn.
253
254
11 Complex functions
Im z
z
Re z
0
z
FIGURE 11.1
The function z → z.
Im z
z
1
|z |
ϕ
–ϕ
0
1
|z |
1
Re z
1
z
FIGURE 11.2
The function z → 1/z.
EXAMPLE 11.2
Polynomial
Rational function
Consider the function g assigning the complex number 1/z to z ∈ C, that is, g(z) =
1/z. According to section 2.1.1 the number 1/z exists for every z ∈ C with z = 0.
The domain of g is C {0}. The image of g also is C {0}. In fact, if z ∈ C and
z = 0, then one has for w = 1/z that g(w) = 1/w = z. In figure 11.2 the function
1/z is represented; here φ = arg z.
Just as for real functions, one can use simple complex functions to build ever
more complicated complex functions. The simplest complex functions are of course
the constant functions f (z) = c, where c ∈ C. Next we can consider positive
integer powers of z, so f (z) = z n with n ∈ N. By adding and multiplying by
constants, we obtain the polynomials p(z) = an z n + an−1 z n−1 + · · · + a1 z + a0 ,
where ai ∈ C for each i. Finally, we can divide two of these polynomials to obtain
the rational functions p(z)/q(z), where p(z) and q(z) are polynomials. As in the
real case, these are only defined for those values of z for which q(z) = 0.
In section 2.1.1 we have seen that C can also be represented as points in R2 : the
complex number z = x + i y is then identified with the point (x, y) ∈ R2 . If we now
write f (z) = f (x + i y) = u + iv with u, v ∈ R, then u and v will be functions of
x and y. Hence,
f (z) = u(x, y) + iv(x, y),
256
11 Complex functions
and cos y = (ei y + e−i y )/2 for y ∈ R. Since ei z is now defined for z ∈ C, this
suggests the following definition.
DEFINITION 11.3
Complex sine and cosine
For z ∈ C we define sin z and cos z by
sin z =
ei z − e−i z
,
2i
cos z =
ei z + e−i z
.
2
When z = y ∈ R, these definitions are of course in accordance with the real
sine and cosine. Many of the well-known trigonometric identities remain valid for
the complex sine and cosine. For example, by expanding the squares it immediately
follows that sin2 z + cos2 z = 1. Similarly one obtains for instance the formulas
cos 2z = cos2 z − sin2 z and sin 2z = 2 sin z cos z. However, not all results are the
same! For example, it is not true that | sin z | ≤ 1: for z = 2i we have | sin 2i | =
(e2 − e−2 )/2 > 3.
EXERCISES
11.1
11.2
Determine the real and imaginary part of the complex functions in exercise 11.1.
11.3
Show that sin2 z + cos2 z = 1 and that sin 2z = 2 sin z cos z.
11.4
Prove that sin(−z) = − sin z and that sin(z + w) = sin z cos w + cos z sin w.
11.5
Hyperbolic sine
Hyperbolic cosine
Determine the domain and range of the following complex functions:
a f (z) = z,
b f (z) = z 3 ,
c f (z) = z − 4 + i,
d f (z) = (3i − 2)/(z + 3).
For x ∈ R the functions hyperbolic sine (sinh) and hyperbolic cosine (cosh) are
defined by sinh x = (e x − e−x )/2 and cosh x = (e x + e−x )/2.
a Prove that sin(i y) = i sinh y and cos(i y) = cosh y.
b Use part a and exercise 11.4 to show that the real part of sin z equals sin x cosh y
and that the imaginary part of sin z equals cos x sinh y.
11.2
Unit disc
Continuity
In this section the concept of continuity is treated for complex functions. Just as for
real functions, continuity of a complex function will be defined in terms of limits.
However, in order to talk about limits in C, we will first have to specify exactly what
will be meant by 'complex numbers lying close to each other'. To do so, we start
this section with the notion of a neighbourhood of a complex number z 0 .
In section 2.1.1 we noted that the set of complex numbers on the unit circle is
given by the equation | z | = 1. The set of complex numbers inside the unit circle
will be called the unit disc. The complex numbers in the unit disc are thus given
by the inequality | z | < 1. In the same way all complex numbers inside the circle
with centre 0 and radius δ > 0 are given by the inequality | z | < δ. Finally, if
we shift the centre to the point z 0 ∈ C, then all complex numbers inside the circle
with centre z 0 and radius δ > 0 are given by the inequality | z − z 0 | < δ. We call
this a neighbourhood of the point z 0 . When the point z 0 itself is removed from a
neighbourhood of z 0 , then we call this a reduced neighbourhood of z 0 ; it is given by
the inequalities 0 < | z − z 0 | < δ. See figure 11.3. We summarize these concepts
in definition 11.4.
11.2 Continuity
257
a
b
Im z
Im z
z0
z0
y0
y0
x0
0
Re z
0
x0
Re z
FIGURE 11.3
A neighbourhood (a) and a reduced neighbourhood (b) of z 0 .
DEFINITION 11.4
Neighbourhood
Let δ > 0. A neighbourhood of z 0 is defined as the set
Reduced neighbourhood
A reduced neighbourhood of z 0 is defined as the set
z ∈ C | z − z0 | < δ .
z ∈ C 0 < | z − z0 | < δ .
Continuity of a complex function can now be defined precisely as for real functions. First the notion of a limit is introduced and subsequently continuity is defined
in terms of limits.
DEFINITION 11.5
Limit in C
Let f be a complex function defined in a reduced neighbourhood of z 0 . Then
limz→z 0 f (z) = w means that for all > 0 there exists a δ > 0 such that for
0 < | z − z 0 | < δ one has | f (z) − w | < .
Hence, the value f (z) is close to w when z is close to z 0 . Geometrically this
means that the numbers f (z) will be lying in a disc which is centred around the
point w and which is getting smaller and smaller as z tends to z 0 .
DEFINITION 11.6
Continuity in C
Let f be a complex function defined in a neighbourhood of z 0 ∈ C. Then f is called
continuous at z 0 if limz→z 0 f (z) = f (z 0 ). The function f is called continuous on
a subset G in C when f is continuous at all points z 0 of G; this subset G should be
a set such that every point z 0 ∈ G has a neighbourhood belonging entirely to G.
Loosely speaking: when f is continuous at z 0 , then the value f (z) is close to
f (z 0 ) when z is close to z 0 . The condition on the subset G in definition 11.6 is quite
natural. This is because continuity at a point z 0 can only be defined if the function is
defined in a neighbourhood of the point z 0 ; hence, together with the point z 0 there
should also be a neighbourhood of z 0 belonging entirely to G. This explains the
condition on the set G. Definitions 11.5 and 11.6 are completely analogous to the
definitions of limit and continuity in the real case. Considering this great similarity
with the real case, a warning is justified. In the real case there are only two different
directions from which a limit can be taken, namely from the right and from the left.
Continuity at a point x0 of a function f defined on R is thus equivalent to
lim f (x) = f (x0 ) = lim f (x).
x↑x0
x↓x0
258
11 Complex functions
For a complex function the situation is completely different. The points z may
approach z 0 in a completely arbitrary fashion in the complex plane, as long as
the distance | z − z 0 | from z to z 0 decreases. Hence, for a complex function it is
often much harder to prove continuity using definition 11.6. However, the rules
for limits and continuity of complex functions are indeed precisely the same as for
real functions. Using these rules it is in most cases easy to verify the continuity
of complex functions. The proofs of the rules are also exactly the same as for real
functions. This is why we state the following theorem without proof.
Let f and g be complex functions defined in a reduced neighbourhood of z 0 . If
limz→z 0 f (z) = a and limz→z 0 g(z) = b, then
THEOREM 11.2
lim ( f (z) ± g(z)) = a ± b,
z→z 0
lim ( f (z) · g(z)) = a · b,
z→z 0
lim ( f (z)/g(z)) = a/b
z→z 0
if b = 0.
If limw→a h(w) = c and if the function h is defined in a reduced neighbourhood of
the point a, then limz→z 0 h( f (z)) = c.
From theorem 11.2 one then obtains, as in the real case, the following results.
Here the set G is as in definition 11.6.
When f and g are continuous functions on a subset G of C, then f + g and f · g
are continuous on G. Moreover, f /g is continuous on G, provided that g(z) = 0 for
all z ∈ G. If h is a continuous function defined on the range of f , then (h ◦ f )(z) =
h( f (z)) is also a continuous function on G.
THEOREM 11.3
As for real functions, theorem 11.3 is used to prove the continuity for ever more
complicated functions. The constant function f (z) = c (c ∈ C) and the function
f (z) = z are certainly continuous on C (the proof is the same as for real functions).
According to theorem 11.3, the product z · z = z 2 is then also continuous. Repeated
application then establishes that z n is continuous on C for any n ∈ N and hence
also that any polynomial is continuous on C. By theorem 11.3 rational functions are
then continuous as well, as long as the denominator is unequal to 0; in this case one
has to take for the subset G the set C with the roots of the denominator removed
(see section 2.1.2 for the concept of root or zero of a polynomial). Without proof
we mention that such a set satisfies the conditions of definition 11.6.
EXERCISES
11.6
a
Show that lim ( f (z) + g(z)) = a + b if lim f (z) = a and lim g(z) = b.
z→z 0
z→z 0
z→z 0
b Use part a to prove that f + g is continuous on G if f and g are continuous on
G (G is a subset of C as in definition 11.6).
11.7
Use the definition to show that the following complex functions are continuous on
C:
a f (z) = c, where c ∈ C is a constant,
b f (z) = z.
11.8
On which subset G of C is the following function continuous?
g(z) =
3z − 4
.
(z − 1)2 (z + i)(z − 2i)
11.3 Differentiability
11.3
259
Differentiability
Just as for continuity, the definition of differentiability of a complex function can be
copied straight from the real case.
DEFINITION 11.7
Differentiability in C
Let f be a complex function defined in a neighbourhood of z 0 ∈ C. Then f is called
differentiable at z 0 if
lim
z→z 0
f (z) − f (z 0 )
z − z0
exists as a finite number. In this case the limit is denoted by f (z 0 ) or by (d f /dz)(z 0 ).
When a complex function is differentiable for every z ∈ C, then f is not called
'differentiable on C', but analytic on C. The following definition is somewhat more
general.
DEFINITION 11.8
Analytic function
Derivative
Let f be a complex function, defined on a subset G of C. Then f is called analytic
on G if f is differentiable at every point z 0 of G (here the subset G should again
be a set as in definition 11.6). The function f (now defined on G) is called the
derivative of f .
Although these definitions closely resemble the definitions of differentiability and
derivative for real functions, there still is a major difference. Existence of the limit
in definition 11.7 is much more demanding than in the real case; this is because the
limit now has to exist no matter how z approaches z 0 . In the real case there are
only two possible directions, namely from the right or from the left. In the complex
case there is much more freedom, since only the distance | z − z 0 | from z to z 0
has to decrease and nothing else is assumed about directions (compare this with the
remarks following definition 11.6). Yet, here we will again see that for the calculation of derivatives of complex functions one has precisely the same rules as for real
functions (see theorem 11.5). As soon as we have calculated a number of standard
derivatives, these rules enable us to determine the derivative of more complicated
functions. In the following examples we use definition 11.7 to determine our first
two standard derivatives.
EXAMPLE 11.5
Consider the constant function f (z) = c, where c ∈ C. Let z 0 ∈ C. Since f (z) −
f (z 0 ) = c − c = 0 for each z ∈ C, it follows that f (z 0 ) = 0. As for real constants
we thus have that the derivative of a constant equals 0. Put differently, the function
f (z) = c is analytic on C and f (z) = 0.
EXAMPLE 11.6
The function f (z) = z is analytic on C and has as its derivative the function 1. In
fact, for z 0 ∈ C one has f (z) − f (z 0 ) = z − z 0 and so ( f (z) − f (z 0 ))/(z − z 0 ) = 1
for each z ∈ C. This shows that f (z 0 ) = 1 for each z 0 ∈ C.
Quite a few of the well-known results for the differentiation of real functions
remain valid for complex functions. The proofs of the following two theorems are
exactly the same as for the real case and are therefore omitted.
THEOREM 11.4
Let f be a complex function and assume that f (z 0 ) exists at the point z 0 . Then f
is continuous at z 0 .
THEOREM 11.5
Let f and g be analytic on a subset G of C (G as in definition 11.6). Then the
following properties hold:
a a f + bg is analytic on G for arbitrary a, b ∈ C and (a f + bg) (z) = a f (z) +
bg (z);
b f · g is analytic on G and the product rule holds: ( f · g) (z) = f (z)g(z) +
f (z)g (z);
Linearity
Product rule
11.4 The Cauchy–Riemann equations∗
11.13
263
We have seen that many of the rules for complex functions are precisely the same
as for real functions. That not all rules remain the same is shown by the following
√
√
√
√
Bernoulli paradox: 1 = 1 = (−1)(−1) = (−1) (−1) = i · i = i 2 = −1.
Which step in this argument is apparently not allowed?
The Cauchy–Riemann equations∗
11.4
The material in this section will not be used in the remainder of the book and can
thus be omitted without any consequences.
The proof of theorem 11.6 clearly shows that in some cases it may not be easy
to show whether or not a function is differentiable and, in case of differentiability,
to determine the derivative. The reason for this is the fact, mentioned earlier, that
a limit in C is of a quite different nature from a limit in R. In order to illustrate
this once more, we close with a theorem whose proof cleverly uses the fact that a
limit in C should not depend on the way in which z approaches z 0 in the expression
limz→z 0 . The theorem also provides us with a quick and easy way to show that a
function is not analytic.
THEOREM 11.7
Let f (z) = u(x, y) + iv(x, y) be a complex function. Assume that f (z 0 ) exists at
a point z 0 = x0 + i y0 . Then ∂u/∂ x, ∂u/∂ y, ∂v/∂ x and ∂v/∂ y exist at the point
(x0 , y0 ) and at (x0 , y0 ) one has
∂u
∂v
=
∂x
∂y
and
∂u
∂v
=− .
∂y
∂x
(11.4)
Proof
In the proof we first let z = x + i y tend to z 0 = x0 + i y0 in the real direction
and subsequently in the imaginary direction. See figure 11.4. It is given that the
Im z
z = x + iy 0
z 0 = x 0 + iy 0
Re z
0
z = x 0 + iy
FIGURE 11.4
The limit in the real and in the imaginary direction.
following limit exists:
f (z 0 ) = lim
z→z 0
f (z) − f (z 0 )
.
z − z0
We first study the limit in the real direction. We thus take z = x + i y0 , which means
that z → z 0 is equivalent to x → x0 since z − z 0 = x + i y0 − x0 − i y0 = x − x0 .
11.4 The Cauchy–Riemann equations∗
265
Moreover, they satisfy the Cauchy–Riemann equations. Hence, the function e z is
analytic on C and f (z) = ∂u/∂ x + i∂v/∂ x = e x cos y + ie x sin y = e z . Compare
the ease of these arguments with the proof of theorem 11.6.
EXERCISES
11.14∗
We know that the function f (z) = z 2 is analytic on C (see exercise 11.9 or the text
following theorem 11.5). Verify the Cauchy–Riemann equations for f (see example
11.4 for the real and imaginary part of f ).
11.15∗
Use the Cauchy–Riemann equations (and the results of example 11.3) to show that
f (z) = 1/z is analytic on C − {0}.
SUMMARY
Complex functions are functions from (a subset of) C to C. By identifying C with
R2 , one can split a complex function into a real part and an imaginary part; these are
then functions from R2 to R. A very important complex function is e z = e x+i y =
e x (cos y + i sin y). It has the characteristic property e z+w = e z ew for all w, z ∈
C. Using e z one can extend the sine and cosine functions from real to complex
functions.
Continuity and differentiability of a complex function f can be defined by means
of limits, just as for the real case: f is called continuous at z 0 ∈ C when
limz→z 0 f (z) = f (z 0 ); f is called differentiable when
lim
z→z 0
f (z) − f (z 0 )
z − z0
exists as a finite number. A complex function f is called analytic on a subset G
of C when f is differentiable at every point of G. The well-known rules from real
analysis remain valid for the complex functions treated here. The function e z , for
example, is analytic on C and has as derivative the function e z again.
SELFTEST
11.16
Consider the complex function f (z) = cos z.
a Is it true that | cos z | ≤ 1? If so, give a proof. If not, give a counter-example.
b Show that cos(w + z) = cos w cos z − sin w sin z.
c Determine the real and imaginary part of cos z.
d Give the largest subset of C on which f is analytic. Justify your answer.
11.17
Determine on which subset G of C the following functions are analytic and give
their derivative:
a (z 3 + 1)/(z − 1),
b 1/(z 4 + 16)10 ,
c e z /(z 2 + 3),
d sin(e z ).
CHAPTER 12
The Laplace transform: definition and
properties
INTRODUCTION
Signals occurring in practice will always have been switched on at a certain moment
in time. Choosing this switch-on moment equal to t = 0, we are then dealing with
functions that are zero for t < 0. If, moreover, such a function is multiplied by a
damping factor e−at (a > 0), then it is not unreasonable to expect that the Fourier
transform of e−at f (t) will exist. As we have seen in the introduction to part 4, this
leads to a new transformation, the so-called Laplace transform. In section 12.1 the
Laplace transform F(s) of a causal function f (t) will be defined by
F(s) =
∞
0
f (t)e−st dt.
Here s ∈ C is 'arbitrary' and F(s) thus becomes a complex function. One of the
major advantages of the Laplace transform is the fact that the integral is convergent
for 'a lot of' functions (which is in contrast to the Fourier transform). For example,
the Laplace transform of the unit step function exists, while this is not the case for
the Fourier transform.
In section 12.1 we consider in detail the conditions under which the Laplace
transform of a function exists. This is illustrated by a number of standard examples
of Laplace transforms. Because of the close connection with the Fourier transform,
it will hardly be a surprise that for the Laplace transform similar properties hold.
A number of elementary properties are treated in section 12.2: linearity, rules for a
shift in the time domain as well as in the s-domain, and the rule for scaling.
In section 12.3 the differentiation and integration rules are treated. These are
harder to prove, but of great importance in applications. In particular, the rule for
differentiation in the time domain proves essential for the application to differential
equations in chapter 14. The differentiation rule in the s-domain will in particular show that the Laplace transform F(s) of a causal function f (t) is an analytic
function on a certain subset of C.
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know and can apply the definition of the Laplace transform
- know the concepts of abscissa of convergence and of absolute convergence
- know the concept of exponential order
- know and can apply the standard examples of Laplace transforms
- know and can apply the properties of linearity, shifting and scaling
- know and can apply the rules for differentiation and integration.
267
268
12 The Laplace transform: definition and properties
12.1
Definition and existence of the Laplace transform
The following definition has been justified in the introduction (and in the introduction to part 4). For the notion 'causal function' or 'causal signal' we refer to section
1.2.4.
DEFINITION 12.1
Laplace transform
Let f (t) be a causal function, so f (t) = 0 for t < 0. The Laplace transform F(s)
of f (t) is the complex function defined for s ∈ C by
F(s) =
∞
0
f (t)e−st dt,
(12.1)
provided the integral exists.
Laplace transform
s-domain
We will see in a moment that for many functions f (t) the Laplace transform F(s)
exists (on a certain subset of C). The mapping assigning the Laplace transform F(s)
to a function f (t) in the time domain will also be called the Laplace transform.
Furthermore, we will say that F(s) is defined in the s-domain; one sometimes calls
this s-domain the 'complex frequency domain' (although a physical interpretation
can hardly be given for arbitrary s ∈ C). Besides the notation F(s) we will also
use (L f )(s), so (L f )(s) = F(s). Often the notation (L f (t))(s), although not very
elegant, will be useful in the case of a concrete function.
In this part of the book we will always tacitly assume that the functions are causal.
The function t, for example, will in this part always stand for (t)t; it is equal to
0 for t < 0 and equal to t for t ≥ 0. In particular, the constant function 1 is
equal to (t) in this part. Moreover, for all functions it will always be assumed
that they are piecewise smooth (see definition 2.4). In particular it then follows that
R
−st dt will certainly exist for any R > 0. The existence of the integral in
0 f (t)e
(12.1) then boils down to the fact that the improper Riemann integral over R, that
is, lim R→∞ 0R f (t)e−st dt, has to exist.
Note also that for s = σ + iω with σ, ω ∈ R it immediately follows from the
definition of the complex exponential that
F(s) =
∞
0
f (t)e−σ t e−iωt dt.
(12.2)
This is an interesting formula, for it shows that the Laplace transform F(s) of f (t)
at the point s = σ +iω is equal to the Fourier transform of (t) f (t)e−σ t at the point
ω, provided all the integrals exist. This is the case, for example, if (t) f (t)e−σ t is
absolutely integrable (see definition 6.2). For the moment, we leave this connection
between the Laplace and Fourier transform for what it is, and return to the question
of the existence of the integral in (12.1), in other words, the existence of the Laplace
transform. As far as the general theory is concerned, we will confine ourselves to
absolute convergence.
DEFINITION 12.2
Absolutely convergent
b
An integral a g(t) dt with −∞ ≤ a < b ≤ ∞ is called absolutely convergent if
b
a
| g(t) | dt < ∞.
This concept is entirely analogous to the concept of absolute convergence for
series (see section 2.4.2). Since
g(t) dt ≤ | g(t) | dt (as in (2.22)), we see
that an absolutely convergent integral is also convergent in the ordinary sense. The
converse, however, need not to be true, just as for series. Note that for the absolute
convergence of the integral in (12.1) only the value of σ = Re s is of importance,
since |eiωt | = 1 for all ω ∈ R. In the general theory we will confine ourselves to
12.1 The Laplace transform: definition and existence
269
absolute convergence, since proofs are easier than for the case of ordinary convergence. For concrete cases it is usually quite easy to treat ordinary convergence as
well. To get some feeling for the absolute and ordinary convergence of the integral
in (12.1), we will first treat two examples.
EXAMPLE 12.1
Laplace transform of 1
Let (t) be the unit step function. The Laplace transform of (t) (or the function 1)
is given by the integral
∞
e−st dt = lim
R→∞ 0
0
R
e−σ t e−iωt dt
if s = σ + iω. We first consider absolute convergence. The absolute value of the
integrand is equal to e−σ t and for σ = 0 one has
R
1
1
e−σ t dt = − e−σ t
= (1 − e−σ R ).
σ
σ
0
0
R
The integral thus converges absolutely if lim R→∞ e−σ R exists and of course this
is the case only for σ > 0. Hence, the Laplace transform of (t) certainly exists
for σ > 0, so for Re s > 0. We will now show that the integral also converges in
the ordinary sense precisely for all s ∈ C with Re s > 0. For s = 0 we have that
lim R→∞ 0R 1 dt does not exist. Since (e−st ) = −se−st (here we differentiate
with respect to the real variable t; see example 2.11), it follows for s = 0 that
R
1
1
e−st dt = − e−st
= (1 − e−s R ).
s
s
0
0
R
Hence, the Laplace transform of (t) will only exist when lim R→∞ e−s R =
lim R→∞ e−σ R e−iω R exists. Since |e−iω R | = 1, the limit will exist precisely
for σ > 0 and in this case the limit will of course be 0 since lim R→∞ e−σ R = 0 for
σ > 0. We conclude that the integral converges in the ordinary sense precisely for
all s ∈ C with Re s > 0 and that the Laplace transform of 1 (or (t)) is given by 1/s
for these values of s. Note that in this example the regions of absolute and ordinary
convergence are the same.
EXAMPLE 12.2
Laplace transform of eat
Let a ∈ R. The Laplace transform of the function eat (hence of (t)eat ) is given by
∞
eat e−st dt = lim
R
R→∞ 0
0
e−(σ −a)t e−iωt dt
if s = σ + iω. Again we first look at absolute convergence. The absolute value of
the integrand is e−(σ −a)t and for σ = a one has
R
0
e−(σ −a)t dt =
1
1 − e−(σ −a)R .
σ −a
Hence, the integral converges absolutely when lim R→∞ e−(σ −a)R exists and this is
precisely the case when σ −a > 0, or Re s > a. We will now determine the Laplace
transform of eat and moreover show that the integral also converges in the ordinary
sense for precisely all s ∈ C with Re s > a. For s = a the Laplace transform will
certainly not exist. Since (e−(s−a)t ) = −(s − a)e−(s−a)t , it follows for s = a that
∞
0
eat e−st dt = lim
R→∞
−
1
1 −(s−a)t R
1
=
e
−
lim e−(s−a)R .
s−a
s−a
s − a R→∞
0
As in example 12.1, we have lim R→∞ e−(s−a)R = lim R→∞ e−(σ −a)R e−iω R = 0
precisely when σ − a > 0. Hence, there is ordinary convergence for Re s > a and
270
12 The Laplace transform: definition and properties
the Laplace transform of e−at is given by 1/(s − a) for these values of s. For a = 0
we recover the results of example 12.1 again.
It is not hard to prove the following general result on the absolute convergence of
the integral in definition 12.1.
THEOREM 12.1
Let f (t) be a causal function and consider the integral in (12.1).
a If the integral is absolutely convergent for a certain value s = σ0 ∈ R, then the
integral is absolutely convergent for all s ∈ C with Re s ≥ σ0 .
b If the integral is not absolutely convergent for a certain value s = σ1 ∈ R, then
the integral is not absolutely convergent for all s ∈ C with Re s ≤ σ1 .
Proof
We first prove part a. Write s = σ + iω. Since eiωt = 1 and e−σ t > 0, it follows
from 12.2 that
∞
0
f (t)e−st dt =
∞
0
| f (t) | e−σ t dt.
For Re s = σ ≥ σ0 one has that e−σ t ≤ e−σ0 t for all t ≥ 0. Hence,
∞
0
f (t)e−st dt ≤
∞
0
| f (t) | e−σ0 t dt.
According to the statement in part a, the integral in the right-hand side of this inequality exists (as a finite number). The integral in (12.1) is thus indeed absolutely
convergent for all s ∈ C with Re s ≥ σ0 . This proves part a.
Part b immediately follows from part a. Let us assume that there exists an s0 ∈ C
with Re s ≤ σ1 such that the integral is absolutely convergent after all. According
to part a, the integral will then be absolutely convergent for all s ∈ C with Re s ≥
Re s0 . But Re σ1 = σ1 ≥ Re s0 and hence the integral should in particular be
absolutely convergent for s = σ1 . This contradicts the statement in part b.
From theorem 12.1 we see that for the absolute convergence only the value
Re s = σ matters. Note that the set { s ∈ C | Re s = σ } is a straight line perpendicular to the real axis. See figure 12.1.
Using theorem 12.1 one can, moreover, show that there are precisely three possibilities regarding the absolute convergence of the integral in (12.1):
the integral is absolutely convergent for all s ∈ C;
the integral is absolutely convergent for no s ∈ C whatsoever;
a
b
Im s
0
σ
Re s = σ
FIGURE 12.1
The straight line Re s = σ (for a σ > 0).
Re s
12.1 The Laplace transform: definition and existence
271
c there exists a number σa ∈ R such that the integral is absolutely convergent for
Re s > σa and not absolutely convergent for Re s < σa .
In case c there is no statement about the absolute convergence for Re s = σa . It is
possible that there is absolute convergence on the line Re s = σa , but it is equally
possible that there is no absolute convergence. In example 12.1 we have seen that
σa = 0; in this case there is no absolute convergence for Re s = σa = 0.
Strictly speaking, we have not given a proof that these three possibilities are the
only ones. Intuitively, however, this seems quite obvious. Suppose that possibilities
a and b do not occur. We then have to show that only possibility c can occur. Now
if a does not hold, then there exists a σ1 ∈ R such that the integral is not absolutely
convergent (we may assume that σ1 is real, since only the real part is relevant for
the absolute convergence). According to theorem 12.1b, the integral is then also not
absolutely convergent for all s ∈ C with Re s ≤ σ1 . Since also b does not hold,
there similarly exists a σ2 ∈ R such that the integral is absolutely convergent for all
s ∈ C with Re s ≥ σ2 (we now use theorem 12.1a). In the region σ1 < Re s < σ2 ,
where nothing yet is known about the absolute convergence, we now choose an
arbitrary σ3 ∈ R. See figure 12.2. If we have absolute convergence for σ3 , then we
Im s
no
absolute
convergence
0
absolute
convergence
σ1
Re s = σ 1
σ3
σ2
Re s
Re s = σ 2
FIGURE 12.2
Regions of absolute convergence.
can extend the region of absolute convergence to Re s ≥ σ3 . If there is no absolute
convergence, then we can extend the region where there is no absolute convergence
to Re s ≤ σ3 . This process can be continued indefinitely and our intuition tells us
that at some point the two regions will have to meet, in other words, that possibility
c will occur. That this will indeed happen rests upon a fundamental property of the
real numbers which we will not go into any further. The above is summarized in the
following theorem.
THEOREM 12.2
For a given causal function f (t) there exists a number σa ∈ R with −∞ ≤ σa ≤
∞
∞ such that the integral 0 f (t)e−st dt is absolutely convergent for all s ∈ C
with Re s > σa , and not absolutely convergent for all s ∈ C with Re s < σa . By
σa = −∞ we will mean that the integral is absolutely convergent for all s ∈ C.
By σa = ∞ we will mean that the integral is absolutely convergent for no s ∈ C
whatsoever.
Abscissa of absolute
convergence
The number σa in theorem 12.2 is called the abscissa of absolute convergence.
The region of absolute convergence is a half-plane Re s > σa in the complex plane.
See figure 12.3. The case σa = ∞ (possibility b) almost never occurs in practice.
272
12 The Laplace transform: definition and properties
Im s
Re s > σ a
σa
0
Re s
Re s = σ a
FIGURE 12.3
The half-plane Re s > σa of absolute convergence.
This is because functions occurring in practice are almost always of 'exponential
order'.
DEFINITION 12.3
Exponential order
The causal function f : R → C is called of exponential order if there are constants
α ∈ R and M > 0 such that | f (t) | ≤ Meαt for all t ≥ 0.
Functions of exponential order will not assume very large values too quickly. The
number α in definition 12.3 is by no means unique, since for any β ≥ α one has
eαt ≤ eβt for t ≥ 0.
EXAMPLE 12.3
The unit step function (t) is of exponential order with M = 1 and α = 0 since
| (t) | ≤ 1.
EXAMPLE
Let f (t) be a bounded function, so | f (t) | ≤ M for some M > 0. Then f is of
exponential order with α = 0. The function (t) is a special case and has M = 1.
EXAMPLE 12.4
Consider the function f (t) = t. From the well-known limit limt→∞ te−αt = 0,
for α > 0, it follows that | f (t) | ≤ Meαt for any α > 0 and some constant M > 0.
Hence, this function is of exponential order with α > 0 arbitrary. However, one
cannot claim that f is of exponential order with α = 0, since it is not true that
| t | ≤ M.
We now recall example 12.2, where it was shown that the Laplace transform of
eat exists for Re s > a. The following result will now come as no surprise.
THEOREM 12.3
Let f be a function of exponential order as in definition 12.3. Then the integral in
(12.1) is absolutely convergent (and so the Laplace transform of f will certainly
exist) for Re s > α. In particular one has for the abscissa of absolute convergence
σa that σa ≤ α.
Proof
Since | f (t) | ≤ Meαt for all t ≥ 0, it follows that
∞
0
f (t)e−st dt =
∞
0
| f (t) | e−σ t dt ≤ M
∞
e−(σ −α)t dt,
0
where s = σ + iω. As in example 12.2 it follows that the latter integral exists for
σ − α > 0, so for Re s > α. Finally, we then have for σa that σa ≤ α. For if σa > α,
then there will certainly exist a number β with α < β < σa . On the one hand it
12.1 The Laplace transform: definition and existence
273
would then follow from β < σa and the definition of σa that the integral does not
converge absolutely for s = β, while on the other hand β > α would imply that
there is absolute convergence.
EXAMPLE
The unit step function (t) is of exponential order with α = 0. From theorem 12.3 it
follows that the Laplace transform will certainly exist for Re s > 0 and that σa ≤ 0.
As was shown in example 12.1, we even have σa = 0.
EXAMPLE 12.5
In example 12.4 it was shown that the function t is of exponential order for arbitrary
α > 0. From theorem 12.3 it then follows that the Laplace transform of t certainly
exists for Re s > 0 and that σa ≤ 0. In example 12.9 we will see that σa = 0.
Abscissa of convergence
In the preceding general discussion of convergence issues, we have confined ourselves to absolute convergence, since the treatment of this type of convergence is
relatively easy. Of course one can wonder whether similar results as in theorem
12.3 can also be derived for ordinary convergence. This is indeed the case, but these
results are much harder to prove. We will merely state without proof that there
exists a number σc , the so-called abscissa of convergence, such that the integral
∞
−st dt converges for Re s > σ and does not converge for Re s < σ .
c
c
0 f (t)e
Since absolute convergence certainly implies ordinary convergence, we see that
σc ≤ σa . In most concrete examples one can easily obtain σc .
EXAMPLE 12.6
In example 12.1 it was shown that the Laplace transform of (t) exists precisely
when Re s > 0. Hence, in this case we have σa = σc = 0.
EXAMPLE 12.7
The shifted unit step function (t − b) in figure 12.4 is defined by
(t − b) =
1
0
where b ≥ 0.
for t ≥ b,
for t < b.
The Laplace transform of (t − b) can be determined as in
⑀ (t – b)
1
0
b
t
FIGURE 12.4
The shifted unit step function (t − b) with b ≥ 0.
example 12.1. For s = 0 the Laplace transform does not exist, while for s = 0
it follows that
∞
0
e−st (t − b) dt =
∞
b
R
1
− e−st
s
R→∞
b
e−st dt = lim
1
e−bs
−
lim e−s R .
=
s
s R→∞
Again the limit exists precisely for Re s > 0 and it equals 0 then. Here we again
have σa = σc = 0. Furthermore, we see that for Re s > 0 the Laplace transform of
(t − b) is given by e−bs /s.
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12 The Laplace transform: definition and properties
EXAMPLE 12.8
From example 12.2 it follows that for the function ebt we have that σa = σc = b.
For Re s > b the Laplace transform is given by 1/(s − b).
EXAMPLE 12.9
We now determine the Laplace transform of the function f (t) = t. For s = 0 the
Laplace transform does not exist, while for s = 0 it follows from integration by
parts that (from example 12.5 we know that σa ≤ 0):
F(s) =
∞
0
te−st dt =
∞
0
R
t
− (e−st ) dt
s
1 ∞ −st
t
+
e
dt
− e−st
s
s 0
R→∞
0
1
1
= − lim Re−s R + lim − 2 e−st
s R→∞
R→∞
s
1
1
= − lim Re−s R − 2 lim e−s R +
s R→∞
s R→∞
= lim
R
0
1
.
s2
As before, one has for Re s = σ > 0 that lim R→∞ e−s R = 0. Since for σ > 0 we
have lim R→∞ Re−σ R = 0 as well, it also follows that lim R→∞ Re−s R = 0 for
Re s > 0. This shows that F(s) = 1/s 2 for Re s > 0. We also see that the limits do
not exist if Re s ≤ 0; hence, σa = σc = 0.
Two-sided Laplace transform
The previous example is a prototype of the kind of calculations that are usually
necessary in order to calculate the Laplace transform of a function: performing an
integration by parts (sometimes more than once) and determining limits. These
limits will in general only exist under certain conditions on Re s. Usually this will
also immediately give us the abscissa of convergence, as well as the abscissa of
absolute convergence. In all the examples we have seen so far, we had σa = σc . In
general, this is certainly not the case; for example, for the function et sin(et ) one
has σa = 1 while σc = 0 (the proof of these facts will be omitted). There are even
examples (which are not very easy) of functions for which the integral in (12.1)
converges for all s ∈ C (so σc = −∞), but converges absolutely for no s ∈ C
whatsoever (so σa = ∞)! As a matter of fact, for the application of the Laplace
transform it almost always suffices to know that some half-plane of convergence
exists; the precise value of σa or σc is in many cases less important.
We close this section by noting that besides the Laplace transform from definition
12.1, there also exists a so-called two-sided Laplace transform. For functions on R
that are not necessarily causal, this two-sided Laplace transform is defined by
∞
−∞
f (t)e−st dt,
for those s ∈ C for which the integral exists. Since in most applications it is assumed that the functions involved are causal, we have limited ourselves to the 'onesided' Laplace transform from definition 12.1. Indeed, the one-sided and two-sided
Laplace transforms coincide for causal functions. Also note that
∞
−∞
f (t)e−st dt =
∞
0
f (t)e−st dt +
∞
0
f (−t)e−(−s)t dt.
Hence, the two-sided Laplace transform of f (t) is equal to the (one-sided) Laplace
transform of f (t) plus the (one-sided) Laplace transform at −s of the function
f (−t). There is thus a close relationship between the two forms of the Laplace
transform.
12.2 Linearity, shifting and scaling
275
EXERCISES
Indicate why the limit lim R→∞ e−iω R does not exist for ω ∈ R.
For which σ, ω ∈ R is it true that lim R→∞ e−σ R e−iω R = 0?
12.1
a
b
12.2
In example 12.2 it was shown that the Laplace transform of eat (a ∈ R) is given by
1/(s − a) for Re s > a. Show that for a ∈ C the same result holds for Re s > Re a.
12.3
Consider the causal function f defined by f (t) = 1 − (t − b) for b ≥ 0.
a Sketch the graph of f .
b Determine the Laplace transform F(s) of f (t) and give the abscissa of absolute
and ordinary convergence.
12.4
Show that the Laplace transform of the function t 2 (hence, of (t)t 2 ) is given by
2/s 3 for Re s > 0.
12.5
Determine for the following functions the Laplace transform and the abscissa of
absolute and ordinary convergence:
a e−2t ,
b
(t − 4),
c e(2+3i)t .
12.6
Determine a function f (t) whose Laplace transform is given by the following functions:
a 1/s,
b e−3s /s,
c 1/(s − 7),
d 1/s 3 .
12.2
Linearity, shifting and scaling
From (12.2) it follows that (L f )(σ + iω) = (F (t) f (t)e−σ t )(ω), provided that all
the integrals exist. It will then come as no surprise that for the Laplace transform L
and the Fourier transform F similar properties hold. In this section we will examine
a number of elementary properties: linearity, shifting, and scaling. It will turn out
that these properties are quite useful in order to determine the Laplace transform of
all kinds of functions.
12.2.1
Linearity
As for the Fourier transform, the linearity of the Laplace transform follows immediately from the linearity of integration (see section 6.4.1). For α, β ∈ C one thus
has
Linearity of L
L(α f + βg) = αL f + βLg
in the half-plane where L f and Lg both exist. Put differently, when F and G are the
Laplace transforms of f and g respectively, then α f + βg has the Laplace transform
α F + βG; if F(s) exists for Re s > σ1 , and G(s) for Re s > σ2 , then (α F + βG)(s)
exists for Re s > max(σ1 , σ2 ). This simple rule enables us to find a number of
important Laplace transforms.
EXAMPLE 12.10
The Laplace transform of eit is given by 1/(s − i) for Re s > Re i, hence for
Re s > 0 (see exercise 12.2). Similarly one has that (Le−it )(s) = 1/(s + i) for
280
12 The Laplace transform: definition and properties
12.16
Determine the Laplace transform F(s) of the following functions:
a f (t) = te2t ,
b f (t) = (t − 1)(t − 1)2 ,
c f (t) = e−3t sin 5t,
d f (t) = ebt cos at for a ∈ R and b ∈ C,
e f (t) = (t − 3) cosh(t − 3),
f f (t) = t 2 et−3 .
12.17
Draw the graph of the following functions and determine their Laplace transform
F(s):
a f (t) = (t − 1)(t − 1),
b f (t) = (t)(t − 1),
c f (t) = (t − 1)t.
12.18
Consider the (causal) function
f (t) =
t
0
for 0 ≤ t < 1,
for t ≥ 1.
Write f as a combination of shifted unit step functions and determine the Laplace
transform F(s).
12.19
Determine a function f (t) whose Laplace transform F(s) is given by:
a F(s) = 2/(s − 3),
b F(s) = 3/(s 2 + 1),
c F(s) = 4s/(s 2 + 4),
d F(s) = 1/(s 2 − 4),
e F(s) = e−2s /s 2 ,
f F(s) = se−3s /(s 2 + 1),
g F(s) = 1/((s − 1)2 + 16),
h F(s) = (3s + 2)/((s + 1)2 + 1),
i F(s) = −6/(s − 3)3 ,
j F(s) = (s − 2)/(s 2 − 4s + 8),
k F(s) = se−s /(4s 2 + 9).
12.3
Differentiation and integration
In this section we continue our investigation into the properties of the Laplace transform with the treatment of the differentiation and integration rules. We will examine
differentiation both in the time domain and in the s-domain. For the application of
the Laplace transform to differential equations (see chapter 14) it is especially important to know how the Laplace transform behaves with respect to differentiation
in the time domain. Differentiation in the s-domain is complex differentiation. In
particular we will show that F(s) is analytic in a certain half-plane in C. And finally
an integration rule in the time domain will be derived from the differentiation rule in
the time domain. The rule for integration in the s-domain will not be treated. This
rule isn't used very often and a proper treatment would moreover require a thorough
understanding of integration of complex functions over curves in C.
12.3.1
Differentiation in the time domain
In section 6.4.8 we have seen that differentiation in the time domain and multiplication by a factor iω in the frequency domain correspond with each other under the
12.3 Differentiation and integration
281
Fourier transform. A similar correspondence, but now involving the factor s, exists
for the Laplace transform.
THEOREM 12.7
Differentiation in the time
domain
Let f be a causal function which, in addition, is differentiable on R. In a half-plane
where L f and L f both exist one has
(L f )(s) = s(L f )(s).
(12.9)
Proof
Let s ∈ C be such that L f and L f both exist at s. By applying integration by parts
we obtain that
(L f )(s) =
∞
0
∞
∞
f (t)e−st dt = f (t)e−st 0 + s
0
f (t)e−st dt.
Since f is differentiable on R, f is certainly continuous on R. But f is also causal
and so we must have f (0) = limt↑0 f (t) = 0. From this it follows that
(L f )(s) = lim f (R)e−s R + s(L f )(s).
R→∞
Since (L f )(s) and s(L f )(s) exist, the limit lim R→∞ f (R)e−s R must also exist.
∞
But then this limit has to equal 0, for (L f )(s) = 0 f (t)e−st dt exists (here we
use the simple fact that for a continuous function g(t) with lim R→∞ g(R) = a,
∞
where a ∈ R and a = 0, the integral 0 g(t) dt does not exist; see exercise 12.20).
This proves the theorem.
Using the concept of a 'function of exponential order' (see definition 12.3), one is
able to specify the half-planes where L f and L f both exist. If we assume that the
function f (t) from theorem 12.7 is of exponential order for a certain value α ∈ R,
then L f exists for Re s > α (see theorem 12.3). One can show that in this case L f
also exists for Re s > α. We will not go into this any further.
By repeatedly applying theorem 12.7, one can obtain the Laplace transform of the
higher derivatives of a function. Of course, the conditions of theorem 12.7 should
then be satisfied throughout. Suppose, for example, that a causal function f (t) is
continuously differentiable on R (so f exists and is continuous on R) and that f
is differentiable on R. By applying theorem 12.7 twice in a half-plane where all
Laplace transforms exist, it then follows that
(L f )(s) = s(L f )(s) = s 2 (L f )(s).
Differentiation in the time
domain
EXAMPLE
Now, more generally, let f (t) be a causal function which is n −1 times continuously
differentiable on R (so the (n − 1)th derivative f (n−1) (t) of f (t) exists and is continuous on R) and let f (n−1) (t) be differentiable on R (in the case n = 1 we have
f (0) (t) = f (t) and by a '0 times continuously differentiable function' we simply
mean a continuous function). In a half-plane where all Laplace transforms exist, we
then have the following differentiation rule in the time domain:
(L f (n) )(s) = s n (L f )(s).
(12.10)
For f (t) = t 2 we have f (t) = 2t. The function (t)t 2 is indeed differentiable on
R and according to (12.9) we thus have s(Lt 2 )(s) = 2(Lt)(s). From example 12.9
we know that (Lt)(s) = 1/s 2 for Re s > 0, and so (Lt 2 )(s) = 2/s 3 for Re s > 0.
Compare this method with exercise 12.4.
The method from the example above can be used to determine Lt n for every
n ∈ N with n ≥ 2. In fact, the function (t)t n satisfies the conditions of
282
12 The Laplace transform: definition and properties
theorem 12.7 for n ∈ N with n ≥ 2 and so it follows from (12.9) that
(Lnt n−1 )(s) = s(Lt n )(s).
Since limt→∞ t n e−αt = 0 for every α > 0, it follows just as in the examples 12.4
and 12.5 that Lt n exists for Re s > 0 (n ∈ N). Hence,
n
for Re s > 0
(Lt n )(s) = (Lt n−1 )(s)
s
(this result can also easily be derived by a direct calculation). Applying this result
repeatedly, we find that
n n−1
1
·
· . . . · (L1)(s)
for Re s > 0.
s
s
s
Now finally use that (L1)(s) = 1/s for Re s > 0 to establish the following important
result:
n!
for Re s > 0,
(12.11)
(Lt n )(s) = n+1
s
where n! = 1 · 2 · 3 · . . . · (n − 1) · n. Note that (12.11) is also valid for n = 1 and
n = 0 (0! = 1 by convention).
Theorem 12.7 cannot be applied to an arbitrary piecewise smooth function. This
is because the function in theorem 12.7 has to be differentiable on R and so it can
certainly have no jump discontinuities. In section 13.4 we will derive a differentiation rule for distributions, which in particular can then be applied to piecewise
smooth functions.
(Lt n )(s) =
12.3.2
Differentiation in the s-domain
On the basis of the properties of the Fourier transform (see section 6.4.9) we expect
that here differentiation in the s-domain will again correspond to a multiplication by
a factor in the time domain. It will turn out that this is indeed the case. Still, for the
Laplace transform this result is of a quite different nature, since we are dealing with
a complex function F(s) here and so with complex differentiation. As in the case of
the Fourier transform we will thus have to show that F(s) is in fact differentiable.
Put differently, we will first have to show that F(s) is an analytic function on a
certain subset of C. One has the following result.
THEOREM 12.8
Differentiation in the
s-domain
Let f be a function with Laplace transform F(s) and let σa be the abscissa of
absolute convergence.Then F(s) is an analytic function of s for Re s > σa and
d
F(s) = −(Lt f (t))(s).
(12.12)
ds
Proof
We have to show that limh→0 (F(s + h) − F(s))/ h (with h → 0 in C; see section
11.2) exists for s ∈ C with Re s > σa . Now
F(s + h) − F(s) =
=
∞
0
0
∞
f (t)e−(s+h)t dt −
∞
h→0 0
f (t)
0
f (t)(e−ht − 1)e−st dt,
which means that we have to show that
lim
∞
e−ht − 1 −st
dt
e
h
f (t)e−st dt
12.3 Differentiation and integration
283
exists. We now assume that under the condition mentioned in the theorem we may
interchange the limit and the integral. Note that we thus assume in particular that
the integral resulting from the interchange will again exist. From definition 11.7,
theorem 11.6, and the chain rule it follows that
e−ht − 1
=
h
h→0
lim
d −st
e
(0) = −t.
ds
This shows that
∞
F(s + h) − F(s)
=
(−t f (t))e−st dt,
h
h→0
0
lim
or
d
F(s) = −(Lt f (t))(s).
ds
The difficult step in this theorem is precisely interchanging the limit and the integral, which in fact proves the existence of the derivative (and thus proves that F(s)
is analytic). It takes quite some effort to actually show that the interchanging is
allowed (see e.g. K¨ rner, Fourier analysis, 1990, Theorem 7.5.2). If we compare
o
theorem 12.8 with theorem 6.8 for the Fourier transform, then it is remarkable that
in theorem 12.8 we do not require in advance that the Laplace transform of t f (t)
exists, but that this fact follows as a side result of the theorem. It also means that the
theorem can again be applied to the function −t f (t), resulting in
d2
F(s) = (Lt 2 f (t))(s)
ds 2
for Re s > σa . Repeated application then leads to the remarkable result that F(s) is
arbitrarily often differentiable and that for Re s > σa
Differentiation in the
s-domain
dk
F(s) = (−1)k (Lt k f (t))(s)
ds k
for k ∈ N.
(12.13)
We will call this the differentiation rule in the s-domain. Usually, (12.13) is applied
in the following way: let F(s) = (L f (t))(s), then
(Lt k f (t))(s) = (−1)k
dk
F(s)
ds k
for k ∈ N.
Multiplication by t k
Hence, this rule is sometimes referred to as multiplication by t k .
EXAMPLE 12.12
Since (L sin t)(s) = 1/(s 2 + 1) for Re s > 0, it follows that
(Lt sin t)(s) = −
EXAMPLE
d
ds
1
s2 + 1
=
2s
(s 2 + 1)2
for Re s > 0.
We know that (Le−3t )(s) = 1/(s + 3), so
(Lt 2 e−3t )(s) =
d2
ds 2
1
s+3
=
2
.
(s + 3)3
This result also follows by noting that (Lt 2 )(s) = 2/s 3 and subsequently using the
shift property from theorem 12.5.
284
12 The Laplace transform: definition and properties
12.3.3
Integration in the time domain
From the differentiation rule in the time domain one quickly obtains the following
result.
THEOREM 12.9
Integration in the time
domain
Let f be a causal function which is continuous on R and has Laplace transform
F(s). Then one has in a half-plane contained in the region Re s > 0
L
t
0
f (τ ) dτ (s) =
1
F(s).
s
(12.14)
Proof
t
t
Define the function g(t) by g(t) = −∞ f (τ ) dτ = 0 f (τ ) dτ , then g is the
primitive of f with g(0) = 0 and g(t) = 0 for t < 0. Since f is continuous, it
follows that g is differentiable on R. According to theorem 12.7 one then has, in
a half-plane where both Laplace transforms exist, that (Lg )(s) = s(Lg)(s). But
g = f and so
s L
t
0
f (τ ) dτ (s) = F(s).
In a half-plane where both Laplace transforms exist and which lies to the right of
Re s = 0, one may divide by s and so the result follows.
t
The causal function sin t is continuous on R and since 0 sin τ dτ = 1 − cos t, it
then follows from theorem 12.9 that
EXAMPLE 12.13
(L(1 − cos t))(s) =
1
1
.
(L sin t)(s) =
2 + 1)
s
s(s
This result is easy to verify since we know from table 7 that (L cos t)(s) = s/(s 2 +1)
and (L1)(s) = 1/s. Hence,
(L(1 − cos t))(s) =
(s 2 + 1) − s 2
1
s
1
=
=
.
− 2
s
s +1
s(s 2 + 1)
s(s 2 + 1)
As in the case of theorem 12.7 one can use the concept 'function of exponential
order' to specify the half-planes where the Laplace transforms exist. If a function
f (t) is of exponential order for a certain α ∈ R, then one can show that the result
from theorem 12.9 is correct for Re s > max(0, α). We will not go into this matter
any further.
Theorem 12.9 can also be applied in the 'opposite direction'. When we are looking for a function f (t) whose Laplace transform is F(s), then we could start by first
ignoring factors 1/s that might occur in F(s). In fact, such factors can afterwards
be re-introduced by an integration.
Let g(t) be a function with Laplace transform G(s) = 4/(s(s 2 + 4)). If we ignore
the factor 1/s, then we are looking for a function h(t) having Laplace transform
H (s) = 4/(s 2 + 4). This is easy: h(t) = 2 sin 2t. Integrating h(t) we find g(t):
EXAMPLE 12.14
g(t) =
t
0
2 sin 2τ dτ = [− cos 2τ ]t = 1 − cos 2t.
0
EXERCISES
12.20
∞
Let g(t) be a continuous function on R. Show that 0 g(t) dt does not exist if
lim R→∞ g(R) = a where a ∈ R and a = 0.
12.3 Differentiation and integration
285
12.21
Show that limt→∞ t n e−αt = 0 for any n ∈ N and α > 0 and use this to show that
the function t n is of exponential order with α > 0 arbitrary. Conclude that (Lt n )(s)
exists for Re s > 0.
12.22
Use the definition to show that for any n ∈ N one has
n
(Lt n )(s) = (Lt n−1 )(s)
for Re s > 0.
s
12.23
One has that (L1)(s) = 1/s for Re s > 0. Use the differentiation rule in the sdomain to show that (Lt n )(s) = n!/s n+1 for Re s > 0.
12.24
In example 12.12 we used the differentiation rule in the s-domain to show that
(Lt sin t)(s) = 2s/(s 2 + 1)2 for Re s > 0. Since t sin t = (teit − te−it )/2i, one
can also derive this result quite easily using the shift property. Give this derivation.
12.25
Consider the function f (t) = t n eat for a ∈ C and let F(s) = (L f (t))(s).
a Determine F(s) using a shift property.
b Determine F(s) using a differentiation rule.
12.26
t
Determine the Laplace transform G(s) of g(t) = 0 τ cos 2τ dτ .
12.27
One has that (L sinh at)(s) = a/(s 2 − a 2 ) for Re s > a (a ∈ R). Which function
f (t) has a/(s(s 2 − a 2 )) as its Laplace transform?
12.28
Determine the Laplace transform F(s) of the following functions:
a f (t) = t 2 cos at,
b f (t) = (t 2 − 3t + 2) sinh 3t.
12.29
Determine a function f (t) whose Laplace transform F(s) is given by:
1
d2
,
a F(s) = 2
ds
s2 + 1
b
1
F(s) = 2 2
.
s (s − 1)
SUMMARY
The Laplace transform F(s) of a causal function f (t) is defined for s ∈ C by
F(s) =
∞
0
f (t)e−st dt.
There exists a number σa ∈ R with −∞ ≤ σa ≤ ∞, such that the integral is absolutely convergent for all s ∈ C with Re s > σa and is not absolutely convergent
for all s ∈ C with Re s < σa . The number σa is called the abscissa of absolute
convergence. The case σa = ∞ almost never occurs in practice, since most functions are of exponential order, so | f (t) | ≤ Meαt for certain M > 0 and α ∈ R.
For ordinary convergence there are similar results; in this case we have a abscissa of
convergence σc . For the unit step function (t) one has, for example, σa = σc = 0;
for Re s > 0 the Laplace transform of (t) is given by 1/s. A number of standard
Laplace transforms, together with their abscissa of convergence, are given in table 7.
There is a simple relationship between the Laplace and the Fourier transform:
(L f )(σ + iω) = (F (t) f (t)e−σ t )(ω). Therefore, the properties of the Laplace
transform are very similar to the properties of the Fourier transform. In this chapter the following properties were treated: linearity, shifting in the time and the sdomain, scaling in the time domain, differentiation in the time and the s-domain,
and integration in the time domain. These properties are summarized in table 8. In
CHAPTER 13
Further properties, distributions, and
the fundamental theorem
INTRODUCTION
In the first three sections of this chapter the number of properties of the Laplace
transform will be extended even further. We start in section 13.1 with the treatment
of the by now well-known convolution product. As for the Fourier transform, the
convolution product is transformed into an ordinary product by the Laplace transform.
In section 13.2 we treat two theorems that have not been encountered earlier in
the Fourier transform: the so-called initial and final value theorems for the Laplace
transform. The initial value theorem relates the 'initial value' f (0+) of a function
f (t) to the behaviour of the Laplace transform F(s) for s → ∞. Similarly, the final
value theorem relates the 'final value' limt→∞ f (t) to the behaviour of F(s) for
s → 0. Hence, the function F(s) can provide information about the behaviour of
the original function f (t) shortly after switching on (the value f (0+)) and 'after a
considerable amount of time' (the value limt→∞ f (t)).
In section 13.3 we will see how the Laplace transform of a periodic function can
be determined. It will turn out that this is closely related to the Laplace transform of
the function which arises when we limit the periodic function to one period.
In order to determine the Laplace transform of a periodic function, it is not necessary to turn to the theory of distributions. This is in contrast to the Fourier transform
(see section 9.1.2). Still, a limited theory of the Laplace transform of distributions
will be needed. The delta function, for example, remains an important tool as a
model for a strong signal with a short duration (a 'pulse'). Moreover, the response
to the delta function is essential in the theory of linear systems (see chapter 14). The
theory of the Laplace transform of distributions will be developed in section 13.4.
In particular it will be shown that the Laplace transform of the delta function is
the constant function 1, just as for the Fourier transform. We will also go into
the relationship between the Laplace transform of distributions and differentiation,
and we will treat some simple results on the Laplace transform and convolution of
distributions.
In section 13.5 the fundamental theorem of the Laplace transform is proven. In
the theory of the Laplace transform this is an important theorem; it implies, for
example, that the Laplace transform is one-to-one. However, in order to apply the
fundamental theorem in practice (and so recover the function f (t) from F(s)), a
fair amount of knowledge of the theory of complex integration is needed. This
theory is beyond the scope of this book. Therefore, if we want to recover f (t) from
F(s), we will confine ourselves to the use of tables, the properties of the Laplace
transform, and partial fraction expansions. This method will be illustrated by means
of examples.
288
13.1 Convolution
289
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know and can apply the convolution of causal functions and the convolution theorem of the Laplace transform
- know and can apply the initial and final value theorems
- can determine the Laplace transform of a periodic function
- know and can apply the Laplace transform of some simple distributions
- know and can apply the differentiation rule for the Laplace transform of distributions
- can apply the convolution theorem for distributions in simple cases
- know the uniqueness theorem for the Laplace transform
- can find the inverse Laplace transform of complex functions by using a table, applying the properties of the Laplace transform, and applying partial fraction expansions.
13.1
Convolution
We have already encountered the convolution product of two functions f and g :
R → C in definition 6.4. When, moreover, f and g are both causal (which is
assumed throughout part 4), then one has for t > 0 that
Convolution of causal
functions
( f ∗ g)(t) =
∞
−∞
f (τ )g(t − τ ) dτ =
t
0
f (τ )g(t − τ ) dτ,
since the integrand is zero for both τ < 0 and t − τ < 0. For the same reason one
has ( f ∗ g)(t) = 0 if t ≤ 0 (also see exercise 6.25). If, moreover, we assume that the
causal functions f and g are piecewise smooth, then the existence of the convolution
product is easy to prove. In fact, for fixed t > 0 the function τ → f (τ )g(t − τ )
is then again piecewise smooth as a function of τ and such a function is always
integrable over the bounded interval [0, t]. Hence, for two piecewise smooth causal
functions f and g, the convolution product exists for every t ∈ R and f ∗ g is again
a causal function. One now has the following convolution theorem (compare with
theorem 6.13).
THEOREM 13.1
Convolution theorem for L
Let f and g be piecewise smooth and causal functions. Let the Laplace transforms
F = L f and G = Lg exist as absolutely convergent integrals in a half-plane
Re s > ρ. Then L( f ∗ g) exists for Re s > ρ and
L( f ∗ g)(s) = F(s)G(s).
(13.1)
Proof
Since f (t) = g(t) = 0 for t < 0, it follows for Re s > ρ that
F(s)G(s) =
∞
−∞
f (t)e−st dt
∞
−∞
g(u)e−su du.
Since the second integral does not depend on t, we can write
F(s)G(s) =
∞
∞
−∞
−∞
f (t)g(u)e−s(t+u) du dt.
Now change to the new variable t + u = τ , then
F(s)G(s) =
∞
∞
−∞
−∞
f (t)g(τ − t)e−sτ dτ dt.
13.2 Initial and final value theorems
291
(Lg)(s) = 1/((s − a)(s − b)). Calculate the obtained convolution product explicitly
and verify the convolution theorem for this case.
13.3
Verify the convolution theorem for the functions f (t) = t 2 and g(t) = et .
13.4
Determine a convolution product ( f ∗ g)(t) whose Laplace transform is given by
the following complex function:
a 1/(s 2 (s + 1)),
b s/((s + 2)(s 2 + 4)),
c s/(s 2 − 1)2 ,
d 1/(s 2 − 16)2 .
13.2
Initial and final value theorems
In this section we treat two theorems that can give us information about a function
f (t) straight from its Laplace transform F(s), without the need to determine f (t)
explicitly. The issues at stake are the limiting value of f at the point t = 0, so
immediately after the 'switching on', and the limiting value of f for t → ∞, that
is, the final value after 'a long period of time'. These two results are therefore called
the initial and final value theorems. As a matter of fact, they can also be used in the
opposite direction: given f (t), one obtains from these theorems information about
the behaviour of F(s) for s → 0 and for s → ∞, without having to determine F(s)
explicitly.
A brief explanation of the notation 's → ∞' is appropriate here, since s is complex. In general it will mean that | s | → ∞ (so the modulus of s keeps increasing).
In most cases this cannot be allowed for a Laplace transform F(s) since we might
end up outside the half-plane of convergence. By lims→∞ F(s) we will therefore
always mean that | s | → ∞ and that simultaneously Re s → ∞. In particular, s will
lie in the half-plane of convergence for sufficiently large values of Re s (specifically,
for Re s > σc ).
Similar remarks apply to the limit s → 0, which will again mean that | s | → 0
(as in section 11.2). If the limit for s → 0 of a Laplace transform F(s) is to exist,
then F(s) will certainly have to exist in the half-plane Re s > 0. When the halfplane of convergence is precisely Re s > 0, then the limit for s → 0 has to be taken
in such a way that Re s > 0 as well. By lims→0 F(s) we will therefore always mean
that | s | → 0 and that simultaneously Re s ↓ 0.
Before we start our treatment of the initial value theorem, we will first derive the
following result, which, for that matter, is also useful in other situations and will
therefore be formulated for a somewhat larger class of functions.
THEOREM 13.2
For the Laplace transform F(s) of a piecewise continuous function f (t) we have
lim F(s) = 0,
s→∞
where the limit s → ∞ has to be taken in such a way that Re s → ∞ as well.
Proof
If s → ∞ such that Re s → ∞ as well, then lims→∞ e−st = 0 for any fixed t > 0.
In fact, for s = σ + iω we have e−st = e−σ t e−iωt and limσ →∞ e−σ t = 0 for any
t > 0. Hence we obtain that
lim F(s) = lim
s→∞
s→∞ 0
∞
f (t)e−st dt = 0,
292
13 Fundamental theorem, properties, distributions
if we assume that we may interchange the integral and the limit. When the function
f (t) is of exponential order, then the problem of the interchanging of the limit and
the integral can be avoided. For if | f (t) | ≤ Meαt , then
∞
| F(s) | ≤
0
| f (t) | e−σ t dt ≤ M
∞
0
e(α−σ )t dt =
∞
M
e(α−σ )t
0
α−σ
for all σ = α. We also agreed that Re s = σ → ∞ and for sufficiently large σ one
will have that σ > α, so α − σ < 0. It thus follows that
e(α−σ )t
∞
0
= lim e(α−σ )R − 1 = −1,
R→∞
which proves that | F(s) | ≤ M/(σ − α). In the limit s → ∞ with Re s = σ →
∞, the right-hand side of this inequality tends to zero, from which it follows that
lims→∞ F(s) = 0 as well.
EXAMPLE
For f (t) = (t) we have F(s) = 1/s. Indeed, lims→∞ F(s) = 0.
EXAMPLE 13.2
The constant function F(s) = 1 cannot be the Laplace transform of a piecewise
continuous function f (t). This is because lims→∞ F(s) = 1.
We recall that for a piecewise smooth function f (t) the limit f (0+) = limt↓0 f (t)
will always exist. The initial value theorem is a stronger version of theorem 13.2
and reads as follows.
THEOREM 13.3
Initial value theorem
Let f (t) be a piecewise smooth function with Laplace transform F(s). Then
lim s F(s) = f (0+),
s→∞
(13.2)
where the limit s → ∞ has to be taken in such a way that Re s → ∞ as well.
Proof
We will not prove the theorem in its full generality. However, if we impose an
additional condition on the function f (t), then a simpler proof of the initial value
theorem can be given using theorem 13.2. We therefore assume that in addition f (t)
is continuous for t > 0. Let f be the derivative of f at all points where f exists.
As in the proof of theorem 12.7, it then follows from an integration by parts that
(L f )(s) = lim f (R)e−s R − f (0+) + s F(s) = s F(s) − f (0+),
R→∞
since lim R→∞ e−s R = 0 (see the proof of theorem 12.7). The difference with
theorem 12.7 is the appearance of the value f (0+) because f is not necessarily
continuous at t = 0. If we now apply theorem 13.2 to f (t), then it follows that
lims→∞ (L f )(s) = 0. Hence we obtain that lims→∞ (s F(s) − f (0+)) = 0,
which proves the theorem, under the additional condition mentioned earlier.
Theorem 13.3 can be used in both directions. When F(s) is known and f (t) is
hard to determine explicitly, then one can still determine f (0+), provided that we
know that f is piecewise smooth. When on the other hand f (t) is known and F(s)
is hard to determine, then theorem 13.3 reveals information about the behaviour of
F(s) for s → ∞.
EXAMPLE
The function (t) has the function F(s) = 1/s as Laplace transform. Indeed, 1 =
(0+) = lims→∞ s F(s).
EXAMPLE
Consider the function f (t) = e−bt cosh at. Then f (0+) = 1 and the Laplace
transform F(s) exists, so lims→∞ s F(s) = 1. This can easily be verified since
F(s) = (s + b)/((s + b)2 − a 2 ) (see table 7).
13.2 Initial and final value theorems
293
We now move on to the final value theorem, which relates the final value f (∞) =
limt→∞ f (t) (a notation which will be used henceforth) to the behaviour of F(s)
for s → 0.
THEOREM 13.4
Final value theorem
Let f (t) be a piecewise smooth function with Laplace transform F(s). When
f (∞) = limt→∞ f (t) exists, then
lim s F(s) = f (∞),
(13.3)
s→0
where the limit s → 0 has to be taken in such a way that Re s ↓ 0 as well.
Proof
Again, theorem 13.4 will not be proven in full generality. If we impose a number of
additional conditions on the function, then a simpler proof can be given, as was the
case for theorem 13.3. We first of all assume that in addition f (t) is continuous for
t > 0. As in the proof of theorem 13.3, it then follows that (L f )(s) = s F(s) −
f (0+). Next we will assume that this result is valid in the half-plane Re s > 0. For
the limit s → 0 (with Re s ↓ 0) we then have
lim s F(s) = f (0+) + lim
s→0
∞
s→0 0
f (t)e−st dt.
Now, finally, assume that the limit and the integral may be interchanged, then we
obtain that
lim s F(s) = f (0+) +
s→0
∞
0
f (t) dt.
Since f is piecewise smooth and continuous for t > 0 and since, moreover,
limt→∞ f (t) exists, we have finally established that
lim s F(s) = f (0+) + [ f (t)]∞ = lim f (t) = f (∞).
0
s→0
t→∞
This proves theorem 13.4, using quite a few additional conditions.
We note once again that F(s) in theorem 13.4 must surely exist for Re s > 0, because otherwise one cannot take the limit s → 0. When F(s) is a rational function,
then this means in particular that the denominator cannot have any zero for Re s > 0
(see example 13.3).
One cannot omit the condition that limt→∞ f (t) should exist. This can be shown
using a simple example. The function f (t) = sin t has Laplace transform F(s) =
1/(s 2 + 1). We have lims→0 s F(s) = 0, but limt→∞ f (t) does not exist.
Theorem 13.4 can again be applied in two directions. When F(s) is known,
f (∞) can be determined, provided that we know that f (∞) exists. If, on the other
hand, f (t) is known and f (∞) exists, then theorem 13.4 reveals information about
the behaviour of F(s) for s → 0.
EXAMPLE
The function (t) has Laplace transform F(s) = 1/s and indeed we have 1 =
limt→∞ (t) = lims→0 s F(s).
EXAMPLE 13.3
Consider the function f (t) = e−at with a > 0. Then limt→∞ f (t) = 0 and so
lims→0 s F(s) = 0. This can easily be verified since F(s) = 1/(s + a). Note that
for a < 0 the denominator of F(s) has a zero for s = −a > 0, which means that in
this case F(s) does not exist in the half-plane Re s > 0. This is in agreement with
the fact that limt→∞ f (t) does not exist for a < 0. Of course, the complex function
F(s) = 1/(s + a) remains well-defined for all s = −a and in particular one has
for a = 0 that lims→0 s F(s) = 0. However, the function F(s) is not the Laplace
transform of the function f (t) for Re s ≤ (−a).
294
13 Fundamental theorem, properties, distributions
EXERCISES
13.5
Can the function F(s) = s n (n ∈ N) be the Laplace transform of a piecewise
continuous function f (t)? Justify your answer.
13.6
Determine the Laplace transform F(s) of the following functions f (t) and verify
the initial value theorem:
a f (t) = cosh 3t,
b f (t) = 2 + t sin t,
t
c f (t) = 0 g(τ ) dτ , where g is a continuous function on R.
13.7
In the proof of theorem 13.3 we used the property (L f )(s) = s(L f )(s) − f (0+).
Now consider the function f (t) = (t − 1) cos(t − 1).
a Verify that the stated property does not hold for f .
b Show that the initial value theorem does apply.
13.8
Determine the Laplace transform F(s) of the following functions f (t) and verify
the final value theorem:
a f (t) = e−3t ,
b f (t) = e−t sin 2t,
c f (t) = 1 − (t − 1).
13.9
Determine whether the final value theorem can be applied to the functions cos t and
sinh t.
13.10
For the complex function F(s) = 1/(s(s − 1)) one has lims→0 s F(s) = −1. Let
f (t) be the function with Laplace transform F(s) in a certain half-plane in C.
a Explain, without determining f (t), why the final value theorem cannot be
applied.
b Verify that F(s) = 1/(s − 1) − 1/s. Subsequently determine f (t) and check
that f (∞) = limt→∞ f (t) does not exist.
13.3
Causal periodic function
Periodic functions
In general, the Laplace transform is more comfortable to use than the Fourier transform since many of the elementary functions possess a Laplace transform, while
on the contrary the Fourier transform often only exists when the function is considered as a distribution. For a periodic function the Fourier transform also exists only
if it is considered as a distribution (see section 9.1.2). In this section we will see
that the Laplace transform of a periodic function can easily be determined without
distribution theory. Since we are only working with causal functions in the Laplace
transform, a periodic function f (t) with period T > 0 will from now on be a function on [0, ∞) for which f (t + T ) = f (t) for all t ≥ 0. In figure 13.1a a periodic
function with period T is drawn. Now consider the function φ(t) obtained from
f (t) by restricting f (t) to one period T , so
φ(t) =
f (t)
0
for 0 ≤ t < T ,
elsewhere.
See figure 13.1b. Using the shifted unit step function, the function φ(t) can be
written as
φ(t) = f (t) − (t − T ) f (t − T ).
13.3 Periodic functions
295
a
f (t )
T
0
2T
3T
t
b
ϕ (t )
T
0
t
FIGURE 13.1
A periodic function f (a) and its restriction φ (b).
If we now apply the shift property in the time domain (theorem 12.4), then it follows
that
(s) = F(s) − e−sT F(s) = (1 − e−sT )F(s),
where (s) and F(s) are the Laplace transforms of φ and f respectively. If s =
σ + iω and σ > 0, then e−sT = e−σ T < 1, and hence 1 − e−sT = 0. For
Re s > 0 we may thus divide by 1 − e−sT and it then follows that
F(s) =
(s)
,
1 − e−sT
where
(s) =
∞
0
φ(t)e−st dt =
T
0
f (t)e−st dt.
Note that for a piecewise continuous function the preceding integral over the
bounded interval [0, T ] exists for every s ∈ C; for the function φ the abscissa
of convergence is thus equal to −∞. For the periodic function f the abscissa of
convergence is equal to 0 and hence the Laplace transform F(s) exists for Re s > 0.
We see here that the Laplace transform of a periodic function can be expressed in
a simple way in terms of the Laplace transform of the function restricted to one
period. These results are summarized in the following theorem.
THEOREM 13.5
Laplace transform of periodic
functions
Let f be a piecewise smooth and periodic function with period T and let (s) be the
Laplace transform of φ(t) = f (t) − (t − T ) f (t − T ). Then the Laplace transform
F(s) of f (t) is for Re s > 0 given by
F(s) =
EXAMPLE 13.4
(s)
,
1 − e−sT
where
(s) =
T
0
f (t)e−st dt.
(13.4)
Consider the periodic block function f (t) with period 2 defined by f (t) = 1− (t −
1) for 0 ≤ t < 2; so f (t) = 1 for 0 ≤ t < 1 and f (t) = 0 for 1 ≤ t < 2. See
figure 13.2. We then have φ(t) = 1 − (t − 1) and from tables 7 and 8 we see
13.4 Laplace transform of distributions
297
1
a
0
2a
3a
4a
5a
t
–1
FIGURE 13.3
Periodic block function of exercise 13.12.
13.4
Laplace transform of distributions
Up till now we have been able to avoid the use of the theory of distributions in
the Laplace transform. For every function we could always calculate the Laplace
transform using the defining integral (12.1). Still, we will need a limited theory of
the Laplace transform of distributions. This is because the delta function will remain
an important tool in the theory of linear systems: in the application of the Laplace
transform the impulse response again plays an essential role (see chapter 14).
In section 13.4.1 the main results will be derived in an intuitive way. For the
remainder of this book it will suffice to accept these results as being correct. In
section 13.4.2 we treat the mathematical background necessary to give a rigorous
definition of the Laplace transform of a distribution. This will enable us to prove
the results from section 13.4.1. Section 13.4.2 may be omitted without any consequences for the remainder of the book.
13.4.1
Intuitive derivation
To get an intuitive idea of the Laplace transform of the delta function δ(t), we consider the causal rectangular pulse rb (t) of height 1/b and duration b > 0. Hence,
∞
rb (t) = ( (t) − (t − b))/b. See figure 13.4. Note that −∞ rb (t) dt = 1 for every
b > 0. For b ↓ 0 we thus obtain an object which, intuitively, will be an approximation for the delta function (see section 8.1). Since (Lrb )(s) = (1 − e−bs )/sb, we
expect that for b ↓ 0 this will give us the Laplace transform of the delta function.
When b ↓ 0, then also −sb → 0 for any s ∈ C. Now write z = −sb, then we
have to determine the limit limz→0 (e z − 1)/z. But this is precisely the derivative
of the analytic function e z at z = 0. Since (e z ) = e z , we obtain for z = 0 that
(e z ) (0) = 1 and hence limb↓0 (Lrb )(s) = 1. As for the Fourier transform, we
thus expect that the Laplace transform of the delta function will equal the constant
function 1.
We will now try to find a possible definition for the Laplace transform of a distribution. First we recall that a function f (t) can be considered as a distribution T f
by means of the rule
Tf ,φ =
∞
−∞
f (t)φ(t) dt
for φ ∈ S.
(13.5)
298
13 Fundamental theorem, properties, distributions
4
r 1/4(t )
3
r 1/2(t )
2
r 1(t )
1
r2(t )
1/2
0
1 1
4 2
1
2
t
FIGURE 13.4
The rectangular pulse function rb (t) for some values of b.
(See (8.12).) If we now take for φ the function e−st , then it follows for a causal
function f (t) that
T f , e−st =
∞
0
f (t)e−st dt = F(s).
(13.6)
The definition of the Laplace transform U = LT of a distribution T now seems
quite obvious, namely as the complex function
U (s) = T (t), e−st .
(13.7)
Note that T (t) acts on the variable t and that s is always an (arbitrary) fixed complex number. Furthermore, the Laplace transform of a distribution is no longer a
distribution, but just a complex function. When f (t) is a causal function defining a
distribution T f , then it follows from (13.6) that the definition in (13.7) results in the
ordinary Laplace transform F(s) of f (t) again:
(LT f )(s) = F(s)
(13.8)
(assuming that F(s) exists in a certain half-plane of convergence).
Two problems arise from definition (13.7). First of all it is easy to see that the
function e−st is not an element of the space S(R) of rapidly decreasing functions;
hence, T, e−st is not well-defined for an arbitrary distribution T . A second problem concerns the analogue of the notion 'causal function' for a distribution, since
(13.6) is only valid for causal functions. Of course we would like to call the distribution T f 'causal' if f is a causal function. But what shall we mean in general by
a 'causal distribution'? This will have to be a distribution being 'zero for t < 0'.
In section 13.4.2 we will return to these problems and turn definition (13.7) into a
rigorous one. In this section we use (13.7) for all distributions that are 'zero for
t < 0' according to our intuition. Examples of such distributions are the delta function δ(t), the derivatives δ (n) (t) (n ∈ N), the delta function δ(t − a) with a > 0, and
the derivatives δ (n) (t − a) (a > 0 and n ∈ N).
EXAMPLE 13.5
From formula (13.7) and definition 8.2 of δ(t) it follows that (Lδ(t))(s) =
δ(t), e−st = 1 since e−st = 1 for t = 0. Hence, Lδ = 1.
13.4 Laplace transform of distributions
299
EXAMPLE 13.6
For the delta function δ(t − a) with a > 0 it follows from formulas (13.7) and (8.10)
that (Lδ(t − a))(s) = δ(t − a), e−st = e−as . Hence, (Lδ(t − a))(s) = e−as .
EXAMPLE 13.7
For the derivative δ (n) (t) it follows from formulas (13.7) and (8.17) that
(Lδ (n) (t))(s) = δ (n) (t), e−st = (−1)n δ(t), (e−st )(n) . But (e−st )(n) =
(−s)n e−st and so (Lδ (n) (t))(s) = s n δ(t), e−st
(Lδ (n) (t))(s) = s n .
Linearity
= sn .
This proves that
The properties of the Laplace transform of distributions are similar to the properties of the Laplace transform of functions (and to the properties of the Fourier transform). The simplest property is linearity. It follows immediately from the definition
in (13.7) and definition 8.5 of the addition of distributions and the multiplication of
a distribution by a complex constant.
EXAMPLE
The Laplace transform of 3iδ(t − 4) + 5 sin t is given by the complex function
3ie−4s + 5/(s 2 + 1).
Differentiation in the time
domain
Besides linearity, the most important property will be the differentiation rule in
the time domain: when T is a distribution with Laplace transform LT , then
(LT (n) )(s) = s n (LT )(s)
(13.9)
for n ∈ N. (Compare this with the differentiation rule in the time domain in (12.10).)
The proof of (13.9) is easy and follows just as in example 13.7 from formulas (13.7)
and (8.17):
(LT (n) )(s) = T (n) , e−st = (−1)n T, (e−st )(n) = s n T, e−st
= s n (LT )(s).
EXAMPLE
We know from example 13.5 that Lδ = 1. From (13.9) it then follows that
(Lδ (n) (t))(s) = s n , in agreement with example 13.7.
Formula (13.9) can in particular be applied to a causal function f defining a
distribution T f , and so it is much more general than the differentiation rule in the
time domain from theorem 12.7. We will give some examples.
EXAMPLE
The Laplace transform F(s) of f (t) = (t − 1) is given by F(s) = e−s /s. Of
course, the function (t − 1) is not differentiable on R, but considered as a distribution we have that (t − 1) = δ(t − 1). According to (13.9) with n = 1 one then
obtains that (Lδ(t − 1))(s) = s(L (t − 1))(s) = e−s . This is in accordance with
example 13.6. Applying (13.9) for n ∈ N it follows that (Lδ (n−1) (t − 1))(s) =
s n (L (t − 1))(s) = s n−1 e−s . It is not hard to obtain this result in a direct way (see
exercise 13.16).
EXAMPLE 13.8
For the (causal) function f (t) = cos t one has (cos t) = δ(t) − sin t, considered as
a distribution (see example 8.10). From formula (13.9) with n = 1 it then follows
that (L(δ(t) − sin t))(s) = s(L cos t)(s). This identity can easily be verified since
Lδ = 1, (L sin t)(s) = 1/(s 2 + 1) and (L cos t)(s) = s/(s 2 + 1).
In example 13.8 we encounter a situation that occurs quite often: a causal function f (t) having a jump at the point t = 0, being continuously differentiable
otherwise, and defining a distribution T f . Let us assume for convenience that
f (0) = f (0+), in other words, let us take the function value at t = 0 equal to
the limiting value f (0+). The magnitude of the jump at t = 0 is then given by
300
13 Fundamental theorem, properties, distributions
the value f (0). We moreover assume that the Laplace transforms of f and f exist (in the ordinary sense), and according to (13.8) we thus have LT f = L f and
LT f = L f . For the derivative T f of f , considered as a distribution, one has,
according to the jump formula (8.21), that T f = f (t) + f (0)δ(t), where f (t) is
the derivative of f (t) for t = 0. Hence,
(LT f )(s) = (L f )(s) + f (0)(Lδ)(s).
Since Lδ = 1 it then follows from (13.9) that
(L f )(s) = (LT f )(s) − f (0) = s(L f )(s) − f (0),
where we used LT f = L f . Applying this rule repeatedly, we obtain for a causal
function f (t) being n times continuously differentiable for t ≥ 0 that
(L f (n) )(s) = s n (L f )(s) −
n
s n−k f (k−1) (0).
(13.10)
k=1
Laplace transform and
convolution
Here f (k) is the kth derivative of f (t) for t = 0 and it is assumed that all Laplace
transforms exist in the ordinary sense. Formula (13.10) is used especially for solving differential equations by means of the Laplace transform. In addition to an
unknown function f (t) satisfying a differential equation, only the values f (k) (0)
(k = 0, 1, . . . , n − 1) are given (see chapter 14).
We close with some elementary results on the Laplace transform of a convolution
product of distributions. From section 9.3 it is known that the convolution product
δ ∗ T exists for any distribution T and that δ ∗ T = T (see (9.19)). If the Laplace
transform LT of T exists, then this implies that L(δ ∗ T ) = LT , and since Lδ = 1
we thus see that L(δ ∗ T ) = Lδ · LT . This shows that in this particular case the
convolution theorem for the Laplace transform also holds for distributions. Using
the same method one can verify in a direct way the convolution theorem for distributions for a limited number of other cases as well. Let us give a second example.
In (9.20) we saw that δ ∗ T = T for a distribution T . Since (LT )(s) = s(LT )(s),
it follows that L(δ ∗ T )(s) = s(LT )(s). But (Lδ )(s) = s and so we indeed have
L(δ ∗ T ) = Lδ · LT .
In most cases these simple results on the Laplace transform of convolution products will suffice in the applications.
13.4.2
Mathematical treatment∗
In section 13.4.1 it was pointed out that the definition in (13.7) of the Laplace transform of a distribution gives rise to two problems. First of all it was noted that the
function e−st is not an element of the space S. For Re s < 0, for example, we have
that e−st → ∞ for t → ∞. This problem is solved by simply allowing a larger
class of functions φ in (13.5). To this end we replace S by the space E defined as the
set of all C ∞ -functions on R (this space of arbitrarily often differentiable functions
has previously been used towards the end of section 9.3). As a consequence of this
change, the number of distributions for which we can define the Laplace transform
is reduced considerably. However, this is unimportant to us, since we will only need
a very limited theory of the Laplace transform of distributions (in fact, only the delta
function and its derivatives are needed). Note that the complex-valued function e−st
indeed belongs to E.
The second problem involved finding the analogue of the notion of causality for
a distribution. This should be a distribution being 'zero for t < 0'. If the function
13.4 Laplace transform of distributions
301
f in (13.5) is causal and we choose a φ ∈ S such that φ(t) = 0 for t ≥ 0, then it
follows that
Tf ,φ =
∞
−∞
f (t)φ(t) dt =
∞
0
f (t)φ(t) dt = 0.
We can now see that the definition of a causal distribution should be as follows.
DEFINITION 13.1
Causal distribution
Let T be a distribution. We say that T = 0 on the interval (−∞, 0) when T, φ = 0
for every φ ∈ S with φ(t) = 0 for t ≥ 0. Such a distribution is called a causal
distribution.
EXAMPLE
If f is a causal function defining a distribution T f , then T f is a causal distribution.
This has been shown earlier.
EXAMPLE
The delta function δ is a causal distribution since δ, φ = φ(0) = 0 for φ ∈ S with
φ(t) = 0 for t ≥ 0. More generally we have that the delta function δ(t − a) with
a ≥ 0 is a causal distribution. In fact, it follows for φ ∈ S with φ(t) = 0 for t ≥ 0
that δ(t − a), φ = φ(a) = 0 since a ≥ 0.
The causal distributions, which moreover can be defined on the space E, will now
form the set of distributions for which the definition of the Laplace transform of a
distribution in (13.7) makes sense.
DEFINITION 13.2
Laplace transform of a
distribution
Let T be a distribution which can be defined on the space E of all C ∞ -functions on
R. Assume moreover that T is causal. Then the Laplace transform U = LT of T is
defined as the complex function U (s) = T (t), e−st .
As was noted following (13.7), the Laplace transform U (s) of a distribution is a
complex function. From definition 13.2 we see that U (s) is defined on the whole
of C. One even has that U (s) is an analytic function on C! The proof of this result
is outside the scope of this book; we will not need it anyway. Also, in concrete
examples this result will follow from the calculations. We will give some examples.
EXAMPLE
The delta function is a causal distribution which can be defined on the space E, since
δ, φ = φ(0) has a meaning for every continuous function φ (see section 8.2.2).
Hence, the Laplace transform of δ is well-defined and as in example 13.5 it follows
that Lδ = 1. Note that the constant function 1 is an analytic function on C.
EXAMPLE
Consider the delta function δ(t − a) at the point a for a > 0. Then Lδ(t − a) is
again well-defined and (Lδ(t − a))(s) = e−as (see example 13.6). This is again an
analytic function on C.
EXAMPLE
For all derivatives of the delta function δ(t) the Laplace transform is also welldefined and as in example 13.7 it follows that (Lδ (n) (t))(s) = s n . The function s n
is again an analytic function on C.
Many of the properties that hold for the Laplace transform of functions, can be
translated into properties of the Laplace transform of distributions. The linearity and
the differentiation rule for the Laplace transform of distributions have already been
treated in section 13.4.1. The shift property in the s-domain and the scaling rule
also remain valid for distributions, but because of the limited applicability of these
rules, we will not prove them. As an illustration we will prove the shift property in
the time domain here.
Let T (t) be a distribution whose Laplace transform U (s) exists (so T is causal
and defined on E). Then one has for a ≥ 0 that
Shift in the time domain
(LT (t − a))(s) = e−as U (s),
(13.11)
302
13 Fundamental theorem, properties, distributions
where T (t − a) is the distribution shifted over a (see definition 9.2). In order to
prove this rule, we first show that T (t − a) is causal. So let φ ∈ S with φ(t) = 0 for
t ≥ 0. Then T (t − a), φ(t) = T (t), φ(t + a) = 0 for t ≥ 0 since T is causal
and t + a ≥ 0 (because a ≥ 0 and t ≥ 0). Hence, T (t − a) is causal. It also follows
immediately that T (t − a) is defined on E, since T is defined on E and the function
ψ(t) = φ(t + a) belongs to E when φ(t) ∈ E. Hence, the Laplace transform of
T (t − a) exists and from definition 13.2 it then follows that
(LT (t − a))(s) = T (t − a), e−st = T (t), e−s(t+a)
= e−as T (t), e−st = e−as U (s),
proving (13.11).
We know that Lδ = 1. From (13.11) it then follows that (Lδ(t − a))(s) = e−as ,
which is in accordance with example 13.6.
EXAMPLE
Some simple results on the convolution in relation to the Laplace transform of
distributions have already been treated in section 13.4.1. As for the Fourier transform, there are of course general convolution theorems for the Laplace transform of
distributions. A theorem comprising all the examples we have treated earlier reads
as follows.
THEOREM 13.6
Convolution theorem
Let S and T be causal distributions which can be defined on the space E. Then S ∗ T
is a causal distribution which can again be defined on the space E and L(S ∗ T ) =
LS · LT .
Proof
The proof that S ∗ T is a causal distribution which can be defined on E is beyond the scope of this book. Assuming this result, it is not hard to prove that
L(S ∗ T ) = LS · LT . For s ∈ C one has L(S ∗ T )(s) = (S ∗ T )(t), e−st =
S(τ ), T (t), e−s(τ +t) , where we used definition 9.3 of convolution. It then follows that L(S ∗ T )(s) = S(τ ), T (t), e−st e−sτ . For fixed τ , the complex number
e−sτ does not depend on t. The number e−sτ can thus be taken outside of the action of the distribution T (t). This results in L(S ∗ T )(s) = S(τ ), T (t), e−st e−sτ .
But now T (t), e−st is, for fixed t, a complex number which does not depend on τ
and so it can be taken outside of the action of the distribution S(τ ). This gives the
desired result: L(S ∗ T )(s) = S(τ ), e−sτ T (t), e−st = (LS)(s) · (LT )(s).
All the examples in section 13.4.1 satisfy the conditions of theorem 13.6. This
is because the delta function and all of its derivatives are causal distributions which
can be defined on the space E.
EXERCISES
13.15
Consider the function f a (t) = ae−at (t) for a > 0 and let Fa (s) = (L f a )(s).
∞
a Sketch the graph of f a (t) and show that −∞ f a (t) dt = 1.
b Show that lima→∞ f a (t) = 0 for every t > 0 and that lima→∞ f a (t) = ∞ for
t = 0. Conclude from parts a and b that the function f a (t) is an approximation of
the delta function δ(t) for a → ∞.
c Determine Fa (s) and calculate lima→∞ Fa (s). Explain your answer.
13.16
Use formula (13.7) to determine the Laplace transform of the nth derivative
δ (n) (t − a) of the delta function at the point a for a ≥ 0.
13.17
Show that the Laplace transform of distributions is linear.
304
13 Fundamental theorem, properties, distributions
If we now multiply the left- and right-hand sides by eσ t , then (13.12) indeed follows,
valid for σ > α and t ≥ 0.
Note that in (13.12) we only integrate over ω and that the value of σ is irrelevant
(as long as σ > α). As for the Fourier transform (see section 7.2.1), the fundamental
theorem immediately implies that the Laplace transform is one-to-one.
THEOREM 13.8
The Laplace transform is
one-to-one
Let f (t) and g(t) be two piecewise smooth functions of exponential order and let
F(s) and G(s) be the Laplace transforms of f (t) and g(t). When F(s) = G(s) in a
half-plane Re s > ρ, then f (t) = g(t) at all points where f and g are continuous.
Proof
Let t ∈ R be a point where both f and g are continuous. Since F(s) = G(s) for
Re s > ρ, it follows from the fundamental theorem that (in the following integrals
we have s = σ + iω with σ > ρ)
A
A
1
1
F(s)est dω = lim
G(s)est dω = g(t).
A→∞ 2π −A
A→∞ 2π −A
f (t) = lim
Inversion theorem
Inversion formula
Inverse Laplace transform
This theorem is often used implicitly if we are asked to determine the function
f (t) whose Laplace transform F(s) is given. Suppose that an f (t) is found within
the class of piecewise smooth functions of exponential order. Then we know by
theorem 13.8 that this is the only possible function within this class, except for
a finite number of points on a bounded interval (also see the similar remarks on
the uniqueness of the Fourier transform in section 7.2.1). Without proof we also
mention that the Laplace transform of distributions is one-to-one as well.
Theorem 13.7, and the resulting theorem 13.8, are important results in the theory of the Laplace transform. As for the Fourier transform, theorem 13.7 tells us
precisely how we can recover the function f (t) from F(s). Obtaining f from F is
called the inverse problem and therefore theorem 13.7 is also known as the inversion
theorem and (13.12) as the inversion formula. We will call the function f the inverse
Laplace transform of F. Still, (13.12) will not be used for this purpose. In fact, calculating the integral in (13.12) requires a thorough knowledge of the integration of
complex functions over lines in C, an extensive subject which is outside the scope
of this book. Hence, the fundamental theorem of the Laplace transform will not be
used in the remainder of this book, except in the form of the frequent (implicit) application of the fact that the Laplace transform is one-to-one. Moreover, in practice
it is often a lot easier to determine the inverse Laplace transform of a function F(s)
by using tables, applying the properties of the Laplace transform, and using partial
fraction expansions.
Partial fraction expansions have been treated in detail in section 2.2 and will be
used to obtain the inverse Laplace transform of a rational function F(s). It will
be assumed that F(s) has real coefficients; in practice this is usually the case. We
will now describe in a number of steps how the inverse Laplace transform of such a
rational function F(s) can be determined.
Step 1
If the degree of the numerator is greater than or equal to the degree of the denominator, then we perform a division. The function F(s) is then the sum of a polynomial
and a rational function for which the degree of the numerator is smaller than the
degree of the denominator. The polynomial gives rise to distributions in the inverse
Laplace transform since s n = (Lδ (n) (t))(s).
EXAMPLE
We want to determine the function/distribution f (t) having Laplace transform
F(s) = (s 3 − s 2 + s)/(s 2 + 1). Since the degree of the numerator is greater than
CHAPTER 14
Applications of the Laplace transform
INTRODUCTION
The greater part of this chapter consists of section 14.1 on linear time-invariant
continuous-time systems (LTC-systems). The Laplace transform is very well suited
for the study of causal LTC-systems where switch-on phenomena occur as well: at
time t = 0 'a switch is thrown' and a process starts, while prior to time t = 0 the
system was at rest. The input u(t) will thus be a causal signal and since the system
is causal, the output y(t) will be causal as well. Applying the Laplace transform
is then quite natural, especially since the Laplace transform exists for a large class
of inputs u(t) as an ordinary integral in a certain half-plane Re s > ρ. This is in
contrast to the Fourier transform, where distributions are needed more often. For
the Laplace transform we can usually restrict the distribution theory to the delta
functions δ(t − a) with a ≥ 0 (and their derivatives). As in chapter 10, the response
h(t) to the delta function δ(t) again plays an important role. The Laplace transform
H (s) of the impulse response is called the transfer function or system function. An
LTC-system is then described in the s-domain by the simple relationship Y (s) =
H (s)U (s), where Y (s) and U (s) are the Laplace transforms of, respectively, the
output y(t) and the input u(t) (compare this with (10.6)).
In this chapter we will mainly limit ourselves to systems described by ordinary
linear differential equations with constant coefficients and with initial conditions all
equal to zero (since the system is at rest at t = 0). The transfer function H (s) is
then a rational function of s and the impulse response can thus be determined by a
partial fraction expansion and then transforming this back to the time domain. As
we know, the response y(t) of the system to an arbitrary input u(t) is given by the
convolution of h(t) with u(t) (see section 10.1). In order to find the response y(t)
for a given input u(t), it is often easier first to determine the Laplace transform U (s)
of u(t) and subsequently to transform H (s)U (s) back to the time domain. This is
because U (s), and hence Y (s) = H (s)U (s) as well, is a rational function for a large
class of inputs. The inverse Laplace transform y(t) of Y (s) can then immediately be
determined by a partial fraction expansion. This simple standard solution method is
yet another advantage of the Laplace transform over the Fourier transform.
If we compare this with the classical method for solving ordinary linear differential equations with constant coefficients (using the homogeneous and the particular
solution), then the Laplace transform again has the advantage. This is because it
will turn out that the Laplace transform takes the initial conditions immediately into
account in the calculations. This reduces the amount of calculation considerably,
especially for higher order differential equations. As a disadvantage of the Laplace
transform, we mention that in general one cannot give a straightforward interpretation in terms of spectra, as is the case for the Fourier transform.
For the differential equations treated in section 14.1, all the initial conditions will
always be zero. In section 14.2 we will show that the Laplace transform can equally
310
14.1 Linear systems
311
well be applied to ordinary linear differential equations with constant coefficients
and with arbitrary initial conditions. In essence, nothing will change in the solution
method from section 14.1. Even more general are the systems of several coupled
ordinary linear differential equations with constant coefficients from section 14.3.
Again these can be solved using the same method, although in the s-domain we
have not one equation, but a system of several equations. For convenience we confine ourselves to systems of two differential equations. Finally, we briefly describe
in section 14.4 how the Laplace transform can be used to solve partial differential
equations with initial and boundary conditions. By applying the Laplace transform
to one of the variables, the partial differential equation becomes an ordinary differential equation, which is much easier to solve.
LEARNING OBJECTIVES
After studying this chapter it is expected that you
- know the concept of transfer function or system function of an LTC-system
- can determine the transfer function and the impulse and step response of a causal
LTC-system described by an ordinary linear differential equation with constant
coefficients
- know the relation between the input and the output in the s-domain using the
transfer function and can use it to calculate outputs
- can verify the stability using the transfer function
- can apply the Laplace transform in solving ordinary linear differential equations
with constant coefficients and arbitrary initial conditions
- can apply the Laplace transform in solving systems of two coupled ordinary linear
differential equations with constant coefficients
- can apply the Laplace transform in solving partial differential equations with initial
and boundary conditions.
14.1
Linear systems
14.1.1
The transfer function
The basic concepts from the theory of LTC-systems (linear time-invariant continuoustime systems) have been treated extensively in chapter 1 and section 10.1. Let us
summarize the most important concepts.
An LTC-system L associates with any input u(t) an output y(t). One also calls
y(t) the response to u(t). When h(t) is the impulse response, that is, h(t) is the
response to δ(t), then it follows for an arbitrary input u(t) that
y(t) = Lu(t) = (h ∗ u)(t)
(14.1)
(see (10.3)). An LTC-system is thus completely determined by the impulse response
h(t). Besides the impulse response we also introduced in section 10.1 the step response a(t), that is to say, the response of the system to the unit step function (t).
We recall that h(t) is the derivative of a(t) (considered as a distribution, if necessary). Using the convolution theorem of the Laplace transform one can translate
relation (14.1) to the s-domain. To this end we assume, as in section 10.2, that for
the LTC-systems under consideration we may apply the convolution theorem, in the
distribution sense if necessary. If U (s), Y (s) and H (s) are the Laplace transforms
of u(t), y(t) and h(t) respectively, then it follows from the convolution theorem that
Y (s) = H (s)U (s).
(14.2)
312
14 Applications of the Laplace transform
The function H (s) plays the same important role as the frequency response from
chapter 10.
DEFINITION 14.1
System function
Transfer function
Let h(t) be the impulse response of an LTC-system. Then the system function or
transfer function H (s) of the LTC-system is the Laplace transform of h(t) (in the
distribution sense, if necessary).
EXAMPLE 14.1
Consider the integrator from example 10.3 with impulse response h(t) = (t). The
Laplace transform of (t) is 1/s (see table 7) and so H (s) = 1/s. Hence, the
response y(t) of the integrator to an input u(t) is described in the s-domain by
Y (s) = U (s)/s.
Switched-on system
In practical situations we are usually dealing with systems where switch-on phenomena may occur: at time t = 0 a system, being at rest, is switched on. The inputs
are then causal and when the LTC-system is causal, then the output will be causal as
well (theorem 1.2). In this chapter we will limit ourselves to causal LTC-systems
and, moreover, we will always assume that all inputs u(t), and thus all outputs y(t)
as well, are causal: u(t) = y(t) = 0 for t < 0. Here we will also admit distributions 'which are zero for t < 0': the delta functions δ(t − a) with a ≥ 0 and
their derivatives (see section 13.4). In particular the impulse response will also be
a causal signal. If now the Laplace transform H (s) exists |
Math 1428 - College Algebra With Applications
This is an information sheet only, not the course syllabus.
COURSE DESCRIPTION
The study of algebra with emphasis on applications. This course should not be taken
by students planning to enroll in calculus. Topics include, but are not limited to,
matrices, functions, conic sections, polynomials, exponential and logarithmic functions,
and sequences and series. Prerequisite: Demonstrated geometry competency (level 2),
and Mathematics 0482 (or college equivalent) with a grade of "C" or better or a qualifying
score on the mathematics placement test or a qualifying A.C.T math score (3 lecture
hours)
COURSE MATERIALS
Follow the instructions below to locate information on the textbook and other materials
for this course. 1. Click on myACCESS. 2. Click on Search for Credit Classes. 3. From the Term drop-down box select the term. 4. Choose your course from the Subjects drop-down menu. 5. In the Course # field, enter your course number. 6. In the Section field, enter the course section number if known. 7. From the Course Types drop-down menu select Flexible Learning. 8. Scroll to the bottom of the page and click on SUBMIT. 9. Click on the Section Name and Title link. 10. Click on Click here for prices of required textbook(s) and supplies and course material information will be displayed.
Alternatively, you can visit the COD Bookstore website to find this information |
Calculus For Dummies
Book Information
The mere thought of having to take a required calculus course is enough to make legions of students break out in a cold sweat. Others who have no intention of ever studying the subject have this notion that calculus is impossibly difficult unless you happen to be a direct descendant of Einstein.
Well, the good news is that you can master calculus. It's not nearly as tough as its mystique would lead you to think. Much of calculus is really just very advanced algebra, geometry, and trig. It builds upon and is a logical extension of those subjects. If you can do algebra, geometry, and trig, you can do calculus.
Calculus For Dummies is intended for three groups of readers:
Students taking their first calculus course – If you're enrolled in a calculus course and you find your textbook less than crystal clear, this is the book for you. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series.
Students who need to brush up on their calculus to prepare for other studies – If you've had elementary calculus, but it's been a couple of years and you want to review the concepts to prepare for, say, some graduate program, Calculus For Dummies will give you a thorough, no-nonsense refresher course.
Adults of all ages who'd like a good introduction to the subject – Non-student readers will find the book's exposition clear and accessible. Calculus For Dummies takes calculus out of the ivory tower and brings it down to earth.
This is a user-friendly math book. Whenever possible, the author explains the calculus concepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry. Then, you'll see how the calculus concepts work in concrete examples. All explanations are in plain English, not math-speak. Calculus For Dummies covers the following topics and more:
Real-world examples of calculus
The two big ideas of calculus: differentiation and integration
Why calculus works
Pre-algebra and algebra review
Common functions and their graphs
Limits and continuity
Integration and approximating area
Sequences and series
Don't buy the misconception. Sure calculus is difficult – but it's manageable, doable. You made it through algebra, geometry, and trigonometry. Well, calculus just picks up where they leave off – it's simply the next step in a logical progression.
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"... We constit ..."
We constitute a new paradigm where the whole solution process is viewed as an object that can be discussed and manipulated. By using structured derivations and a minimal amount of logical syntax, we can write solution to typical problems in not only algebra and equation solving but also in, e.g., real analysis. We argue why structured derivations give students a better grasp of problem solutions and better possibilities to reread and discuss solutions afterwards, as compared with traditional informal approaches to writing down solutions.
"... Abstract c ..."
Abstract calculational style of reasoning, the emphasis on the algorithmic nature of mathematics, and the promotion of self-discovery by the students. These ideas are discussed and the case is made, through a number of examples that show the teaching style that we want to introduce, for their relevance in shaping mathematics training for the years to come. In our opinion, the education of software engineers that work effectively with formal methods and mathematical abstractions should start before university and would benefit from the ideas discussed here. We are all shaped by the tools we use, in particular the formalisms we use shape our thinking habits, for better or for worse, and that means we have to be very careful in the choice of what we learn and teach, for unlearning is really not possible. — E. W. DIJKSTRA in [15] 1 |
In Grade 8 and 9 all pupils are required to take Mathematics as a subject.
A mark of 40% has to be achieved to pass Mathematics.
The use of calculators in these grades are minimised as pupils tend to rely on them rather than learning important principles. The focus in these grades should be on the process and the techniques that are needed rather than the answers.
Grade10, 11, 12
In Grade 10, 11 and 12 pupils needs to choose between Mathematics and Mathematical Literacy. A pupil is required to pass Mathematics (40%) in Grade 9, in order to choose Mathematics as a subject in Grade 10.
All grades are placed in classes according to their previous year's marks. This allows teachers to gear their lessons towards a specific class. All classes cover the same work and write the same test on the same day.
Maths Olympiad
Pupils take part in the South African Mathematics Olympiad. The first round is written in March. The junior division consists of separate papers for grades 8 and 9 and the senior division of one paper for grades 10 to 12. Each paper consists of 20 questions with multiple-choice answers and learners have one hour to complete the paper.
Learners who attain 50% or higher in the first round qualify for the second round which is written in May. There are separate papers for the juniors and seniors. This time the grade 8 ad 9 learners write the same paper. Learners have two hours to complete twenty questions.
Support lessons
Support lessons are offered to all learners. Please refer to the Academics page for the latest support schedule.
Support lessons are not revision sessions and are there to assist learners with specific problems they have identified in work already covered. Learners are encouraged to inform teachers when they will attend lessons.
Advanced Programme Mathematics
Advanced Programme Mathematics can be taken as an extra subject. It assists gifted students with any mathematical or financial course at tertiary level.
In Gr. 10 the subject will be taught by Ms Collie on a Wednesday morning 6:30 -7:30.
In Gr. 11 Mr. Hall will be teaching the subject on a Monday and Thursday morning 06:30 - 07:30.
In Gr. 12 the subject is taught by Mr Neervoort on Monday and Thursday mornings 06:30 - 07:30.
Grade 10, 11 and 12 Work Plan
To assist learners to plan their year and revise for tests & exams, the following is a rough work plan for Gr. 10 – 12 for 2014. Please note that minor changes could occur: |
Math Center
The Math Center is a non-credit, Community Education class which provides assistance
in mathematics as a completely free service. Current Allan Hancock College students
as well as other individuals who are 18 years or older may fill out a simple registration
form and attend as frequently as they want. Registration forms may be found in the
Math Center or at Community Education in Building S.
The goal of the Math Center (sometimes called the Math Lab) is to assist students
in the successful completion of any Allan Hancock College mathematics class by providing
additional instructional resources. The Math Center offers many resources, including
one-on-one, drop-in tutoring by our staff of instructors and student tutors. Please
see the full list of resources below:
Free, drop-in tutoring
A place to study individually or in small groups
In-house loan of current textbooks and solutions manuals
A library of supplemental books, DVDs, and video tapes for check-out
Computers for mathematical purposes
Calculators
Handouts on math topics, including content from various math courses as well as information
on overcoming math anxiety and preparing for and taking math tests
Two private study rooms
Make-up testing
Workshops
Joining the math center group
Current students may access more detailed information by entering their myHancock
portal and joining the Math Center Group. Details may include information such as
the current schedule of instructors and student tutors who work in the Math Center,
a schedule of instructors and tutors who specialize in statistics, upcoming workshops
on selected topics, etc. To join the Math Center Group:
Enter myHancock
Look at the center of the Home page in the box titled "My Groups." Click on "View
All Groups" at the bottom of the box.
STAFF |
This site contains tutorial lessons for College Algebra, Intermediate Algebra, Beginning Algebra, and Math for the Sciences....
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This site contains tutorial lessons for College Algebra, Intermediate Algebra, Beginning Algebra, and Math for the Sciences. Each lesson contains explanations, examples, and videos. There are also practice problems with complete solutions.
This is a Stand Alone Instructinal Resource (StAIR) designed to teach and test students on their understanding of different...
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This is a Stand Alone Instructinal Resource (StAIR) designed to teach and test students on their understanding of different triangle classifications. It could be used as initial instruction, or as an effective review tool at the end of a unit/lesson on triangle classification. |
Richard Beals' Analysis: An Introduction is a serious textbook for serious students. Intended for advanced undergraduates, this book demands as much personal maturity from the reader as it does mathematical sophistication.
The distinguishing feature of this book is its breadth. It is typical for an introductory analysis text to treat certain fundamental topics with great care, making only passing references (if that) to more sophisticated applications. Beals takes a rather different approach. He clearly views this book as being an introduction to the entire area of analysis, rather than an exposition of a predetermined set of topics. Less than half of the book is dedicated to material which (in the reviewer's experience) would generally appear in a standard introductory course. The remainder deals with more advanced topics, as well as a variety of applications. The last third of the book, in fact, is devoted exclusively to Fourier series and differential equations.
While, technically speaking, this book could be used for a first course in analysis, the title is perhaps something of a misnomer. The important introductory concepts are all discussed, precisely and completely, but often as a stepping-stone to more sophisticated results. Take, for example, the chapter that deals with continuity. Beals spends less than six pages (including exercises) discussing the general properties of continuous functions; after that, he shifts his attention to the spaces C([a,b]) and the Weierstrass Approximation Theorem. While one could argue that six pages are sufficient to his purposes, this transition might seem a bit precipitate to someone encountering these concepts for the first time.
Beals' writing style is characterized by a certain austere elegance. The author has an admirable command of the English language, and he appears unaffected by the excessive informality that has afflicted so many undergraduate textbooks. Apart from a few casual remarks in the introduction, there is virtually no "padding" anywhere in the text. The lemmas, propositions, theorems, and corollaries come in rapid succession, with very little commentary in between. Beals clearly expects a level of discipline from his readers that is comparable to his own.
Analysis: An Introduction is most appropriate for an undergraduate who has already grappled with the main ideas from real analysis, and who is looking for a succinct, well-written treatise that connects these concepts to some of their most powerful applications. Beals' book has the potential to serve this audience very well indeed.
Christopher Hammond is Assistant Professor of Mathematics at Connecticut College. |
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This Textbook
Overview
Packed with examples, this book provides a smooth transition from elementary ordinary differential equations to more advanced concepts. Asmar's relaxed style and emphasis on applications make the material understandable even for readers with limited exposure to topics beyond calculus. Encourages the use of computer resources for illustrating results and applications, but is also suitable for use without computer access. Includes additional specialized topics that can be read as desired, and that can be read independently of each other. Denotes exercises requiring use of a computer with computer icons, asking readers to investigate problems using computer-generated graphics and to generate numerical data that cannot be computed by hand. Offers Mathematica files for download from the author's Web site; can be accessed through the Prentice Hall address better than the first edition
I bought this book specifically because I had not been able to find a good book to explain Green's Functions. A friend of mine had this book and I was impressed by the chapter on Green's Functions so I bought a copy for myself and it's a great addition to my library.
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The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn the connection between a "quadratic function" and a "quadratic equation" along with the three possible cases for the solution(s) of a quadratic equation. Grades 5-9. 30 minutes on DVD. |
Activity Overview
The main objective of the activity is to find an approximation for the value of the mathematical constant e and to apply it to exponential growth and decay problems. To accomplish this, students are asked to search for the base, , that defines a function with the property that at any point on the graph, the slope of the tangent line (instantaneous rate of change) is equal to . The result is approximating the value of e—Euler's number and the base of the natural logarithms.
Topic: Exponential & Logarithmic Functions
Graph exponential functions of the form .
Evaluate the exponential function for any value of .
Calculate the doubling time or half-life in a problem involving exponential growth or decay.
Teacher Preparation and Notes
Students encounter the exponential constant e at various levels in their mathematics schooling. It may happen well before they reach calculus, and it is often used without an appreciation for where it originates (or why it is important). A good time to use this activity is when students first encounter e, but it is also appropriate for Precalculus and Calculus students when they are studying derivatives and instantaneous rate of change.
Prerequisites for the students are: familiarity with graphing and tracing functions on the calculator; an understanding of functions and function notation (both "" and ""); and an intuitive understanding of rate of change.
Problem 1
The activity begins with an investigation of how the value of affects the shape of the graph of . Students will enter several equations with different values of and examine the graphs of these functions. They are asked to make observations and draw conclusions.
The last question posed in this problem asks students to explain why the value of cannot be negative. It may be worthwhile to discuss this in a whole-class setting.
Problem 2
In this problem, students work specifically with the graph of . Students will draw a tangent line to the curve. Students will explore the relationship between the slope of the tangent line and the value of the function at that point. They should examine several different points and different values of .
Problem 3
Students are again asked to observe the changing values of the slope of the tangent line and the value of the function—and how they are related.
Students will discover that there is exactly one value of for which the slope of the tangent and value of the function are equal—and that this value is a very interesting number!
Applications
Students are given a series of application problems to apply the knowledge about what they have learned by doing completing this activity.
Student Solutions
Problem 1
Answers may vary. Possible observations: graph gets "steeper" as increases and "flatter" as decreases; always passes through the point ; increasing when and decreasing when
Answers may vary. Possible explanation: Even roots of negative numbers are not real numbers. Consider, for example, , which is not a real number.
Problem 2
, , and slope will vary
the slope is less than
Answers may vary. Possible observations: slope is always positive; as increases, the slope increases; curve never reaches the axis
Sample table
slope of tangent at
Problem 3
Answers will vary. Possible answer:
Answers will vary. Possible answer:
Answers will vary.
Sample table
Sample table
slope of tangent at
Applications
Modeling equation: (where is the value and is the time in years); one year: ; two years: ; five years:
Modeling equation: (where is population); about
Modeling equation: (where is the volume); about
Modeling equation for growing snowball: (where is the weight and is the time in seconds); : ; : ; : ; :
Possible limitations: the modeling equation might not be appropriate after too long a period of time, for example—the snowball may break apart if it gets too big, or it might reach the end of the hill. |
AGENDAS FOR THE WEEK: March 21 – March 25
MONDAY MAR 21 WEDNESDAY MAR 23 FRIDAY MAR 25
A DAY 9:00-10:30 10:36-12:16 A DAY 9:00-10:30 10:36-12:16 A DAY 9:00-10:30 10:36-12:16
Objective(s): SWBAT Objective(s): SWBAT Objective(s): SWBAT
Identify the x- and y- intercepts for rational Solve rational equations. Identify the nth real roots of integers.
functions. Identify extraneous solutions. Simplify radical expressions.
Identify the x- and y- asymptotes for rational Solve rational inequalities. Translate between expressions rational exponents
functions and radical expressions.
Graph rational functions. Simplify expressions with rational exponents.
Warm Up Warm Up Engage
P Students will review factoring. Specifically,
students will practice using the "Box Method" to
factor trinomial expressions.
Students will graph a single rational function. This
will be a quiz.
Students will review properties of square roots by
simplifying expressions with square roots.
Explore Explore Explore
L 8-4 Challenge: Rational Functions
Students will identify key features in the graph of
a rational function.
8-5 Exploration: Solving Rational Equations and
Inequalities
Students will illustrate how rational expressions
can be used in real life.
8-6 Exploration: Radical Expressions and
Rational Exponents
Students will use graphing calculators to see the
relationship between roots and powers.
Explain
Students will work individually to identify the Explain Explain
holes, intercepts, and asymptotes for "bottom Students will take notes on solving rational Students will discuss the meaning of other kinds
heavy" and "same" rational functions. equalities and rational inequalities. They will also of roots. Students will extend the properties of
practice solving these problems. square roots to these new kinds of roots.
A Elaborate
Students will explore the effect of transformations
of the rational function 1/x.
Elaborate
Students will use the relationship between radicals
and rational exponents to learn the properties of
rational exponents.
Evaluate Evaluate Evaluate
N
Students will be evaluated while graphing the Students will be evaluated based on their work Students will be evaluated based on their
rational expressions during the explanation phase. during the explanation. Also, they will be understanding of the relationships between
Also, students will be evaluated for understanding evaluated based on their responses on the radicals and rational exponents.
on Wednesday by means of a single question quiz. homework 8-5 Practice A/B.
Resources: Graphing Calculators Graphing Calculators Graphing Calculators |
Questions About This Book?
The Used copy of this book is not guaranteed to inclue any supplemental materials. Typically, only the book itself is included.
Summary
For a one semester or one-term algebra-based Introductory Statistics Courses. Drawing on the author's extensive teaching experience and background in statistics and mathematics, this text promotes student success in introductory statistics while maintaining the integrity of the course. Four basic principles characterize the approach of this text: generating and maintaining student interest; promoting student success and confidence; providing extensive and effective opportunity for student practice; and allowing for flexibly of teaching styles. |
History of Mathematics - 3rd edition
Summary: The updated new edition of the classic and comprehensive guide to the history of mathematicsFor more than forty years, A History of Mathematics has been the reference of choice for those looking to learn about the fascinating history of humankind's relationship with numbers, shapes, and patterns. This revised edition features up-to-date coverage of topics such as Fermat's Last Theorem and the Poincar_ Conjecture, in addition to recent advances in areas such as finite group theory and...show more computer-aided proofs. Distills thousands of years of mathematics into a single, approachable volume Covers mathematical discoveries, concepts, and thinkers, from Ancient Egypt to the present Includes up-to-date references and an extensive chronological table of mathematical and general historical developments.Whether you ? re interested in the age of Plato and Aristotle or Poincar_ and Hilbert, whether you want to know more about the Pythagorean theorem or the golden mean, A History of Mathematics is an essential reference that will help you explore the incredible history of mathematics and the men and women who created it. ...show less
0470525487 Brand New. Exact book as advertised. Delivery in 4-14 business days (not calendar days). We are not able to expedite delivery71.50 +$3.99 s/h
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PROFESSIONAL & ACADEMIC BOOKSTORE Dundee, MI
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This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more...
see more
This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more abstract concepts in algebra. Students will gain proficiency in computation with rational numbers (positive and negative fractions, positive and negative decimals, whole numbers, and integers) and algebraic properties. New concepts include solving two-step equations and inequalities, graphing linear equations, simplifying algebraic expressions with exponents, i.e. monomials and polynomials, factoring, solving systems of equations, and using matrices to organize and interpret data course begins by establishing the definitions of the basic trig functions and exploring their properties, and then...
see more
This course begins by establishing the definitions of the basic trig functions and exploring their properties, and then proceeds to use the basic definitions of the functions to study the properties of their graphs, including domain and range, and to define the inverses of these functions and establish the their properties. Through the language of transformation, the student will explore the ideas of period and amplitude and learn how these graphical differences relate to algebraic changes in the function formulas. The student will also learn to solve equations, prove identities using the trig functions, and study several applications of these functions. Note that this courseMathematics 003)
Undergraduate lab series designed to familiarize students with using computer models to answer biochemical questions. ...
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Undergraduate lab series designed to familiarize students with using computer models to answer biochemical questions. Ideally, this lab would be taught as a supplement to a concurrent lecture course. Students are assumed to have completed one year of undergraduate calculus.Topics include acid-base chemistry, Gibbs free energy, Michaelis-Menten kinetics, enzyme inhibition, hemoglobin, and the Bohr effect. Math skills used include graphing (2-D and 3-D), algebra, logarithms, and numerical solutions to systems of equations.The modules are designed to be self-contained lab exercises. They are Mathcad documents that the students complete for credit. Thus, students must have access to Mathcad (version 13 or higher). PDF versions of the modules are also provided for demonstration purposes. |
El Cerrito Statistics, it is intended for math majors. I got "A" in that course. My instructor was a visiting scholar and taught more theorems than other Linear Algebra sections |
Numerical Mathematics and Computing
9780495114758
ISBN:
0495114758
Pub Date: 2007 Publisher: Thomson Learning
Summary: Authors Ward Cheney and David Kincaid show students of science and engineering the potential computers have for solving numerical problems and give them ample opportunities to hone their skills in programming and problem solving. The text also helps students learn about errors that inevitably accompany scientific computations and arms them with methods for detecting, predicting, and controlling these errors. A more t...heoretical text with a different menu of topics is the authors' highly regarded NUMERICAL ANALYSIS: MATHEMATICS OF SCIENTIFIC COMPUTING, THIRD EDITION.
Cheney, Ward is the author of Numerical Mathematics and Computing, published 2007 under ISBN 9780495114758 and 0495114758. Three hundred thirty two Numerical Mathematics and Computing textbooks are available for sale on ValoreBooks.com, one hundred twenty four used from the cheapest price of $13.04, or buy new starting at $41.73 |
1111574618
9781111574611
Student Workbook for Kafmann/Schwitters' Elementary and Intermediate Algebra: A Combined Approach:The Student Workbook contains all of the Assessments, Activities, and Worksheets from the Instructor's Resource Binder for classroom discussions, in-class activities, and group work.
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Rent Student Workbook for Kafmann/Schwitters' Elementary and Intermediate Algebra: A Combined Approach 6th edition today, or search our site for Jerome E. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Brooks Cole. |
Complete Book of Algebra and Geometry Grades 5-6
9780769643304
ISBN:
0769643302
Publisher: Carson-Dellosa Publishing, LLC
Summary: The Complete Book of Algebra and Geometry offers children in grades 5-6 easy-to-understand lessons in higher math concepts, skills, and strategies. This best-selling, 352 page workbook teaches children how to understand algebraic and geometric languages and operations. Children complete a variety of activities that help them develop skills and then complete lessons that apply these skills and concepts to everyday sit...uations. Including a complete answer key this workbook features a user friendly format perfect for browsing, research, and review. Basic Skills Include: -Order of Operations -Numbers -Variables -Expressions -Integers -Powers -Exponents -Points -Lines -Rays -Angles -Area Over 4 million in print! The best-selling "Complete Book series" offers a full complement of instruction, activities, and information about a single topic or subject area. Containing over 30 titles and encompassing preschool to grade 8 this series helps children succeed in every subject area!
Carson-Dellosa Publishing Staff is the author of Complete Book of Algebra and Geometry Grades 5-6, published under ISBN 9780769643304 and 0769643302. Two Complete Book of Algebra and Geometry Grades 5-6 textbooks are available for sale on ValoreBooks.com, one used from the cheapest price of $12.33, or buy new starting at $46.86 |
branches of mathematics come together in harmonic analysis, each adding richness to the subject and each giving insights into this fascinating field. Devito's Harmonic Analysis presents a comprehensive introduction to Fourier analysis and Harmonic analysis and provides numerous examples and models so that students leave with a clear understanding of the theory. |
...Discrete math was a part of my Bachelors in Mathematics. I also have significant amounts of coursework in computer science including optimization of algorithms, and in mathematics including logic, set theory, combinatorics, number theory, and graph theory. The focus of my undergraduate work was geometry and topology |
Elementary Statistics: A Brief Version, is a shorter version of the popular text Elementary Statistics: A Step by Step Approach. This softcover edition includes all the features of the longer book, but it is designed for a course in which the time available limits the number of topics covered. It is for general beginning statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. This edition places more emphasis on conceptual understanding and understanding results. This edition also features increased emphasis on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing technologies commonly used in such courses. |
The concept of a radical (or root) is a familiar one, and was reviewed in the conceptual explanation of logarithms in the previous chapter. In this chapter, we are going to explore some possibly unfamiliar properties of radicals, and solve equations involving radicals |
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This book is designed to help students get a feel for what a career in Management Information System (MIS) would be like. Our students report that they learn more about information systems from their internships than from their IS courses. Consequently, we designed a course that looks very much like an internship—an introduction to the field followed by a substantial project.
A by product of creating an engaging course is increased enrollment in the MIS major. Even students who have never heard of MIS become excited about the major and either switch majors or add it as a double major or minor.
Rather than detailed explanations and worked out examples, this book uses activities intended to be done by the students in order to present the standard concepts and computational techniques of calculus. The student activities provide most of the material to be assigned as homework.
This is a user-friendly math book. Whenever possible, the author explains the calculus concepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry. Then, you'll see how the calculus concepts work in concrete examples. All explanations are in plain English.
Note: We do not host these books and we take no responsibility, whatsoever, in this regard. If you see or find anything objectionable or a violation of copyright, do let us know and we will be happy to remove such content. Contact us: flazxbooks@gmail.com |
Featured Research
from universities, journals, and other organizations
The aftermath of calculator use in college classrooms
Date:
November 12, 2012
Source:
University of Pittsburgh
Summary:
Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, experts say. They have proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students.
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Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, says Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research & Development Center. King has proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students published in the British Journal of Educational Technology.
"We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
Together with Carol Robinson, coauthor and director of the Mathematics Education Centre at Loughborough University in England, King examined whether the inherent characteristics of the mathematics questions presented to students facilitated a deep or surface approach to learning. Using a limited sample size, they interviewed 10 second-year undergraduate students enrolled in a competitive engineering program. The students were given a number of mathematical questions related to sine waves -- a mathematical function that describes a smooth repetitive oscillation -- and were allowed to use calculators to answer them. More than half of the students adopted the option of using the calculators to solve the problem.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," said King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the problems, the students were interviewed about their process. A student who had used a calculator noted that she struggled with the answer because she couldn't remember the "rules" regarding sine and it was "easier" to use a calculator. In contrast, a student who did not use a calculator was asked why someone might have a problem answering this question. The student said he didn't see a reason for a problem. However, he noted that one may have trouble visualizing a sine wave if he/she is told not to use a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes -- especially at the undergraduate level," said King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
King also suggests that relevant research should be done investigating the correlation between how and why students use calculators to evaluate the types of learning approaches that students adopt toward problem solving 5, 2013 — Researchers have developed a classroom design that gives instructors increased flexibility in how to teach their courses and improves accessibility for students, while slashing administrative ... full story
May 21, 2012 — Discipline-based education research has generated insights that could help improve undergraduate education in science and engineering, but these findings have not yet prompted widespread changes in |
This eBook reviews quadratics, cubics and other polynomials, in ways from sketching and understanding their graphs, to factorising quadratics, understanding the quadratic formula, solving quadratic 'like' equations, completing the square, factorising cubics, algebraic division and understanding the remainder and factor theorems. Further, we include an extensive selection of questions for the ...
This eBook introduces the subject of logarithms and exponentials, from the basic definition of logarithm, through the laws of logarithms, undertaking an assessment and an appreciation of exponential graphs, looking at the linear form of exponentials interspersed with a series of questions and worked examples.
This eBook introduces the subject of integration, starting by introducing integration as the inverse of differentiation, with a twist or two. We then introduce the indefinite integral and its constant of integration, and go on to introduce the definite integral. As well as this, we provide ample examples and illustrations to illuminate the prose. We go on to introduce trapeziums linear ...
This eBook introduces the subjects of indices and surds, ranging from introducing both indices and the laws of indices, surds and the laws of surds, to developing the students skills in manipulation such numbers through setting a wide range of questions..
This eBook introduces the subject of differentiation, across this wide-ranging subject, starting with definitions and first principles to developing an understanding and appreciation of the first and second order differentials of the equation y = xn through a development of the equations of the gradient and normal to a curve at a particular point as well as a thorough review of maximum, minimum ..
This eBook introduces co-ordinate geometry and graphs, ranging from finding the equations of the straight-line joining two points for which the co-ordinates are known, to calculating both the mid-point and length of a line between two known co-ordinates to plotting equations of the form y = kx^n where n is even or odd for various values of k, as well as y = k√(x) where x is positive.
This eBook introduces the subject of circle and circle geometry, introduces the equation of a circle, explores circle geometry, examines tangential lines to circles and their properties and equations, as well as exploring arc-length and sector area of circles where angles are represented in radians. Further, we include some elementary questions for the student to enjoy
A simple, step-by-step approach to learning the piano.
Beginner's Piano is designed to make learning the basics of piano playing and music theory as simple as possible for students and teachers alike. A no-nonsense approach with ideas clearly introduced using jargon-free text, the course is dedicated to developing fluency in reading and a thorough foundation in music
This Bible lesson series is unique in that it takes a brand new believer through the Bible while explaining how the individual stories fit together to create the one BIG macro story of the Bible, God's Story. After completing these lessons, you will have more Bible knowledge than most Christians who attend church every week |
Welcome
to the Math Workshop! We are pleased to offer an opportunity for you to
refresh your memory about math skills that will be helpful in your introductory
biostatistics and epidemiology courses. By the end of the course, you
will remember the mathematical concepts you learned earlier in your education,
you will see how these concepts will apply in future courses, and you
will be more comfortable with math and thus better able to concentrate
on the courses' main concepts.
The course
is presented in two parts. First, for those who want to brush up on elementary
concepts, there are three on-line modules. The second part will be conducted
in standard classroom format when you arrive at HSPH in August.
Online
Modules
Each module
has 3 parts:
Warmup
exercise: This will introduce the concepts reviewed in the module. Doing
the exercise is a good way to see how well you know the material. The
solutions are available in a separate file; we suggest you work through
the problems before peeking at the solutions!
Content:
This is a series of "slides" presenting the material, as
if you were attending a regular class. There are practice problems sprinkled
throughout, and the answers to the practice problems are presented at
the end. The content may be split into multiple files so that you can
break up your work sessions. You can either flip through the slides
on the computer, or print them out. There are 2 versions of the slides
provided for each module: 1 with full-size slides shown 1 per page,
and 1 with 4 small slides per page. If you are printing them out, you
may want to select the 4 per page format.
Homework:
The homework problems give you a chance to see if you have mastered
the material. While the solutions are available on-line in a separate
file, you are strongly encouraged to work through the homework carefully
before looking at the solutions.
You
will need Adobe Acrobat Reader installed on your computer to read the
module elements. If you don't already have it, click below to download
it free from Adobe's web site.
Classroom
Component
Class
will be held from 7:00am to 9:00am. Class will be held in Kresge Building Room G-1. The format will be
similar: warmup exercises, content, and homework. There will be opportunities
to ask questions. Homework will be graded and returned. |
Presented by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course is designed to acquaint visitors with "calculus principles such as derivatives, integrals, limits,...
HippoCampus, a project of the Monterey Institute for Technology and Education (MITE), seeks to provide ?high-quality, multimedia content on general education subjects to high school and college students.? This cou...
Once again, the Mathematical Association of America has struck instructional gold with this latest gem from their online collection of resources for mathematics educators. Created by Barbara Margolius, this derivative...
Graph is "an open source application used to draw mathematical graphs in a coordinate system." Anyone who wants to draw graphs of functions will find this program useful. The program makes it very easy to visualize a... |
This tutorial is less about statistics and more about interpreting data--whether it is presented as a table, pictograph, bar graph or line graph. Good for someone new to these ideas. For a student in high school or college looking to learn statistics, it might make sense to skip (although it might not hurt either).
The world seldom gives you two numbers and tells you which operation to perform. More likely, you'll be presented with a bunch of information and you (yes, YOU) need to make sense of them. This tutorial gives you practice doing exactly that. When watching videos, pause and attempt it before Sal. Then work on as many problems as you want in the exercise at the end of the tutorial.
Equality is usually a good thing, but the world is not a perfect place. No matter how hard we try, we can't help but compare one thing to another and realize how unequal they may be.
This tutorial gives you the tools to do these comparisons in the mathematical world (which we call inequalities). You'll become familiar with the "greater than" and "less than symbols" and learn to use them.
You've probably been learning how to do arithmetic for some time and feel pretty good about it. This tutorial will make you feel even better once by showing you a bunch of examples of where it can be applied (using multiple skills at a time). Get through the exercises here and you really are an arithmetic rock star! |
Math Made Nice - N - Easy, Book #1 - 99 edition
Summary: Almost everyone needs some math in everyday life, at work, in a career, for study, for shopping, for paying bills. dealing with a bank, in sports, using credit cards, etc. This series of books simplifies the learning, understanding, and use of math, making it non- threatening, interesting, and even fun. The series develops math skills in an easy-to-follow sequence ranging from basic arithmetic to pre-algebra and beyond. These books draw on material developed by the U...show more.S. Government for the education of government personnel with limited math and technical backgrounds. Volume I covers number systems, sets, integers, fractions, and decimalsBorgasorus Books, Inc. MO Wentzville, MO
PAPERBACK Good 08789120025 |
(Note: As many as possible of the following topics are accompanied by applications.) Calculus in the contexts of conic sections, polar coordinates and parametric curves. Vectors in R^2 and R^3. The dot and cross products. Distance, angles, lines and planes. Quadric surfaces. Cylindrical and spherical coordinates. Vector-valued functions, limits, continuity, derivatives and integrals of the same. Tangential and normal components of acceleration. Arc length as a parameter and curvature. Projectile motion. Functions of several variables, limits, continuity, and partial derivatives of the same. The total differential and error estimation. Tangents, planes, normal lines, the gradient and directional derivatives. Free optimization, absolute extrema and LaGrange multipliers. Multiple and iterated integration, including Fubini's Theorem. Applications of multiple integration such as volume, surface area, the average value of a function, centers of mass and moments of inertia. Parameterized surfaces and integration in cylindrical and spherical coordinates. Vector field, div, grad, curl and their interpretation. Equivalent conditions for conservative vector fields.
GOALS AND OBJECTIVES:
1. Convert between rectangular and polar coordinates and plot several polar curves on a grapher (parabola, circle, ellipse, hyperbola). 2. Calculate the results of vector addition, subtraction, scalar multiplication, dot product and cross product. 3. Perform calculations for geometry in space. 4. Convert between cylindrical, spherical, and rectangular coordinates. 5. Explain the application of differentiation of vector-valued functions to velocity and acceleration. 6. Find arc length and curvature of a given curve. 7. Evaluate partial derivatives and explain their meaning geometrically. 8. Use local linear approximation applications. 9. Produce a directional derivative for any given direction. 10. Locate and classify extrema for functions of two variables. 11. Apply 2- and 3- dimensional integration to geometric and physical problems.
REQUIREMENTS:
All students must have their own text. If a graphing calculator is to be used, the student must obtain their own. Assignments and exams are required by all faculty. Some faculty may require attendance, participation, projects, working in groups, graphing calculators, computer lab assignments or other forms of assessment. |
Find a Parker, CO CalculusCalculus generally begins with the concept of limits and then progressing into derivatives and integrals, along with their respective applications. The student will also learn about more advanced mathematical modeling, differential equations, infinite series, all of which will prepare them for m |
MSM701 and acceptance into the Master of Arts in Teaching Middle School Mathematics program or permission of the Program Coordinator. Not available for degree credit towards the MAT or MS mathematics programs
Description
This course is intended to bridge the gap between algebra and calculus. It will develop a firm understanding of the concept of function, how to graphically represent various functions, analyze their behavior and create new functions from old. The course will look closely at various function classes including polynomials, exponential, logarithmic and trigonometric. Functions will be used to model real-life situations. |
Book summary
The Second Edition of this classic text maintains the clear exposition, logical organization, and accessible breadth of coverage that have been its hallmarks. It plunges directly into algebraic structures and incorporates an unusually large number of examples to clarify abstract concepts as they arise. Proofs of theorems do more than just prove the stated results; Saracino examines them so readers gain a better impression of where the proofs come from and why they proceed as they do. Most of the exercises range from easy to moderately difficult and ask for understanding of ideas rather than flashes of insight. The new edition introduces five new sections on field extensions and Galois theory, increasing its versatility by making it appropriate for a two-semester as well as a one-semester course. [via]
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Once Upon A Time Books via United States
Hardcover, ISBN 0201073919 Publisher: Addison Wesley Longman Publishin, 1980 This is a used book. It may contain highlighting/underlining and/or the book may show heavier signs of wear. It may also be ex-library or without dustjacket. All orders are shipped the same or the next day. |
Subject: Mathematics (9 - 12), or Technology Education (9 - 12) Title: Family Ties: Parabolas Description: This lesson allows students to manipulate the parameters while using the vertex form of the equation of a parabola to see the effects on the graph. The spreadsheet can be altered for other functions.This lesson plan was created as a result of the Girls Engaged in Math and Science University, GEMS-U ProjectThinkfinity Lesson Plans Mathematics Title: Egg Launch ContestAdd Bookmark Description: Students will represent quadratic functions as a table, with a graph, and with an equation. They will compare the data and move between the representations. They will use a calculator as a tool to determine the equations for the functions and view the graphs. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 |
Mathematical Methods for Physicists : A Comprehensive * Updates the leading graduate-level text in mathematical physics * Provides comprehensive coverage of the mathematics necessary for advanced study in physics and engineering * Focuses on problem-solving skills and offers a vast array of exercises * Clearly illustrates and proves mathematical relations New in the Sixth Edition: * Updated content throughout, based on users' feedback * More advanced sections, including differential forms and the elegant forms of Maxwell's equations * A new chapter on probability and statistics * More elementary sections have been deleted |
Abhijit Dasgupta, "Set Theory: With an Introduction to Real Point Sets"
English | 2014 | ISBN-10: 1461488532 | 442 pages | PDF | 3,8 MB
What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner |
An algebra book requires a different type of reading than a novel or a short story. Every sentence in a math book is full of information and logically linked to the surrounding sentences. You should read the sentences carefully and think about their meaning. As you read, remember that algebra builds upon itself; for example, the method of factoring that you'll study on page 188 will be useful to you on page 697. Be sure to read with a pencil in your hand: Do calculations, draw sketches, and take notes.
An algebra book requires a different type of reading than a novel or a short story. Every sentence in a math book is full of information and logically linked to the surrounding sentences. You should read the sentences carefully and think about their meaning |
Microsoft Mathematics provides a set of mathematical tools that help students get school work done quickly and easily. With Microsoft Mathematics, students can learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra, algebra, trigonometry, physics, chemistry, and calculus.vHere are some Free Windows Media Player alternatives that worth a try.
Sunday, October 21, 2012
For a more personal spin in field of blogging, you might want to take a look at Blogging Tips: Confessions of a Six Figure Blogger. It's a new book (Amazon Kindle & Paperback) written by Blogging Tips guru Zac Johnson and in it, he provides all kinds of useful information about how to get into blogging and what you need to do when you get there if you want to be a "six figure blogger" yourself. Let's dive in and have a look at this new book.
A Story of Blogging Success
Zac is not new to the Internet. He has been making money online for over 15 years and he has been blogging professionally for a little over five years. While there are some connections, he recognizes there are differences too.
And that's how he chooses to begin Confessions of a Six Figure Blogger. Available as an e-book for Kindle,Blogging Tips starts with a bit of a personal story about how Zac got into blogging in the first place, expanding beyond Internet marketing and online advertising into content creation by way of ZacJohnson.com.
There are many opportunities available to you and Zac wants to stress that you can "become an authority in any niche." By way of blogging, Zac has been able to create a brand, build a business, and expand his international presence by way of appearances at conferences, as well as on ABC and FOX News.
What Can I Expect to Learn?
While Zac's personal story may be of personal interest to you, there's a better chance that you'd rather learn how to emulate his success and put six figures into your own bank account.
As you dive into Confessions of a Six Figure Blogger, you'll find that the progression is natural and logical. He starts by talking about what you need to know about blogging, like why blogging is the future of the Internet and branding. He then talks about the steps that you need to take to go live with your blog, as well as many of the critical monetization methods that you need to employ.
If you've read other books on the subject, like Make Money Online: Roadmap of a Dot Com Mogul, then some of this material will feel familiar to you. Even so, Zac offers a fresh perspective and it's all laid out in an easy to understand and easy to follow manner. If you're new to blogging, then this e-book is even more valuable. While it is extensive, it is also easily digestible.
Actionable Items You Can Use Right Now
One thing that struck me about this e-book was that it was definitely on the text-heavy side of things. There really isn't anything included by way of graphics or visual aids, but these aren't really needed for the most part anyway.
What Zac does very well is explain what may otherwise be some esoteric information about how to blog successfully in more of an everyday language. More importantly, he doesn't talk too much about what would work in theory or the about the philosophy of blogging. Instead, he provides directly actionable items.
In the chapter with 10 must-dos for serious bloggers, as an example, he stress the importance of having a great logo created for yourself, for your blog, or for your brand as a whole. He also suggests some logo creation services that he recommends, as well as the typical pricing from these designers. It also helps that, along the way, he also links to some of the blog posts from his archive that you may find useful and relevant.
How Much Does It Cost?
Considering the wealth of information contained in this book, you're probably going to be surprised by the price. It's available now in the Kindle Store for just $2.99. That's less than five bucks and you get nearly 100 pages of incredibly useful, easy-to-read, and actionable knowledge about what it takes to be a six figure blogger. Remember that you don't need to have a Kindle to read Kindle e-books; there are Kindle apps for PC, iPhone, iPad, Android, and several other platforms. Go get it. This is well worth more than the three dollar asking price.
Friday, October 05, 2012
Many users use online data backup tool like Dropbox, Skydrive and many more, though they are the best to keep your data on cloud, but how do you deal with corrupt data on your hard-disk. How will you fix a video file which is partially downloaded and how about recovering your convocation pictures from a badly scratched CD/DVD.
Such situations keep coming and to make sure that we don't lose out important data and recover them quickly, there are many paid and free software's available in the market, which helps you to recover different types of data files on your system. Here I'm compiling a list of such data recovery software's that you can use and recover your data without any pain or loss. Though, by the end it all depends upon how badly your data is corrupted and if it's possible to recover it or not.
Anyhow, before we move ahead, here is a quick tip: Make sure you maintain 2-3 backup of your important data. You can take help of online storage websites like you can store all your images on sites like Flickr, Picasa. You can use Amazon S3 backup to store your important files and keep them safe and secure. You can use Evernote to keep a backup of important notes and web clips.
5 of The Best data recovery tools available:
1. Undela : Free Data recovery tool
Undela is an exceptional, free-to-use program that has the ability to recover deleted files very quickly. It is even possible to recover data from damaged hard disks using Undela. In the event of a virus attack or power failure, Undela can recover any deleted files. As well as hard drives, you can recover data from a variety of media types, including MP3 players, SD Cards for your digital camera and USB memory sticks.
No matter what document type has been lost, whether an e-mail, image, audio or video file Undela can easily recover it. Even if the file has been deleted from the Recycle Bin you may still be able to retrieve it. Supported file allocation types include NTFS, NTFS5, FAT12, FAT16, and FAT32 in Windows, and XFS in Linux.
In addition the software is easily accessible to all users due to its multilingual interface. Currently supported languages are English, German, French, Italian, Korean, Japanese, Chinese, Spanish, Portuguese and Russian. Deleted files are easily searchable by file name or mask. The tool also presents previews of any deleted document.
CD Recovery Toolbox is another free data recovery tool, ideal for recovering data from damaged, corrupt, scratched CDs or DVDs. In addition this tool is also able to recover files from HD DVD and Blu-Ray discs. The tool is very easy to use: simply select the drive and the files you wish to recover to start the process. The recovery is usually very quick, though the effectiveness of the recovery may depend on the extent of the damage to the source media.
If your CD is severely scratched, the recovery process will take longer, and may sometimes fail to recover all the lost files. It does however help to recover a large amount of data that may otherwise be lost forever.
Key Advantage:
This tool is really easy to use and ensures quick data retrieval from any scratched and damaged CD/DVD.
To ensure effective backup of any e-mail information, use Mail PassView to export your email account settings. A number of email applications are supported, including Microsoft Outlook and Outlook Express, Windows Live Mail (Hotmail), Incredimail and Eudora. Run the software to find your Email login details (username and password) and any other details from your e-mail client.
The recovered information can be exported as a text file or copied onto the clipboard to be pasted into a Word Document or any other text file. A word of warning: your antivirus software may consider an e-mail retrieval tool as a hacking tool and as such might alert you when you attempt to run the program.
Key Advantage:
Another piece of software that is really simple to use, supporting a range of email clients, easily exporting your account details and passwords should you require it.
If you're looking to recover corrupt MP3 files, MP3val is the ideal piece of data recovery tool for media files. This tool enables you to scan and fix MP3 files that have lost data. Occasionally MP3 files are rendered unplayable due to an incomplete download or sometimes because of other issues.
MP3val is able to fix the majority of these issues and will also automatically create a backup file of the original file. In addition to restoring MP3 files, you can also use this software to fix MPEG files. Unfortunately these are the only types of files that can be restored. The software gives very fast results. As standalone software there is no installation necessary. MP3val uses a front-end GUI with a command-line version.
Key advantage:
MP3Val fixes corrupt MP3 and MPEG files, automatically creating a backup of the original files. Crucially no installation is necessary.
If you're looking to restore data from partially downloaded video files, look no further than DivFix++. You can use this tool to fix any DivX (AVI) movie files. The tool will build the index element of the file, which is the part that frequently goes missing if a movie download is incomplete. There is a built-in error-detection feature that can identify the problem with the file and fix it. Original files are always backed up.
The software offers Multilingual Support and also supports a number of operative systems – Windows, Linux and Mac OSX.
Well these are just few of many data recovery tools that I can think of now and there are many others for Wi-fi password, exchange password and many more specific software's, which we will cover into next article of our data recovery series. If you want to share any other free software, do let me know via comment. Also, if you find this post useful don't forget to share it on Google plus and Facebook.
Wednesday, October 03, 2012 right click gives a bad user experience and for Bloggers, you can always fight such copy-paste blogger using Google DMCA.
Now for me, when I have to write a tutorial, I take information from the pages on Internet and give proper credentials with link in the post. Now, the problem which I have faced recently is many of these sites have right click disabled and it's pain to copy from these sites normally. So, here I have compiled a series of possible ways to copy content from those pages. FYI, many websites disable CTRL +C options to ensure better security from hackers and malicious sites.
Ways to copy text from Right click Disabled pages:
Most of the bloggers and webmasters uses JavaScript technique to disable right-click, to prevent scrapers sites from stealing their content.
Many times we often come to websites where we found contents useful like how-to , Guides and we copy it into worded or notepad. Generally we select some text and then right click to copy. But on Protected sites a message box appears saying "Right-Click on this site is disabled. Hold Ctrl key and click on link to open in new tab"
But there are numerous way through one can copy contents from Right Click protected sites
In Chrome browser, you can quickly disable JavaScript by going to settings. See the screenshot for better explanation:
Goto Setting >> UnderHood Tab >> Content Settings
or enter chrome://settings/content
Then Select Do not allow any site to run JavaScript
Similarly if you are using Firefox, you can remove the tick from "Enable JavaScript" option.
Using Proxy Sites
There are many proxy sites, which let you disable JS while browsing. All you need to use those proxy sites, which offer such features and you can quickly use right-click on click disabled sites.
If you have to copy the specific text content and you can take care of HTML tags, you can use browser view source options. All the major browser give an option to source of the page, which you can access directly using the format below or by right click. Since, right click is out of question here, we will simply open chrome browser and type: view-source: before the post URL Like
Saturday, September 29, 2012
Parents, take heed. Summer means fun and sun for young people, but there could be a dark side to all that free time. Is your teen home alone? Are the computers in your house protected from porn sites? Unfortunately, it's possible your students are finding sexually explicit material online that could hurt their emotional, spiritual and even physical health.
According to TopTenREVIEWS Google Analytics data reveals a dramatic increase – indeed a 4,700% jump – in searches for the term "porn" in the days immediately following the end of school for most students. While falling short of scientific proof, it's a strong indicator of what many youngsters may be doing on their summer vacation.
There could be many explanations for this dramatic change in numbers, but there is no question it coincides with the time most schools get out.
Since parents are busy and cannot constantly stand guard over computer use, a little electronic help might come in handy. Internet filter software can offer some reassurance to parents that their children are protected from material that parents deem objectionable.
The internet filter software on the market today allows parents to block websites and chat rooms that parents deem inappropriate. This software can do much more including such things as filtering emails, monitoring social media sites and sending parents email alerts if someone using a computer is accessing objectionable content.
Academic studies have shown that young people who are exposed to sexually explicit material before age 18 are more likely to become promiscuous, get pregnant, test positive for a sexually transmitted disease and engage in forced sex.
Nothing takes the place of heart-to-heart talks between parents and children about values, human sexuality and the things that are considered healthy, respectful and worthwhile according to a particular family's principles.
However, internet filter software can shield children from images, language, videos and other depictions of behaviors that are contrary to the parents' standards. It could be another tool to help parents get involved in the already complicated task of trying to raise healthy, well-adjusted children.
Dr. Mary Anne Layden, director of the Sexual Trauma and Psychopathology Program,
Center for Cognitive Therapy in the Department of Psychiatry at the University of Pennsylvania, is the author of "The Social Costs of Pornography: A Statement of Findings and Recommendations."
"There is evidence that the prevalence of pornography in the lives of many children and adolescents is far more significant than most adults realize, that pornography is deforming the healthy sexual development of these young viewers, and that it is used to exploit children and adolescents," Layden wrote.
How does it deform them?
In a telephone interview, Layden said academic studies show that there are 23 unhealthy behaviors that people exposed to "sexualized media" before the age of 18 are more likely to display. These can include a greater likelihood to have sex earlier in life, have multiple partners, engage in forced sex, test positive for Chlamydia, be more accepting of sexual harassment and become juvenile offenders.
"Are any of those things the kind of things we want for our kids?" Layden said. "My own research shows that pornography is mis-education about sex. It lies about sex."
For example, she said pornography shows that women love to be degraded and violently hurt, which is not true in real life. It also depicts men as vicious, narcissistic and out of control – which is also not true, she said.
"This is hate speech against men and hate speech against women," Layden said. "It sends the wrong message about people, relationships and functions. Porn doesn't say anything about love or commitment or caring. It also doesn't say anything about producing children."
The blunt-talking and often controversial Layden said she tells parents, "You've got to say to children: 'There won't be any porn in this home'" and then take strong measures to keep it away from impressionable youngsters.
"It's good to talk to the kids, but I think prevention is 100 percent better," Layden said.
What is Internet filter software?
Schools and businesses can easily have dozens of computers operating at once, with many or even all of them providing Internet access. Wherever Internet access is available, there's always the possibility that members of an organization may visit websites you'd rather they stay away from, as well as the ever-present threat of the Internet being used as a tool for Cyber bullying or sexual harassment. With the abundance of inappropriate material available online, such as pornography, games and gambling, it can be a challenge to effectively filter and limit access to those sites.
Download to your phone
Q: What is TrueCaller? A: TrueCaller offers a form of Web-enabled caller ID for your Android, iPhone, Symbian and Windows Phone device! It enables you to match phone numbers to names, addresses, social networks, and even pictures! With our unique phone number database TrueCaller enables you to identify numbers like no other service is able to.
Q: I can´t find TrueCaller on Android Market. How can I get it? A: Visit truecaller.com from your phone´s browser to get it.
Q: What phones are supported? A: Mostly all smartphones but check our Download page for a list of supported devices
Q: What countries are supported? A: All countries!
Q: If I hard-reset my phone and re-install TrueCaller, do I have to pay the license again? A: No, your IMEI is connected to our system so it will automatically detect that your payment has already been done during initialization.
Q: How can I add support for my country? A: Please ask us using the sticky post in our blog and we will try adding the requested country.
Q: The phone number of a caller isn't shown, why doesn't Truecaller show that to me? A: TrueCaller will only be activated on incoming calls where the number isn't previously stored on your cellphone and if the number isn't anonymous.
If a number has been saved twice in your phone book on a Symbian device (such as Nokia devices) the contact will not be displayed when a call is received and TrueCaller will not make a lookup because the number is still saved in your phone book.
Q: "Error: Please select your country in settings" A: Open settings and select a new country and save. Then change back to your country and save. Make sure you have Internet connection available. In Symbian version, check that your Access Point is correct in Settings.
Q: My requests are being processed slow, why? A: If you have a very poor network connection it could take longer. A normal request time is between 1 and 5 seconds depending on country. Unfortunately the searchspeed in some countries may be slow because of the service provider.
Q: How much data-traffic does TrueCaller use? A: The data is very optimized and therefor it will never be bigger than a 1 kb per request. To find out exactly how much that would cost you in your currency, please contact your operator.
Q: My phone is not supported, how can i get it supported? A: Please leave us a message using the sticky post in our blog. If there is enough requests for a specific application, we may develop it! If you are using a Windows Mobile Pro device and your phone is not in the list, you can try to select by OS.
Q: Does TrueCaller always work? A: TrueCaller will always work provided you have 3G or EDGE or WLAN internet connection on your cellphone and the number is available on reverse phone number lookup services.
Q: The application cannot find any phone numbers? A: Make sure you have selected your local country in Settings. Otherwise Truecaller will not be able to lookup local numbers. If you still experience problem with finding all numbers, please leave us a comment on the sticky post on our blog.
Q: When will UK be supported? A: Due to some law restrictions TrueCaller will not be supported in UK. Hopefully we will be able to support UK in a near future but what we do support at the moment is a area localization function which shows you which area the area code belongs to. Update: You can now use our CallerID+ in all countries including UK.
Q: I have bought a license of TrueCaller. Can I download a updated version from your website without losing my license? A: Yes. Your phone is registred in our system as paid so you will always be able to get new updates. Even if you uninstall TrueCaller and install it later on it will still work. (This answer is only for users in countries where TrueCaller is not free) |
"Applied Mathematics"
free workshop was held on June 20, 2003 at Allan Hancock College in Santa Maria, California. Instructor and student materials are available for online viewing and for downloading. A Microsoft PowerPoint Presentation is used in conjunction with hands-on laboratory exercises. The exercises include activities requiring the use of spreadsheets and simulation software. Also required are a few electronic components and a proto-board trainer with onboard power supplies. Circuit component layouts are shown in the PowerPoint presentation to help first-time users setup circuits. A student version simulation software package is available for downloading, so is an e-book on electronics. A parts list is included with each laboratory exercise indicating what equipment and electronic components that are required for there completion.Tue, 19 Jul 2011 12:34:28 -0500Modeling Orbital Debris Problems
algebra lesson from Illuminations helps students develop their understanding of mathematical functions and modeling using spreadsheets, graphing calculators, and computer graphing utilities. The differences between linear, quadratic and exponential models are described. Students will also improve their understanding of how to choose the appropriate graphical representations for data. The material is intended for grades 9-12 and should require 5 class periods to complete.Mon, 24 Jan 2011Introductory Algebra: Algebra 1B
by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course follows up on a previous course, Algebra 1A, which "develops algebraic fluency by providing students with the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. In addition, the course develops proficiency with operations involving monomial and polynomial expressions." Along with providing a syllabus, the Course View section of the site is broken into three units: Exponents, monomials, and polynomials; Relations, functions, & quadratic equations; and Rational & radical expressions & equations. Each unit has five lessons, and each lesson has objectives, readings, multimedia components, assessments, and answers. Also, for instructors looking for more targeted teaching tools, the Topic View of the course presents both a sequential and alphabetical list of individual concepts covered in the course.Wed, 16 Jul 2008 03:00:03 -0500Integrating Mathematical and Biological Concepts
by Tina Fujita, James Hawker, and John Whitlock of Hillsborough Community College, these five curriculum guides integrate mathematical and biological concepts. These guides can be used in mathematics courses to illustrate biological applications or in biology courses to reinforce mathematics as a tool used in scientific analysis and to explain biological phenomena. Each modular unit, available on this page as PDFs, incorporates background information from both a mathematical and biological perspective and engages students with various assignments. Topics include calculating infant mortality rates, metabolic rate, surface to volume ratios, length tension curves, and preparing chemical solutions.Tue, 22 Apr 2008 11:37Maths Online Gallery
by the University of Vienna's futureMedia initiative, the Maths Online Gallery consists of a large collection of extremely useful interactive learning units that demonstrate mathematical concepts. A large number of interactive modules exist in such areas as analytic geometry, trigonometric functions, probability and statistics, integration, Fourier series, as well as model-building and simulation. The gallery was started in 1998 and new learning units are being continually added. This is an especially good resource for college and university teachers looking for in-class interactive illustrations of a large array of basic and advanced mathematical concepts.Wed, 14 Nov 2007 03:00:01 Nov 2007 03:00:03 -0600Algebra Quiz [java required]
solving algebraic equations with this interactive quiz brought to you by Interactivate and the Computational Science Education Reference Desk. The quiz allows you to select the difficulty level, time limit and type of equation you want to practice solving. Students can choose to include any or all of the following problem types in their quiz: variable on both sides, distributive property, quadratic, one-step problems, and two-step problems. This is an excellent activity for students to use in the classroom, at home to review or to study for upcoming tests.Mon, 5 NovCollege Algebra Handouts
by Professor Jody Harris at Broward Community College, these handouts are an excellent resource to print and give to community and technical college students in the algebra classroom. The subjects of the handouts include: quadratic equations; functions, domain, and range; inverse functions; compound interest; and exponential growth and decay. Each PDF explains, in graphic or equation form, the algebraic principle and some contain homework problems for students to complete to improve their algebra skills. [ASC]Mon, 22 Oct 2007 03:00:02 |
This classic, newly-revised book presents fundamental mathematics in the context of business and consumer applications to help put readers on the path to success. The all-new 1997 edition improves upon previous editions with a wealth of updated features. The material in Applied Business Mathematics will teach the student, the mathematical and critical thinking skills that will help him be a smart shopper, an informed citizen, and a valued employee.
Math is an integral part of business. Let the 15th Edition of Applied Business Math provide comprehensive coverage of personal and business-related finance. Learn the curriculum by lesson presentations that include examples with solutions. Basic math skills, spreadsheet applications, a simulation, and e-commerce are presented within each chapter. Upon completion you are prepared to understand and manage your personal finances, as well as grasp the fundamentals of business finance.
BUSINESS MATH provides comprehensive coverage of personal and business-related mathematics. In addition to reviewing the basic operations of arithmetic, students are prepared to understand and manage their personal finances, as well as grasp the fundamentals of business finances. Basic math skills are covered in a step-by-step manner, building confidence in users before they try it alone. Spreadsheet applications are available on the Data CD and a simulation activity begins every chapter. Chapters are organized into short lessons for ease of instruction and ease of learning.
Keeping Financial Records for Business 9E will give your students a broad knowledge of business operations and the basic skills they need to keep better financial records. The text contains a colorful graphic design and features that will capture students? interest, such as multicultural insights and interviews with individuals who use record keeping in their daily lives. A step-by-step approach to each new task makes it easier for students to master the job skills of record keeping |
More About
This Textbook
Overview
James Stewart's well-received CALCULUS: CONCEPTS AND CONTEXTS, Second Edition follows in the path of the other best-selling books by this remarkable author. The First Edition of this book was highly successful because it reconciled two schools of thought: it skillfully merged the best of traditional calculus with the best of the reform movement. This new edition continues to offer the balanced approach along with Stewart's hallmark features: meticulous accuracy, patient explanations, and carefully graded problems. The content has been refined and the examples and exercises have been updated. In addition, CALCULUS: CONCEPTS AND CONTEXTS, Second Edition now includes a free CD-ROM for students that contains animations, activities, and homework hints. The book integrates the use of the CD throughout by using icons that show students when to use the CD to deepen their understanding of a difficult concept.
In CALCULUS: CONCEPTS AND CONTEXTS, this well respected author emphasizes conceptual understanding - motivating students with real world applications and stressing the Rule of Four in numerical, visual, algebraic, and verbal interpretations. All concepts are presented in the classic Stewart style: with simplicity, character, and attention to detail. In addition to his clear exposition, Stewart also creates well thought-out problems and exercises. The definitions are precise and the problems create an ideal balance between conceptual understanding and algebraic skills.
Editorial Reviews
Booknews
New edition of a text that synthesizes reform and traditional approaches to calculus instruction, with emphasis on conceptual understanding through visual, numerical, and algebraic approaches. It's more streamlined than the author's traditional texts, with some topics interwoven throughout the book instead of being treated separately and fewer theorem proofs, among other differences. The included CD-ROM is intended to enrich and enhance the text with modules that allow students to explore topics in a "laboratory" environment. Annotation c. Book News, Inc., Portland, OR (booknews.com) October 15, 2002
Disappointed teacher
I have been teaching math on the college level for over 20 yrs. I am disappointed to say that there are much better texts that should be used in a classroom setting.
1 out of 1 people found this review helpful.
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Anonymous
Posted September 7, 2002
Calculus
Textbooks are supposed to explain concepts as well as give examples. This book approaches learning by example, but along with that should come explanation. This book is too expensive for what it dosen't deliver, knowledge.
1 out of 1 people found this review helpful.
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Anonymous
Posted July 15, 2002
There has to be better
This text did a very good job of explaining the material in the early chapters, but once you reach the later chapters that include multivariable calculus it goes downhill. The examples to some lessons don't help explain anything. Most of the lessons in the last 5 chapters are only good for providing practice problems. You must have a good teacher to understand the material though.
1 out of 1 people found this review helpful.
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Anonymous
Posted November 7, 2001
Needs Improvement
His book is not real clear and if one spends time doing most of the problems listed for each section, they will find that he does not present a clear explanation on how to solve most of the higher numbered problems. He really needs to work on his presentation.
1 out of 1 people found this review helpful.
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Anonymous
Posted July 8, 2002
Good enough
This is one of the better written text books that i have seen in a very long time, and i think it is very good, and very read-able. I am a math major and i hope that i will be able to find more books like it because it is wonderful. High school they were aweful and didn't explain anything and yours is given with many examples and is kinda easy to pick up on with the steps that you use. Just be sure next time to have examples for all of your practice problems, and don't assume that just because they are in calculus class they know what was supposed to be learned before. Maybe just add a little something from before in some of the exericises that you work the problem out so that one can recoginize and see what you are actually starting from. But besides that it is a class you can teach yourself with your book!! Keep it up and I can't wait for the next one.
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Anonymous
Posted September 30, 2001
good book, cant look at it!!
i have to agree with the archemides... its a good book, but i cant stand the very very dull, and plain explanations that are given here!! but it def. teaches wat it should with explanations, and good figues....
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Anonymous
Posted September 22, 2001
THIS IS A GREAT BOOK!
I use this book for my Multivariate Calculus class at Stuyvesant High School in NYC. It is a great book. Although for this class, we only use the second portion of the book, this year, my school has ordered more copies for the Calculus BC students to use the first portion of the book that does not deal with Multivariabe.
0 out of 1 people found this review helpful.
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Anonymous
Posted December 11, 2000
Nasty Calculus with 0 stars
I HATE CALCULUS!!!!!!!!! IT IS THE WORST SUBJECT IN THE WORLD!!! EVEN THOUGH THIS IS A GOOD BOOK, THE SUBJECT IS SO HORRID THAT THE BOOK BECOMES HORRIBLE TOO.
0 out of 2 people found this review helpful.
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Anonymous
Posted December 15, 2000
A very excellent textbook
I would like all of you to enjoy the excitements in discovering calculus when you read this excellent calculus textbook. I am using it right now and I am really amazed how wonderful the author's explanations about the concepts.This book hads a tons of great figures and lots of examples to illustrate the concept. I would like all of you to buy this book and enjoy the beautiful of this subject
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Summary
Designed specifically for the Calculus III course, Multivariable Calculus, 8/e, contains chapters 10 through 14 of the full Calculus, 8/e, text. The text continues a two-semester Calculus I with Precalculus text. Every edition of the series has made the mastery of traditional calculus skills a priority, while embracing the best features of new technology and, when appropriate, calculus reform ideas. Now, the Eighth Edition is the first calculus program to offer algorithmic homework and testing created in Maple so that answers can be evaluated with complete mathematical accuracy The Eighth Edition continues to provide an evolving range of conceptual, technological, and creative tools that enable instructors to teach the way they want to teach and students to learn they way they learn best.
Table of Contents
Vectors and the Geometry of Space
Vectors in the Plane
Space Coordinates and Vectors in Space
The Dot Product of Two Vectors
The Cross Product of Two Vectors in Space
Lines and Planes in Space Section Project: Distances in Space
Surfaces in Space
Cylindrical and Spherical Coordinates
Vector-Valued Functions
Vector-Valued Functions Section Project: Witch of Agnesi
Differentiation and Integration of Vector-Valued Functions
Velocity and Acceleration
Tangent Vectors and Normal Vectors
Arc Length and Curvature
Functions of Several Variables
Introduction to Functions of Several Variables
Limits and Continuity
Partial Derivatives Section Project: Moireacute; Fringes
Differentials
Chain Rules for Functions of Several Variables
Directional Derivatives and Gradients
Tangent Planes and Normal Lines Section Project: Wildflowers
Extrema of Functions of Two Variables
Applications of Extrema of Functions of Two Variables Section Project: Building a Pipeline
Lagrange Multipliers
Multiple Integration
Iterated Integrals and Area in the Plane
Double Integrals and Volume
Change of Variables: Polar Coordinates
Center of Mass and Moments of Inertia Section Project: Center of Pressure on a Sail |
Reading the text: Read each section of a text a number
of times a) skim before class b) read after class c) refer to while doing
homework d) review before test and final e) refer back to for review of
concepts used in later sections. Chapter Summaries and Link Its, Review
Chapters (7,17,22), and summaries in the front of the
book are available for studying for tests.The review chapters have summaries of the preceding chapters and Review
and Supplementary Exercises. Use the index at the end of the book to locate definitions.
Problems: Start the problems listed here
immediately after the section is started in class. As you read a chapter, do
the odd number"Apply
your Knowledge" and all "Check your Skills" multiple choice problems. Check the
"Apply your Knowledge" and"Check your Skills" problems and the odd number exercises against
the answers in the back of the book, and keep working on them until you can get
that answer. You may refer to the student solution guide, or discuss the
problems with your classmates or a tutor or me.Test questions will be based on the three odd number exercises for
each chapter assigned here. The exercise to be handed in is shown for each
chapter. You will be able to ask about a few homework problems at the beginning
of each class. |
First Course in Abstract Algebra
Considered a classic by many, John Fraleigh's A First Course in Abstract Algebra is an in-depth introductory text for the Abstract Algebra course. ...Show synopsisConsidered a classic by many, John Fraleigh's A First Course in Abstract Algebra is an in-depth introductory text for the Abstract Algebra course. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. It is geared towards high level courses at schools with strong math programs. *New exercises have been written, while past exercises have been revised and modifed. *Classical approach to abstract algebra. *Focus on applications of abstract algebra. *Classic text for the high end of the market - known and loved in this discipline. It is geared towards high level courses at schools with strong maths programs *Accessible pedagogy includes historical notes written by Victor Katz (author of another AWL book The History of Mathematics), an authority in this area *By opening with a study of group theory, this text provides students with an easy transition to axiomatic mathematics 1292024968 Brand new book. International Edition. Ship...New. 1292024968 |
elementary introductions to mathematical analysis, the treatment of the logical and algebraic foundations of the subject is necessarily rather skeletal. This book attempts to flesh out the bones of such treatment by providing an informal but systematic account of the foundations of mathematical analysis written at an elementary level. This book is entirely self-contained but, as indicated above, it will be of most use to university or college students who are taking, or who have taken, an introductory course in analysis. Such a course will not automatically cover all the material dealt with in this book and so particular care has been taken to present the material in a manner which makes it suitable for self-study. In a particular, there are a large number of examples and exercises and, where necessary, hints to the solutions are provided. This style of presentation, of course, will also make the book useful for those studying the subject independently of taught course. less |
The goals of this course are to develop knowledge of the contributions made by mathematicians and the influence these contributions have made to the development of human thought and culture over time. The course provides a chronological tracing of mathematics from the ancient Chinese into modern times, with an emphasis on problems and the individuals who formulated and solved them. Prerequisite: Grade of C or higher in MATH 222. Course meets World/Eastern Culture graduation requirement.
Prerequisite(s) / Corequisite(s):
Grade of C or higher in MATH 222.
Course Rotation for Day Program:
Offered odd Fall.
Text(s):
Most current editions of the following:
The History of Mathematics
By Burton, D.M. (Allyn and Bacon) Recommended
An Introduction to the History of Mathematics
By Eves, H. (Saunders College Publishing) Recommended
A Contextual History of Mathematics
By Calinger, R. (Prentice Hall) Recommended
A History of Mathematics
By Katz, V.J. (Addison Wesley) Recommended
A History of Mathematics
By Suzuki, J. (Prentice Hall) Recommended
Course Objectives
To acquaint the student with the impact of historical mathematics on contemporary society, and the impact of social, economic and cultural forces on the development of mathematics.
To study mathematics from the perspective of those who developed it.
To examine the influence of earlier mathematics upon contemporary theory, and to examine the interrelations among the various branches of mathematics.
Measurable Learning Outcomes:
Provide an overview of the development of mathematics throughout history and its interaction with the culture.
Provide historical perspective for the discipline and thus place significant topics and concepts within their appropriate historical context.
Find approximations to π using historical methods, including the method of Archimedes.
Know the development of Calculus and the major contributors to this area.
Explain the emergence of mathematical ideas within the context of the societies in which they first appeared.
Describe and explain the nature of proof and its relationship to mathematics.
Explain the importance of mathematics to science and modern society.
Give examples of significant historical applications of mathematics to astronomy, geography, timekeeping and everyday life.
Provide an explanation relating to the diversity of cultures that developed common mathematical concepts.
Explain the major role of religions in influencing mathematical thought.
Topical Outline:
Early numerical systems and the development of Hindu-Arabic numeration
Ancient mathematics: Babylonian, Egyptian, Greek, Chinese, and Hindu
Medieval and Renaissance mathematics
The seventeenth century: development of analytic geometry, probability, modern number theory, and calculus
The eighteenth century: exploitation of calculus
The nineteenth century: contributions of Gauss, the emergence of algebraic structure
Topics from 20th century mathematics |
Precalculus (4th Edition)
Book Description: Bob Blitzer's unique background in mathematics and behavioral sciences, along with his commitment to teaching, inspired him to develop a precalculus series that gets readers engaged and keeps them engaged. Presenting the full scope of the mathematics is just the first step. Blitzer draws in the reader with vivid applications that use math to solve real-life problems. These applications help answer the question "When will I ever use this?" Readers stay engaged because the book helps them remain focused as they study. The three-step learning system–See It, Hear It, Try It–makes examples easy to follow, while frequent annotations offer the support and guidance of an instructor's voice. Every page is interesting and relevant, ensuring that readers will actually use their textbook to achieve success! Prerequisites: Fundamental Concepts of Algebra; Functions and Graphs; Polynomial and Rational Functions; Exponential and Logarithmic Functions; Trigonometric Functions; Analytic Trigonometry; Additional Topics in Trigonometry; Systems of Equations and Inequalities; Matrices and Determinants; Conic Sections and Analytic Geometry; Sequences, Induction, and Probability; Introduction to Calculus For all readers interested in college algebra |
Precalculus
Overview
The academic standards for the precalculus core area establish the process skills and core
content for Precalculus, which should provide students with the mathematics skills and
conceptual understanding necessary for them to further their mathematical education or
to pursue mathematics-related technical careers.
The content of the precalculus standards encompasses characteristics and behaviors of
functions, operations on functions, behaviors of polynomial functions and rational
functions, behaviors of exponential and logarithmic functions, behaviors of trigonometric
functions, and behaviors of conic sections. Teachers, schools, and districts should use the
precalculus standards to make decisions concerning the structure and content of
Precalculus. Content in this course may go beyond the precalculus standards.
All courses based on the academic standards for precalculus must include instruction
using the mathematics process standards, allowing students to engage in problem solving,
decision making, critical thinking, and applied learning. Educators must determine the
extent to which such courses or individual classes may go beyond these standards. Such
decisions will involve choices regarding additional content, activities, and learning
strategies and will depend on the objectives of the particular courses or individual classes.
In all courses based on the precalculus standards, hand-held graphing calculators are
required for instruction and assessment. Students should learn to use a variety of ways to
represent data, to use a variety of mathematical tools such as graph paper, and to use
technologies such as graphing calculators to solve problems.
Note: The term including appears in parenthetical statements in the high school
mathematics indicators to introduce a list of specifics that are intended to clarify and
focus the teaching and learning of the particular concept. That is, within these
parenthetical including statements are specified the components of the indicator that are
critical for the particular core area with regard both to the state assessments and to the
management of time in the classroom. While instruction must focus on the entire
indicator, educators must be certain to cover the components specified in the
parenthetical including statements.
Precalculus
The mathematical processes provide the framework for teaching, learning, and assessing in
all high school mathematics core courses. Instructional programs should be built around
these processes.
Standard PC-1: The student will understand and utilize the mathematical processes of
problem solving, reasoning and proof, communication, connections,
and representation.
Indicators
PC-1.1 Communicate knowledge of algebraic and trigonometric relationships by using
mathematical terminology appropriately.
PC-1.2 Connect algebra and trigonometry with other branches of mathematics.
PC-1.3 Apply algebraic methods to solve problems in real-world contexts.
PC-1.4 Judge the reasonableness of mathematical solutions.
PC-1.5 Demonstrate an understanding of algebraic and trigonometric relationships by
using a variety of representations (including verbal, graphic, numerical, and
symbolic).
PC-1.6 Understand how algebraic and trigonometric relationships can be represented
in concrete models, pictorial models, and diagrams.
PC-1.7 Understand how to represent algebraic and trigonometric relationships by using
tools such as handheld computing devices, spreadsheets, and computer algebra
systems (CASs).
Precalculus
Standard PC-2: The student will demonstrate through the mathematical processes an
understanding of the characteristics and behaviors of functions and
the effect of operations on functions.
Indicators
PC-2.1 Carry out a procedure to graph parent functions (including y = xn, y = loga x, y
1
= ln x, y = , y = ex, y = ax, y = sin x, y = cos x, y = tan x, y = csc x, y = sec x,
x
and y = cot x).
PC-2.2 Carry out a procedure to graph transformations and combinations of
transformations.
PC-2.3 Analyze a graph to describe the transformation.
PC-2.4 Carry out procedures to algebraically solve equations involving parent
functions or transformations of parent functions (including y = xn, y = loga x, y
1
= ln x, y = ,
x
y = ex, y = ax, y = sin x, y = cos x, y = tan x, y = csc x, y = sec x, and y = cot x).
PC-2.5 Analyze graphs, tables, and equations to determine the domain and range of
parent functions or transformations of parent functions (including y = xn, y =
1
loga x, y = ln x, y = , y = ex, y = ax, y = sin x, y = cos x, y = tan x, y = csc x, y
x
= sec x, and y = cot x).
PC-2.6 Analyze a function or the symmetry of its graph to determine whether the
function is even, odd, or neither.
PC-2.7 Recognize and use connections among significant points of a function
(including roots, maximum points, and minimum points), the graph of a
function, and the algebraic representation of a function.
PC-2.8 Carry out a procedure to determine whether the inverse of a function exists.
PC-2.9 Carry out a procedure to write a rule for the inverse of a function, if it exists.
Precalculus
Standard PC-3: The student will demonstrate through the mathematical processes an
understanding of the behaviors of polynomial and rational functions.
Indicators
PC-3.1 Carry out a procedure to graph quadratic and higher-order polynomial
functions by analyzing intercepts and end behavior.
PC-3.2 Apply the rational root theorem to determine a set of possible rational roots of
a polynomial equation.
PC-3.3 Carry out a procedure to calculate the zeros of polynomial functions when
given a set of possible zeros.
PC-3.4 Carry out procedures to determine characteristics of rational functions
(including domain, range, intercepts, asymptotes, and discontinuities).
PC-3.5 Analyze given information to write a polynomial function that models a given
problem situation.
PC-3.6 Carry out a procedure to solve polynomial equations algebraically.
PC-3.7 Carry out a procedure to solve polynomial equations graphically.
PC-3.8 Carry out a procedure to solve rational equations algebraically.
PC-3.9 Carry out a procedure to solve rational equations graphically.
PC-3.10 Carry out a procedure to solve polynomial inequalities algebraically.
PC-3.11 Carry out a procedure to solve polynomial inequalities graphically.
Precalculus
Standard PC-4: The student will demonstrate through the mathematical processes an
understanding of the behaviors of exponential and logarithmic
functions.
Indicators
PC-4.1 Carry out a procedure to graph exponential functions by analyzing intercepts
and end behavior.
PC-4.2 Carry out a procedure to graph logarithmic functions by analyzing intercepts
and end behavior.
PC-4.3 Carry out procedures to determine characteristics of exponential functions
(including domain, range, intercepts, and asymptotes).
PC-4.4 Carry out procedures to determine characteristics of logarithmic functions
(including domain, range, intercepts, and asymptotes).
PC-4.5 Apply the laws of exponents to solve problems involving rational exponents.
PC-4.6 Analyze given information to write an exponential function that models a
given problem situation.
PC-4.7 Apply the laws of logarithms to solve problems.
PC-4.8 Carry out a procedure to solve exponential equations algebraically.
PC-4.9 Carry out a procedure to solve exponential equations graphically.
PC-4.10 Carry out a procedure to solve logarithmic equations algebraically.
PC-4.11 Carry out a procedure to solve logarithmic equations graphically.
Precalculus
Standard PC-5: The student will demonstrate through the mathematical processes an
understanding of the behaviors of trigonometric functions.
Indicators
PC-5.1 Understand how angles are measured in either degrees or radians.
PC-5.2 Carry out a procedure to convert between degree and radian measures.
PC-5.3 Carry out a procedure to plot points in the polar coordinate system.
PC-5.4 Carry out a procedure to graph trigonometric functions by analyzing
intercepts, periodic behavior, and graphs of reciprocal functions.
PC-5.5 Carry out procedures to determine the characteristics of trigonometric
functions (including domain, range, intercepts, and asymptotes).
PC-5.6 Apply a procedure to evaluate trigonometric expressions.
PC-5.7 Analyze given information to write a trigonometric function that models a
given problem situation involving periodic phenomena.
PC-5.8 Analyze given information to write a trigonometric equation that models a
given problem situation involving right triangles.
PC-5.9 Carry out a procedure to calculate the area of a triangle when given the
lengths of two sides and the measure of the included angle.
PC-5.10 Carry out a procedure to solve trigonometric equations algebraically.
PC-5.11 Carry out a procedure to solve trigonometric equations graphically.
PC-5.12 Apply the laws of sines and cosines to solve problems.
PC-5.13 Apply a procedure to graph the inverse functions of sine, cosine, and tangent.
PC-5.14 Apply trigonometric relationships (including reciprocal identities;
Pythagorean identities; even and odd identities; addition and subtraction
formulas of sine, cosine, and tangent; and double angle formulas) to verify
other trigonometric identities.
PC-5.15 Carry out a procedure to compute the slope of a line when given the angle of
inclination of the line.
Precalculus
Standard PC-6: The student will demonstrate through the mathematical processes an
understanding of the behavior of conic sections both geometrically
and algebraically.
Indicators
PC-6.1 Carry out a procedure to graph the circle whose equation is the form
(x h)2 (y k)2 r 2 .
PC-6.2 Analyze given information about the center and the radius or the center and
the diameter to write an equation of a circle.
PC-6.3 Apply a procedure to calculate the coordinates of points where a line
intersects a circle.
PC-6.4 Carry out a procedure to graph the ellipse whose equation is the form
x h y k 1.
2 2
a2 b2
PC-6.5 Carry out a procedure to graph the hyperbola whose equation is the form
x h2 y k 2
1.
a2 b2
PC-6.6 Carry out a procedure to graph the parabola whose equation is the form
y k a(x h)2 .
|
Prerequisite: MAT 076 (min grade C) or 1 year of high school geometry (min grade C), and MAT 080 (min grade C), or 2 years of high school algebra (min grade C) or appropriate Placement score or ACT score of 21-22
30201
A
3
TTh
11:00-12:40
1C03
CR. Conderman
17
MAT115 Principles of Modern Math
Prerequisite: MAT 076 (min grade C) or 1 year high school geometry (min grade C), and MAT 080 (min grade C) or 2 years of high school algebra (min grade C) or appropriate Placement score or ACT score of 21-22
IAI#: M1904
30137
A
3
TTh
08:00-09:15
3G06
CR. Conderman
20
30502
B
3
TTh
11:00-12:15
2M05
JL. Horn
16
MAT121 College Algebra
Prerequisite: MAT 076 (min grade C) or 1 year of high school geometry (min grade C) and MAT 080 (min grade C) or 2 years of high school algebra (min grade C) or appropriate Placement or ACT score of 21-22
30139
A
4
MTWTh
08:30-09:20
2M07
EA. Etter
27
$10
30138
B
4
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12:30-2:15
2M05
JL. Horn
18
$10
30140
C
4
MTWTh
11:00-11:50
2M07
EA. Etter
28
$10
30347
D
4
MW
10:50-12:30
2H14
KM. Megill
21
$10
30141
N
4
TTh
6:00-7:40
2L03
RK. Hobson
29
$10
MAT122 Trigonometry
Prerequisite: MAT 121 (min grade C) or appropriate Placement score or 4 years of college preparatory high school mathematics (min grade C) and apprpriate placement score or ACT score of 23-25
30142
N
3
TTh
6:00-7:15
2H14
CR. Conderman
21
$10
MAT203 Calculus & Analytic Geometry I
Prerequisite: MAT 122 (min grade C) or appropriate Placement score or 4 years of college preparatory high school mathematics (min grade C) and appropriate Placement score or ACT score of 23-25 |
Features
Offers a prelude to each chapter that describes topics covered and places material in historical context
Provides many graphical exercises, also available online, that are solved using MATLAB
Summary
An Introduction to Partial Differential Equations with MATLAB exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green's functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB® software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.
The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the "Big Three" PDEs-- the heat, wave, and Laplace equations, and is followed by chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions.
Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials.
Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green's functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.
The Big Three PDEs
Second-Order, Linear, Homogeneous PDEs with Constant Coefficients The Heat Equation and Diffusion The Wave Equation and the Vibrating String Initial and Boundary Conditions for the Heat and Wave Equations Laplace's Equation--The Potential Equation Using Separation of Variables to Solve the Big Three PDEs
Fourier Series
Introduction Properties of Sine and Cosine The Fourier Series The Fourier Series, Continued The Fourier Series---Proof of Pointwise Convergence Fourier Sine and Cosine Series Completeness Solving the Big Three PDEs Solving the Homogeneous Heat Equation for a Finite Rod Solving the Homogeneous Wave Equation for a Finite String Solving the Homogeneous Laplace's Equation on a Rectangular Domain Nonhomogeneous Problems Characteristicsfor Linear PDEs First-Order PDEs with Constant Coefficients First-Order PDEs with Variable Coefficients D'Alembert's Solution for the Wave Equation--The Infinite String Characteristics for Semi-Infinite and Finite String Problems General Second-Order Linear PDEs and Characteristics
Integral Transforms
The Laplace Transform for PDEs Fourier Sine and Cosine Transforms The Fourier Transform The Infinite and Semi-Infinite Heat Equations Distributions, the Dirac Delta Function and Generalized Fourier Transforms Proof of the Fourier Integral Formula Bessel Functions and Orthogonal Polynomials The Special Functions and Their Differential Equations Ordinary Points and Power Series Solutions; Chebyshev, Hermite and Legendre Polynomials The Method of Frobenius; Laguerre Polynomials Interlude: The Gamma Function Bessel Functions Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials Sturm-Liouville Theory and Generalized Fourier Series Sturm-Liouville Problems Regular and Periodic Sturm-Liouville Problems Singular Sturm-Liouville Problems; Self-Adjoint Problems The Mean-Square or L2 Norm and Convergence in the Mean Generalized Fourier Series; Parseval's Equality and Completeness PDEs in Higher Dimensions PDEs in Higher Dimensions: Examples and Derivations The Heat and Wave Equations on a Rectangle; Multiple Fourier Series Laplace's Equation in Polar Coordinates; Poisson's Integral Formula The Wave and Heat Equations in Polar Coordinates Problems in Spherical Coordinates The Infinite Wave Equation and Multiple Fourier Transforms Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green's Identities for the Laplacian Nonhomogeneous Problems and Green's Functions Green's Functions for ODEs Green's Function and the Dirac Delta Function Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation Green's Function's for Equations of Evolution
Editorial Reviews
"The strongest aspect of this text is the very large number of worked boundary value problem examples." SIAM review
"This is a useful introductory text on partial differential equations (PDEs) for advanced undergraduate / beginning graduate students of applied mathematics, physics, or engineering sciences. …It may be said that this is a nice introductory text which certainly is of great use in preparing and delivering courses." —Zentralblatt MATH
"Readers new to the subject will find Coleman's appendix cataloguing important partial differential equations in their natural surroundings quite useful. …Coleman's more explicit, extended style would probably allow its use as an advanced graduate or reference text for UK engineers or physicists." —Times Higher Education
"The book presents very useful material and can be used as a basic text for self-study of PDEs." —EMS Newsletter, Sept., 2005
"Each chapter is introduced by a 'prelude' that describes its content and gives historical background. Each section concludes with a set of exercises, many of which are marked 'MATLAB'." —CMS Notes, Volume 37, No. 2, March 2005 |
Three dimensional - perception: Understanding and appreciation of scale and proportion of objects, building forms and elements, colour texture, harmony and contrast. Design and drawing of geometrical or abstract shapes and patterns in pencil. Transformation of forms both 2 D and 3 D union, substraction, rotation, development of surfaces and volumes, Generation of Plan, elevations and 3 D views of objects. Creating two dimensional and three dimensional compositions using given shapes and forms. Sketching of scenes and activities from memory of urbanscape (public space, market, festivals, street scenes, monuments, recreational spaces etc.), landscape (river fronts, jungles. gardens, tre es, plants etc.) and rural life.
Note:Candidates are advised to bring pencils, own geometry box set, erasers and colour pencils and crayons for the Aptitude Test.
MATHEMATICS
UNIT 1:SETS, RELATIONS AND FUNCTIONS:
Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations, functions;. one-one, into and onto functions, composition of functions.
UNIT 2:COMPLEX NUMBERS AND QUADRATIC EQUATIONS: Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots.
UNIT 3:MATRICES AND DETERMINANTS:
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.
UNIT 4:PERMUTATIONS AND COMBINATIONS:
Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and C (n,r), simple applications.
Integral as an anti - derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities. Evaluation of simple integrals of the type
Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form.
Cartesian system of rectangular co-ordinates in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes. Straight lines Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines. Circles, conic sections Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency.
UNIT 12:Three Dimensional Geometry:
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines. Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.
UNIT 13:Vector Algebra:
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
UNIT 14:STATISTICS AND PROBABILITY:
Measures of Dispersion: Calculation of mean, median, mode of grouped and ungrouped data. Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data. Probability: Probability of an event, addition and multiplication theorems of probability, Baye's theorem, probability distribution of a random variate, Bernoulli trials and Binomial distribution. |
Advanced Calculus : Course in Mathematics Analysis - 2nd edition
ISBN13:978-0821847916 ISBN10: 0821847910 This edition has also been released as: ISBN13: 978-0534376031 ISBN10: 0534376037
Summary: ADVANCED CALCULUS rigorously presents the fundamental concepts of mathematical analysis in the clearest, simplest way, within the context of illuminating examples and stimulating exercises. Emphasizing the unity of the subject, the text shows that mathematical analysis is not a collection of isolated facts and techniques, but rather a coherent body of knowledge. Beyond the intrinsic importance of the actual subject, the author demonstrates that the study of mathemati...show morecal analysis instills habits of thought that are essential for a proper understanding of many areas of pure and applied mathematics. Students gain a precise understanding of the subject, together with an appreciation of its coherence and significance. The full book is suitable for a year-long course; the first nine chapters are suitable for a one-term course on functions of a single variable. This book is included in the Brooks/Cole Series in Advanced Mathematics.
Benefits:
NEW! Many changes, some small and some large, have been made in the proofs to emphasize even more strongly the following point of view: Mathematics students must learn to read mathematics, not just listen to it in class, and this book gives them an opportunity to do so. This is a course for students who are learning the art of proof. When the students read a proof, they should feel they understand how mathematicians discovered the proof and so the text avoids 'rabbit out of the hat' proofs. The proofs seem natural to the students and encourage them to believe they could have discovered the proofs.
NEW! There has been a fundamental conceptual clarification in the development of the material on the structure of the real numbers and the integral.
NEW! Many new exercises, of varying degrees of difficulty, have been added. A Solutions Manual is available with detailed solutions to selected problems.
NEW! A number of non-essential concepts have been eliminated and optional material has been placed in a way to make it possible to cover the essential material more easily and fluidly.
The text's presentation is conversational and mathematically precise.
In addition to the essential topics, the author includes important topics such as the approximation methods for estimating integrals, the Weierstrass Approximation Theorem, and metric spaces, without disturbing the coherence of the course.
A wide variety of exercises helps students gain a genuine understanding of the material. The more challenging problems often stimulate the student to carefully reread the relevant sections in order to properly assimilate the material.
The Algebra of Derivatives. Differentiating Inverses and Compositions. The Mean Value Theorem and Its Geometric Consequences. The Cauchy Mean Value Theorem and Its Analytic Consequences. The Notation of Leibnitz.
5. ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS.
Solutions of Differential Equations. The Natural Logarithm and the Exponential Functions. The Trigonometric Functions. The Inverse Trigonometric Functions.
Solutions of Differential Equations. Integration by Parts and by Substitution. The Convergence of Darboux and Riemann Sums. The Approximation of Integrals.
8. APPROXIMATION BY TAYLOR POLYNOMIALS.
Taylor Polynomials. The Lagrange Remainder Theorem. The Convergence of Taylor Polynomials. A Power Series for the Logarithm. The Cauchy Integral Remainder Theorem. A Non-Analytic, Infinitely Differentiable Function. The Weierstrass Approximation Theorem.
9. SEQUENCES AND SERIES OF FUNCTIONS.
Sequences and Series of Numbers. Pointwise Convergence of Sequences of Functions. Uniform Convergence of Sequences of Functions. The Uniform Limit of Functions. Power Series. A Continuous, Nowhere Differentiable Function.
10. THE EUCLIDEAN SPACE Rn.
The Linear Structure of Rn and the Scalar Product. Convergence of Sequences in Rn. Open Sets and Closed Sets in Rn.
Linear Mappings and Matrices. The Derivative Matrix and the Differential. The Chain Rule.
16. IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM.
Functions of a Single Variable and Maps in the Plane. Stability of Nonlinear Mappings. A Minimization Principle and the General Inverse Function Theorem.
17. THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS.
A Scalar Equation in Two Unknowns: Dini's Theorem. The General Implicit Function Theorem. Equations of Surfaces and Curves in R³. Constrained Extrema Problems and Lagrange Multipliers.
18. INTEGRATING FUNCTIONS OF SEVERAL VARIABLES.
Integration of Functions on Generalized Rectangles. Continuity and Integrability. Integration over Jordan Domains.
19. ITERATED INTEGRATION AND CHANGES OF VARIABLES.
Fubini's Theorem. The Change of Variables Theorem: Statements and Examples. Proof of the Change of Variables Theorem.
20. LINE AND SURFACE INTEGRALS.
Arclength and Line Integrals. Surface Area and Surface Integrals. The Integral Formulas of Green and Stokes. Appendix A: Consequences of the Field and Positivity Axioms. The Field Axioms and Their Consequences. The Positivity Axioms and Their Consequences. Appendix B: Linear Algebra.
All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU:9780821847916-5-0
$78.00 +$3.99 s/h
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Sequitur Books Boonsboro, MD
Brand new. We distribute directly for the publisher. Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. T...show morehe goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables.Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material.Supplemented with numerous exercises, Advanced Calculus is a perfect book for undergraduate students of analysis. ...show less
$82.00 +$3.99 s/h
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Has minor wear and/or markings. SKU:9780821847916-3-0 just |
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More About
This Book
all the puzzling elements of mathematics, making even the most complicated material clear and easy to handle.
Features include:
Completely updated coverage of the new SAT I, with added topics such as Algebra II
A sample exam for the new SAT I math sections developed especially for this book
Bob Miller was a lecturer in mathematics at City College of New York, a branch of the City University of New York, for more than 30 years. His principal goal is to make the study of mathematics both easier and enjoyable 6, 2006
Math Made Easy!
Bob Miller takes very difficult concepts and makes them extremely easy to grasp. It is the one math book that will not make you tear your hair out of your head! The author uses a bit of humor and emotion to make math understandable to the average layman. The books are meant to be a supplement to whatever math text you are using. After Bob explains the concepts you will breeze through your textbook's examples. A great book at a great price!
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. |
Purpose of the unit: This unit is designed to develop students' understanding of basic geometry and math concepts that are needed in order to continue in this class of algebra.
Rationale of the unit: The unit topic is important for building a foundation of knowledge for the following units in algebra. A clear understanding of the objectives will increase their chance of success for the rest of the year in the class of algebra. The integration of the concepts with the other core subjects will not only help the students understand the concepts but also see the significance of math concepts in many areas of study.
Goals of the unit: The goals of this unit are for the students to : 1. To develop an understanding of basic and introductory geometry and math concepts. 2. To look forward to continuing in the study of algebraic concepts. 3. To understand how geometry and algebra relate to many other areas of study such as science, social studies, and language arts. 4. To understand how the concepts of geometry and algebra are relevant to their daily lives. 5. To look forward to living a life of discovery.
Instructional objectives (discoveries) of the unit: 1. To be able to do geometric formulas. 2. To understand tree diagrams. 3. To understand the coordinate plane. 4. To be able to round and estimate. 5. To be able to add and subtract decimals. 6. To be able to multiply and divide decimals. 7. To use divisibility rules to find all factors of a given number and to identify numbers as prime or composite. 8. To write the prime factorization of a number. 9. To find the GCF of two or more numbers and to find the LCM of two or more numbers. 10. To write equivalent fractions and write fractions in simplest form.
Unit Overview: Throughout this unit students will developing an understanding of basic geometry and algebraic concepts through lectures, class discussions, games such as Sudoku, and story problems that incorporate the them e of discovery and integrate the core subjects as well. |
Find a Falls ChurchYou may be presented a problem, where you decide which concept is applicable for its solution. It is helpful to know the following: Prime factorization of the integers, of algebraic polynomials, and the Pythagorean triples. It is helpful to know the various measures associated with geometric figures. |
III.Integrals
Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
Basic properties of definite integrals. (Examples include additivity and linearity.)
Fundamental Theorem of Calculus.
Use of the Fundamental Theorem to evaluate definite integrals.
Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation.
Antiderivatives following directly from derivatives of basic functions.
Citation Manager
"
6. Recommendations ."
Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools: Report of the Content Panel for Mathematics 45
Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics
Computation of derivatives.
Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
Basic rules for the derivative of sums, products, and quotients of functions.
Chain rule and implicit differentiation.
Derivatives of parametric, polar, and vector functions.
III. Integrals
Interpretations and properties of definite integrals.
Computation of Riemann sums using left, right, and midpoint evaluation points.
Definite integral as a limit of Riemann sums over equal subdivisions.
Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
Basic properties of definite integrals. (Examples include additivity and linearity.)Fundamental Theorem of Calculus.
Use of the Fundamental Theorem to evaluate definite integrals.
Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation.
Antiderivatives following directly from derivatives of basic functions.
Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).
Improper integrals (as limits of definite integrals).
Applications of antidifferentiation.
Finding specific antiderivatives using initial conditions, including applications to motion along a line.
Solving separable differential equations and using them in modeling. In particular, studying the equation y' = ky and exponential growth.
OCR for page 45
Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics
Solving logistic differential equations and using them in modeling.
Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
*IV. Polynomial Approximations and Series
* Concept of series. A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.
* Series of constants.
+ Motivating examples, including decimal expansion.
+ Geometric series with applications.
+ The harmonic series.
+ Alternating series with error bound.
+ Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series.
+ The ratio test for convergence and divergence.
+ Comparing series to test for convergence or divergence.
* Taylor series.
+ Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.)
+ Maclaurin series and the general Taylor series centered at x = a.
+ Maclaurin series for the functions ex, sin x, cos x, and
+ Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.
+ Functions defined by power series.
+ Radius and interval of convergence of power series.
+ Lagrange error bound for Taylor polynomials.
SOURCE: College Entrance Examination Board. 1999. Advanced Placement Program Course Description: Calculus – May 2000, May 2001 (pp. 9-14). New York: author. |
High School Math Curriculum Synopsis
Algebra I (Bob Jones University) - Algebra 1 is a required course offered to ninth grade students. Today's society depends on algebra for the development and maintenance of technological tools. Algebra 1 prepares students with tools that help them understand and use technological tools. It enhances their ability to work practical problems by providing methods and practice for solving more complicated problems. The text approaches algebraic concepts systematically, beginning with basic operations on integers and ending with application of quadratic functions. This course works to prepare students for future mathematics courses, and it meets all fourteen goals for high school mathematics that are identified by the NCTM (National Council of Teachers of Mathematics).
Geometry (Bob Jones curriculum) – This required math course for tenth grade students deals primarily with Euclidean geometry. The text is written from a Biblical perspective and is designed to increase the student's ability to work practical problems. It also helps develop thinking processes that are necessary for future math classes and everyday life.
Algebra II (Bob Jones curriculum) – This is an honors math course offered to eleventh grade students and written from a Biblical perspective. It provides general knowledge of basic mathematical concepts including number systems, operations, geometry, and functions. It also provides computational skills necessary for placement and aptitude tests, as well as, advancement to Pre-calculus.
Pre-calculus (Bob Jones curriculum) – This is an honors math course offered to twelfth grade students and written from a Biblical perspective. It provides general knowledge of basic mathematical geometry and functions. It also provides computational skills necessary for placement and aptitude tests.
Consumer Math – offered during a student's eleventh or twelfth grade year.
Business Math (A Beka curriculum) - Business Mathematics in Christian Perspective is offered during a student's eleventh or twelfth grade year. This course reviews all basic mathematical skills while teaching stewardship, business management, investments, income, taxes, and banking. |
Student Resources
As noted in our mission statement, we are strongly committed to providing students with a solid and broad-based mathematical foundation that helps prepare them for a variety of careers as well as graduate study in mathematics and the mathematical sciences.
CSBSJU Mathematics Colloquium
The CSBSJU Mathematics Colloquium meets roughly every other Thursday. Here is a schedule for Spring 2014:
Abstract: To "color" a graph, graph parts (vertices, edges, both, etc.) are partitioned into different colors. We will look at how some classical ideas from graph theory behave once a coloring is applied to the graph. (No prior knowledge of graph theory will be needed.) Along with the founding ideas and new avenues to pursue in this area, some results from CSBSJU students will also be presented.
Mathematical Competitions
Each year students from the College of Saint Benedict and Saint John's University participate in two mathematical competitions. Each November students may participate in the NCS/MAA team competition. This contest consists of ten problems, which are graded with a value of ten points per problem. The problems typically range in difficulty from fairly easy to extremely difficult. Students work in groups of three and submit their work as a team.
Every February students may participate in the Mathematical Contest in Modeling or MCM. The MCM is a contest where teams of undergraduates use mathematical modeling to present their solutions to real world problems. Students in teams of three work on one problem over an entire weekend. Not only is their work graded on mathematical correctness, but also on clarity and ease of understanding. For additional information see the COMAP website.
Actuary
A good number of our students pursue a career as an Actuary, a person who calculates risk for insurance companies. For further information contact Phil Byrne or Kris Nairn.
Student Employment Opportunities
Students with an aptitude for mathematics have the option to work for the department as a course assistant, teaching assistant and work in the Math Skills Center. This opportunity not only help students prepare for teaching mathematics in the secondary and college level, but also they get paid! For more information contact Phil Byrne.
Summer Experiences
Every summer the math department sponsors students to do summer research with an advisor. CSB/SJU also has a strong tradition of students participating in Research Experiences for Undergraduates (REUs) around the country. For further information contact Tom Sibley and visit our page on summer research experiences. |
recommended for secondary cerified teachers hoping to gain a deeper understanding of content and pedagogical techniques in algebra and geometry. The National Council of Teachers of Mathemeatics, Principles and Standards for School Mathematics, and the Pennsylvania State Standards for Mathematics are a major focus. The topics covered include Real and Complex numbers, functions, equations, integers, and polynomials, number system structures, congruence, distance, and similarity, trigonometry, area and volume, axiomatics and Euclidean Geometry. The graphing calculator (TI83) and other technology will be used throughout the course of foster discovery and to gain insights into the fundamental concepts.
Prerequisites: MATH2170 and graduate standing. 3 Credits |
Description:
Under the motto, "Show me how, now!" algebasics is a fine online mathematics instructional resource that takes young and old alike through the basics of algebra. The breadth of the material is divided into sixteen sections, which begin with, appropriately, "the basics", and proceed all the way to a section on applying algebra to real-world situations. Each section asks users to solve a number of problems so that they will gain mastery of each concept. The interface deployed here is quite user-friendly, as each problem is narrated so that users will better understand the process needed to complete each problem successfully. Overall, it's a well-designed introduction to this area of the mathematical universe, and one that is very easy to use. |
More About
This Textbook
Overview
The Barnett, Ziegler, Byleen College Algebra/Precalculus series is designed to be user friendly and to maximize student comprehension. The goal of this series is to emphasize computational skills, ideas, and problem solving rather than mathematical theory. The large number of pedagogical devices employed in this text will guide a student through the course. Integrated throughout the text, the students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematically concepts. In each section, the worked examples are followed by matched problems that reinforce the concept being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A Smart CD is packaged with the seventh edition of the book. This CD tutorial reinforces important concepts, and provides students with extra practice |
Introduction to Proofs, an Inquiry-Based approach
A Free text for a course on proofs
Introduction to Proofs is a Free undergraduate text.
It is inquiry-based, sometimes called the
Moore method or the discovery method.
Highlights
Inquiry-based
The text consists of a sequence of exercises, statements
for students to prove,
along with a few definitions and remarks.
The instructor does not lecture but instead lightly guides as the
class works through the material together.
For these students, this is the
best way to develop mathematical maturity.
Covers needed material
Parity through arity:
number theory to the Fundamental Theorem of Arithmetic,
sets to DeMorgan's Laws,
functions to two-sided inverses,
and
relations to partitions.
There is a brief optional chapter on infinity.
No prerequisite
We enroll sophomore Math majors with
three semesters of calculus.
Even this background isn't a logical prerequisite
but instead
ensures that students have mathematical aptitude.
Free
This includes the LaTeX source so an instructor
can tune the text to the class.
Extras
There are beamer slides on logic topics
that reinforce and organize the discussions that arise naturally
in the first couple of weeks.
There are also answers for all exercises.
Here is Introduction to Proofs
Click to download
Introduction to Proofs.
You can also get a compact version
that copies onto six or seven sheets and so is perfect
for handing out on the first day of class.
There are beamer slides covering enough
logic to get students started.
I find the answers useful in thinking ahead
for class (and they kept me honest in sequencing the material).
You can also
clone the repository,
if you are into LaTeX.
What is "inquiry-based"?
This teaching style has students work directly with the mathematics.
I'll illustrate by discussing my version of this class.
The students have four grades: one each from a mid-term and a final,
a grade from the written work on their hand-in problems,
which are extensively discussed in class,
and a grade reflecting their
contributions to the in-class discussion.
Thus the in-class work is a major part of the course \mdash;
exercises are the core experience.
The best way to see what this means is by seeing a typical day.
As background,
students have pledged not to work together or to use any
resources such as other books, the net, students who have
taken the course before, etc.
I tell them that working with others is a great thing
but that a variety of experiences is also a great thing and
this class proceeds in this way.
Consequently, on each class day
students arrive having thought about and
worked on each exercise, on their own.
Typical is four exercises.
I shuffle index cards to
pick who will go to the board today — the first person picked
gets their choice of the four, the next person chooses from the
remaining three, etc.
(I leave out the people who were picked last time.
In addition, twice in the semester each student can take a pass.)
While those students are at the board there is usually business to do.
In perhaps one class out of three there are hand-in problems to collect.
In the first few weeks of the semester I have beamer slides that
discuss some aspects of logic such as the definition of implication.
After the first few weeks there may be LaTeX questions, since
some hand-ins must be in LaTeX.
(At the start of the course I distribute a
LaTeX for Undergraduates tutorial;
note that some PDF readers have trouble with this document but it
displays correctly in Acrobat Reader.)
And sometimes we just talk about math things, such as the Clay prizes,
or news in the math world, or interesting blogs, or good writing style
for mathematics, or
what math courses are offered next semester.
With the exercises are on the board the work begins.
Students talk through the proposed solutions,
sentence by sentence and sometimes word by word.
This discussion
is typically filled with misconceptions to work out,
bad ideas that take a while to go nowhere,
and good ideas that have initial trouble getting
a hearing, so as an instructor I can struggle with
letting it take its course
(I speak as little as I can stand, try not to nod or shake my head, and
if the room allows it then
I sit in the back so people do not easily look to me).
But the discussion does eventually come to an end and usually it is
the correct end, with the bonus that the process has allowed students
to have come to understand why wrong things are wrong and
well as to see what is right.
(Sometimes I guide a bit, as with speaking up after the class
has decided that the proof is OK to say,
"In the second paragraph,
exhibiting the n=3 and n=5 cases is not enough to show
that in all cases the square of an odd is odd.")
The discussion can take the entire period for just the four problems.
If there is time, as there will be in perhaps a half or a third
of the classes,
then I will propose another exercise for general discussion.
(There are some exercises that traditionally give people trouble
such as that there are infinitely many primes,
or that for a function being a correspondence is equivalent to
being invertible.
I try to arrange the homework so that we do these exercises in class.)
Finally, I assign a new set of problems,
perhaps saying that one of them is to be handed in and
perhaps that it must be done in LaTeX.
That ends the day.
A note about the hand-ins:
I grade them out of 10 and they are due the next class.
A student who does not see how to do it can
choose to hand it in the class after that,
having heard the discussion on the problem, for a maximum grade of
5/10.
On the first day of the semester I comment that
typically every person in the class
uses this option at some point.
To close, there are two things I must say.
First, I find this teaching
harder than lecturing because
I have given up the flow of control
but it is also rewarding.
Students learn better than with other
approaches and that's the point, isn't it?
Second, I must acknowledge my debt on this project
to the Inquiry-based learning community
and particularly to Prof Jacqueline Jensen-Vallin's
wonderful Proofs book.
Can you help with Introduction to Proofs?
If you are an instructor who adopts this book, I'd greatly appreciate an
email.
I'd be glad to hear your comments but I'd also just be glad to hear
about people using it.
(I save any comments,
especially bug reports.
and periodically revise.)
If you have some material that you are able to share back
then I'd be delighted to see it.
Of course, I reserve the ability to choose whether to include it.
I gratefully acknowledge
all the contributions that I include,
or I can keep you anonymous if you prefer.
My email is jhefferon at smcvt.edu.
Various lengths
Introduction to Proofs's LaTeX code allows it to
comes in three lengths.
I teach a four-credit course and so the longest version
has the right number of questions for a fifteen week semester,
with three classes per week, and four questions per class.
The shortest version omits some questions to end with
the right number for three questions per class.
The intermediate version is ... well, I think you know.
If you are not sure then take the longest one.
It is the canonical version of the text and is the one linked-to
above so you probably already did the download.
It has all the questions that the other two have, and in class omitting
is much easier than adding.
License
To bookstores: thank you for your concern about my rights.
I give instructors permission
to make copies of this material, either electronic or paper,
and give or sell those copies to students.
(Instructors may like to make an extra copy and prorate the price of student
copies so that their copy is paid for.)
If you have further questions, don't hesitate to contact me at the email
address on the top of this page.
If you want to modify the text:
feel free — that is one of my motivations for starting the
project.
If you are able to share back your modifications
then I'd be glad to see them.
If not that is fine.
However, I ask as a favor that you make clear which material
is yours and which is from the main version of the text.
When I get questions or bug reports
having to work out what is happening gets frustrating all
around unless authorship is clear.
In particular,
changing the cover to include a statement about your modifications
would help.
Something like this would be great: " \fbox{The material in the second appendix on logic is not from the main version of the text but has been added by Professor Jones of UBU. For this material contact \url{sjones@example.com}.}"
Related work
You may also be interested in my
Linear Algebra
text, also Freely available.
It is for a first US undergraduate course, covering the standard material
while focusing on helping students develop mathematical maturity.
It has been widely used for many years. |
Product description
Math in Focus has numerous instructional advantages that make it a successful program. Carefully paced instruction focuses on teaching fewer math topics per year to a level of mastery. Additionally, consistent use of visual models and manipulatives bridge the concrete and the abstract. Instruction is centered around problem solving, with multiple models that help students visualize and understand math concepts.
Student packs include Student Textbooks and Student Workbooks for both semesters as well as Assessments. Volume A is designed for the first half of the year, and Volume B is designed for the second half. Teacherís Editions contain complete program support, including Chapter Overviews with math background, cross-curricular connections, and a planning guide. Student Books allow for age-appropriate and mathematically sound practice, assessment, and development of problem-solving and thinking skills. Assessments provide a clear picture of student progress. Virtual manipulatives, which are available separately, provide and easy-to-use alternative to traditional manipulatives. (Math in Focus is also compatible with the physical manipulatives that are available for other math programs.)
Type: Paperback ()Category: > Home SchoolingISBN / UPC: 9780547549354/0547549350Publish Date: 6/1/2010Item No: 159446Vendor: Saxon Publishers |
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Starting at $52005Worksheets for Classroom or Lab Practice for Elementary and Intermediate Algebra
Summary
Elementary and Intermediate Algebra is a book for the student. The authors' goal is to help build students' confidence, their understanding and appreciation of math, and their basic skills by presenting an extremely user-friendly text that models a framework in which students can succeed. Unfortunately, students who place into developmental math courses often struggle with math anxiety due to bad experiences in past math courses. Developmental students often have never developed nor applied a study system in mathematics. To address these needs, the authors have framed three goals for Elementary and Intermediate Algebra: 1) reduce math anxiety, 2) teach for understanding, and 3) foster critical thinking and enthusiasm. The authors' writing style is extremely student friendly. They talk to students in their own language and walk them through the concepts, explaining not only how to do the math, but also why it works and where it comes from, rather than using the "monkey-see, monkey-do" approach that some books take.
Table of Contents
Preface
vi
To the Student
xiii
Learning Styles Inventory
xix
Foundations of Algebra
1
(102)
Number Sets and the Structure of Algebra
2
(11)
Fractions
13
(14)
Adding and Subtracting Real Numbers; Properties of Real Numbers
27
(16)
Multiplying and Dividing Real Numbers; Properties of Real Numbers
43
(16)
Exponents, Roots, and Order of Operations
59
(14)
Translating Word Phrases to Expressions
73
(8)
Evaluating and Rewriting Expressions
81
(22)
Summary
91
(6)
Review Exercises
97
(4)
Practice Test
101
(2)
Solving Linear Equations and Inequalities
103
(88)
Equations, Formulas, and the Problem-Solving Process
104
(15)
The Addition Principle
119
(14)
The Multiplication Principle
133
(14)
Applying the Principles to Formulas
147
(7)
Translating Word Sentences to Equations
154
(12)
Solving Linear Inequalities
166
(25)
Summary
180
(5)
Review Exercises
185
(4)
Practice Test
189
(2)
Problem Solving
191
(76)
Ratios and Proportions
192
(15)
Percents
207
(15)
Problems with Two or More Unknowns
222
(14)
Rates
236
(8)
Investment and Mixture
244
(23)
Summary
252
(5)
Review Exercises
257
(5)
Practice Test
262
(2)
Chapters 1--3 Cumulative Review Exercises
264
(3)
Graphing Linear Equations and Inequalities
267
(110)
The Rectangular Coordinate System
268
(10)
Graphing Linear Equations
278
(12)
Graphing Using Intercepts
290
(10)
Slope-Intercept Form
300
(16)
Point-Slope Form
316
(12)
Graphing Linear Inequalities
328
(11)
Introduction to Functions and Function Notation
339
(38)
Summary
359
(8)
Review Exercises
367
(7)
Practice Test
374
(3)
Polynomials
377
(86)
Exponents and Scientific Notation
378
(12)
Introduction to Polynomials
390
(12)
Adding and Subtracting Polynomials
402
(11)
Exponent Rules and Multiplying Monomials
413
(12)
Multiplying Polynomials; Special Products
425
(12)
Exponent Rules and Dividing Polynomials
437
(26)
Summary
454
(3)
Review Exercises
457
(4)
Practice Test
461
(2)
Factoring
463
(82)
Greatest Common Factor and Factoring by Grouping
464
(13)
Factoring Trinomials of the Form x2 + bx + c
477
(7)
Factoring Trinomials of the Form ax2 + bx + c, where a ≠ 1
484
(9)
Factoring Special Products
493
(9)
Strategies for Factoring
502
(6)
Solving Quadratic Equations by Factoring
508
(12)
Graphs of Quadratic Equations and Functions
520
(25)
Summary
532
(5)
Review Exercises
537
(3)
Practice Test
540
(1)
Chapters 1--6 Cumulative Review Exercises
541
(4)
Rational Expressions and Equations
545
(98)
Simplifying Rational Expressions
546
(15)
Multiplying and Dividing Rational Expressions
561
(11)
Adding and Subtracting Rational Expressions with the Same Denominator
572
(7)
Adding and Subtracting Rational Expressions with Different Denominators |
I am sorry - I so disagree with you at every point.
#1 I was very clear that I was using technology to do calculus without calculus
for specific purposes, e.g. find extreme values, find the area under a curve,
find the length of a curve, etc. in order to do real and interesting problems
from life.
#2 Any textbook that has the student use a graphics calculator to solve problems
by tracing out the extreme value (maximum) is doing this, i.e. using calculus
without knowing calculus and almost every textbook in the US does this. This
makes a great deal of interesting problems previously only accessible to
calculus students now available to every student. I am only extending this to
area under a curve and arc length (We do not have access to graphing calculators
where I live so we use freeware like geogebra.)
== So I would ask, are you using graphing calculators or software to graph
functions and find their extreme values? If yes, then you are doing calculus
without calculus. Even if you are just using technology to graph functions, you
are doing calculus without calculus. (In my day, we graphed functions by doing
all the calculus on them. There was no other way. Nowadays, we type the function
into a program and hit "graph". Calculus without calculus and thank
goodness!)
== My point was only that we should extend this process.
#3 Here is a wonderful arc length problem. How far does a projectile travel
(given h0, v0 and angle)? Currently textbooks only ask about its maximum height
and the horizontal distance traveled. Using Length[], you can find the actual
distance traveled and discuss things like average speed, etc. Why should we wait
for Calculus 2 and a very, very difficult solution that only a very, very few
will understand? This is fun! And useful and understandable.
#4 Proving Pythagoras' theorem is interesting. Knowing that its converse also
works is useful. However, knowing how to use the theorem is vital.
Proving the quadratic formula by completing the square is interesting.
Finding roots and the vertex of a quadratic function using completing the square
is a mathematical exercise - fun if you like to do that stuff. Factoring ditto.
However, knowing how to use the quadratic formula completely and understanding
that quadratics are symmetrical, understanding what roots are and not being
afraid of irrational roots and not trying to graph complex roots is vital.
#5 Your BTW is NOT a proof of the formula for the area of a circle in any way,
shape or form. It is an "illustration". It will only "become" a proof (and a
very difficult one to actually write down) if you let the number of pie pieces
-> infinity. Hmm. Think that is called calculus.
My real point is that it seems to me that mathematics is taught the way we
mathematicians think is interesting (I hold a doctorate in theoretical
mathematics). This serves only to propagate mathematicians, i.e. it fulfills the
needs of the very, very tiny group of students who will become mathematicians
and leaves the rest bewildered (and angry).
However, in my opinion the real tragedy is that the math we teach is almost
completely inapplicable to the STE of STEM. For example, sin²x+cos²x=1
(a) Have a student actually use this formula with x=45° to see whether he will
get 1 (technology - does he understand that sin²x=(sin(x))^2?). My college kids
certainly do not.
(b) Have a student use the formula to find the parametric equations for a
circle(engineering/physics - the formula is "bad" because it uses x as the
argument and "backwards" for applying to parametric equations since x "goes
with" cosine), i.e. the parametric form of the unit circle is x=cos(t),
y=sin(t).
So we are teaching mathematics for those students who will become
mathematicians. Doesn't seem to be a really good plan.
We are trying to teach children that need to drive a car safely and properly how
the engine works. Yes, you should be able to fill up the windshield water fluid.
Yes, should should be able to check your tire pressure. But you don't need to
know how the motor works. And for sure, it is more important to know to check
your mirrors before changing lanes (i.e. check your units match before
solving).
Rigor is important. Learning how to look at a problem is important. Logical
thinking is important. Checking units is important. Checking that the answer
makes sense is important. Thinking about how to use technology to check your
solution is important |
Cognition and Learning Background: Absolute Value
Methods of Teaching
The typical method of teaching absolute value involves memorization of the formal definition and application of the case-by-case method and the properties of inequalities in order to solve problems (McLauren, 1985, p. 91).
Typical instruction with absolute value begins with the definition used in conjunction with algebraic representation in simple equality equations (such as or ). While more textbooks are emphasizing conception of absolute value as distance and using number lines as a tool to understanding, the more common approach is procedural solution using algebra and representation of the answers algebraically and on a number line. The prolonged connection of absolute value with distance or number line representation tends to fall by the wayside in favor of procedural fluency as a goal.
California State Standards Involving Absolute Value
Grade Seven
2.5 - Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.
California State Standards Glossary Definition of Absolute Value
A number's distance from zero on the number line. The absolute value of -4 is 4; the absolute value of 4 is 4.
Cognitive Obstacles and Common Misconceptions
Basic misconceptions
Absolute value is "always positive" meaning that the solution of an absolute value problem can never be negative
Absolute value is "always positive" and so any solutions must be made positive
Absolute value problems only have one solution
Absolute value problems have two solutions which are always mirrors (i.e. 4 and -4)
Note that these misconceptions are related but distinct. There is a difference between a negative solution being impossible (i.e. not to be considered) and having only positive solutions, which may involve considering a negative solution but subsequently making it positive.
Cognitive Obstacles with the Definition of Absolute Value
The definition of absolute value is:
Sink (1979) found that students ignore the conditions (the if statements above). Students have been trained since elementary grades to treat the absolute value of a quantity as a nonnegative value. Students will often say that absolute value means "the answer is always positive". However, as they tend to ignore the condition statements and , students become confused with the second sentence of the definition. The idea that contradicts their previous understanding that the absolute value would be nonnegative.
Cognitive Obstacles with Inequalities
Absolute value is often found in conjunction with inequalities, leading to all of the cognitive obstacles that are associated with linear equality and inequality equations. Because absolute value in conjunction with inequalities typically leads to compound inequalities, the number of intertwining cognitive obstacles can quickly increase.
This section can be removed and expanded upon when the Inequalities page is developed.
Students have difficulty reading inequalities correctly
As with equalities, students are confused by variables on the right side of the inequality
is more cognitively demanding than
Compound inequalities necessitate the ability to use variables on the right sight of the inequality sign
Students do not understand that inequalities can not be read left to right and right to left in the same way
Students are used to equality equations, which can be read in any order
Students read as "6 is equal to x" and "x is equal to 6"
Students likewise read as "6 is greater than x" and "x is greater than 6" even though this is incorrect
Conjunction and disjunction of compound inequalities are not well understood
Compound inequalities can be written as one statement or as two statements joined by a keyword
The singular statement guarantees that one part of the inequality must be correctly read right to left (see above)
The joined statements require selection of the correct keyword ("and" or "or")
The keyword may or may not be present in the presentation of the problem
Sometimes it is given in the problem or implied by the section of the book being studied (see below)
Sometimes it must be inferred from the behavior or the graph
Students are encouraged to think of problems as "and" or "or" inequalities
Students do not often understand which is the correct keyword and why
Textbooks often divide these problems artificially into two sections
Students are led to believe they are separate kinds of problems
The cases of overlapping inequalities or all real number solutions are not always presented as possibilities
Students are trained to classify the problems but do not understand the underlying meaning or purpose of the classification
Compound inequalities can be represented with a number line
Early problems often require usage of the number line
Textbooks typically drop the requirement for the solution to be represented as a number line with the more complicated problems
Students are taught to observe the basic characteristics without understanding what they truly mean
Students have difficulty with the various kinds of notation
Inequality notation ()
Used to formulate problems and solutions
Compound inequalities can be represented with inequality notation, but have their own cognitive obstacles (see above)
Inequality notation requires the use of a variable
Interval notation ()is typically introduced earlier in the school year
Interval notation is used only for representing solutions to problems written in other notations
Interval notation does not make use of variables
Compound inequalities can be represented with interval notation, but require special symbols
The union and intersection symbols ( and ) are typically taught during set theory and hastily reviewed (and then forgotten) when teaching interval notation
The union and intersection symbols are another way of representing "and" and "or"
Set notation (
Used to formulate problems and solutions
Less frequently used to describe inequalities before Algebra II
Number Line Graph notation
Can be used as ways to understand and conceptualize problems
Used mainly to describe solutions; usage of the number line as a tool for solving the problem diminishes after early examples
Have two styles that relate to interval notation
Cognitive Obstacles with Absolute Value Inequalities
Students are instructed to solve problems using the case-by-case method
There is a "positive" case and a "negative" case
The "shortcut" for the negative case is to reverse the inequality sign and the sign of the constant on the side opposite the variable
The reason for this "shortcut" is often poorly explained
The quantity inside the absolute value sign can be positive or negative
If this quantity is negative, the resultant equation can be multiplied or divided on both sides by -1
Even if this process is explained, it is often not related back to distance or another way of making sense of the operation
Since students do not understand the procedure, they will often forget to reverse one or both signs
This illustrates a typical example of the case-by-case solution method. The procedure skips right to the "shortcut", where the inequality and the right hand side constant have their signs reversed without explanation. Since the coefficient of the variable is negative, the solution obtained by dividing both sides by -2 requires the inequality to be reversed again. Students will mistakenly think this has already been taken care of since they have already flipped the inequality sign once.
Textbooks sometimes instruct that answers to absolute value inequalities will be "or" if the original problem is > or and "and" if the original problem is < or
This is an additional procedural step to memorize or forget
Sometimes the mnemonics "greatOR than" or "less thAND" are used
This is only true for absolute value inequalities -- it does not transfer to compound inequalities as a whole
Students are often only given this procedural step and mnemonic and do not understand the underlying concepts
Furthermore, students will neglect to check that their answer makes sense and will default to the what the mnemonic or shortcut indicates
Dean (1985) pointed out that students will assume that the solution set always contains only alternating intervals. Multiplicity and the sign of factors can change the solution intervals even when the critical points remain the same.
Pedagogical Tools and Strategies
Ahuja - absolute value as distance
Ahuja (1976) suggests considering absolute value in terms of distance, making the definition of absolute value:
For real numbers and , consider as the distance between and , where and are the representations of the numbers and , respectively, on the real line.
Example: Solve
Since the distance of from 5 is 2, can be either 7 or 3.
Example: Solve
. This equation says that the distance of from -7 is 3. The answer would be either -4 or -10.
Arcidiacono (1983) recommends using piecewise functions to reinforce the algebraic and graphical representations of the absolute value function together from the beginning of instruction. His three stage approach to absolute value is:
Ballowe - use square roots to rewrite absolute value problems
Ballowe (1998) suggests using the definition to rewrite the problem without absolute value signs.
Example:
rewrite using the definition
square both sides
Brumfiel - teach all five definitions of absolute value
Brumfiel (1980) recommends teaching five definitions of absolute value and the investigation of one problem with all five definitions. The definitions are:
Let be any real number. On a coordinate line let be the point whose coordinate is . Then is the undirected distance between and the origin.
Let be any real number. Choose any numbers and so that . Let and be the points whose coordinates are and . Then is the undirected distance between the points and .
is the "larger" of the numbers , . We write this briefly as = Max {, }, that is, is the maximum element of the set that consists of and . Of course we agree that Max {,} = . This takes care of the case .
Note that Definition 2 is essentially the same as Ahuja's(1976) recommendation.
Note that Ballowe (1998) expanded upon the ideas in Definition 4
Note that in Definition 5, Brumfiel (1980) has ordered the definition such that the condition comes first in each statement, addressing Sink's (1979) concern that the condition would be ignored by the student.
McLauren - evaluate the critical points
McLauren (1985) recommends solving absolute value inequalities as equalities, plotting the results on a number line, listing the intervals defined by the critical points, and testing each interval to see if it is a solution. This technique can be extended to quadratic and rational inequalities.
Sink - rooting out definition misconceptions
The teacher must show with examples how substituting values according to the conditions causes the result to be nonnegative.
Example
Example
Stallings-Roberts - the Absolute Value Scale (AVS)
The Absolute Value Scale (AVS) (Stallings-Roberts, 1991) is a manipulative tool that students can create themselves that helps to visually associate absolute value problems with number lines and the concept of absolute value as distance.
Students can make the AVS from a sheet of ruled notebook paper, creased lengthwise, creating two one-inch by eleven-inch strips that are placed next to one another. The bottom strip represents the number line. The top strip represents the distance scale that corresponds to the number line.
AVS can be used to model algebraic problems
AVS can be used to understand how to rewrite some equalities and inequalities in terms of absolute value.
Wagster - solve by intervals
Wagster (1986) recommends a way similar to McLauren's (1985). First, the zeros of each expression in an absolute value inequality are marked on a number line. Then, in each interval on the number line, the equations are solved for that interval only. If the solution is within the interval, it is a solution of the inequality.
Curricula and Technological Resources
Technological Activities
Students can use graphing calculators to plot absolute value equations with two expressions. Students can examine equations with one, two, or no solutions and use the graphing calculator to find precise intersections.
The following links are tools to help visualize absolute value graphs, both on the number line and in the coordinate plane. A indicates the best tools for learning. Many of these are simple tools that allow users to see how a graph of
an absolute value equation behaves. The better tools show different aspects of absolute value or show how intersecting graphs
produce solutions to the equations being graphed.
Websites - Games
The following links are to games that utilize absolute value in some way. None of the games are involved or teach the concepts. They could, however, be used to reinforce lessons or provide extra practice with the basic concepts of absolute value. These are all drill-and-practice
type activities and don't explore the conceptual side of absolute value like the tools above. |
Please Note! The current edition of Saxon Math 8/7 is the 3rd Edition. This 1st edition is offered for families using older versions of Saxon.
Boost your students understanding of Saxon Math with DIVE's easy-to-understand lectures! Each lesson concept in Saxon's textbook is taught step-by-step on a digital whiteboard, averaging about 10-15 minutes in length; and because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching Topics covered in Math 8/7 include arithmetic calculation, measurements, basic geometry, fraction/decimal/percent conversions/unit multipliers, as well as pre-algebra exercises.
System Requirements:
Mac OS 10.3.9-10.4.x
Windows 98, 2000, ME, XP, Vista, 8
Quicktime Download Required.
Customer Reviews for Saxon Math 87, 1st Edition DIVE CD-Rom
This product has not yet been reviewed. Click here to continue to the product details page. |
Modify Your ResultsPrentice Hall Mathematics is designed to enable students tap into the power of mathematics.
The text will help them be successful on the tests they take in class and on high-stakes tests required by your state. The practice in each lesson will prepare them for the format as well as for the content of these tests.
The authors and consulting authors on Prentice Hall Math: Tools for Success team worked with Prentice Hall to develop an instructional approach that addresses the needs of middle grades students with a variety of ability levels and learning styles. Authors also prepared manuscripts for strands across three levels of Middle Grades Math. Consulting authors worked alongside authors throughout program planning and all stages of manuscript development offering advice and suggestions for improving |
...
Show More Spreadsheet Tools for Engineers provides beginning engineering students with a strong foundation in problem solving using Excel as the modern day equivalent of the slide rule. As part of McGraw-Hill's BEST series for freshman engineering curricula, this text is particularly geared toward introductory students. The author provides plenty of background information on technical terms, and numerous examples illustrating both traditional and spreadsheet solutions for a variety of engineering problems. The first three chapters introduce the basics of problem solving and Excel fundamentals. Beyond that, the chapters are largely independent of one another. Topics covered include graphing data, converting units, analyzing data, interpolation and curve fitting, solving equations, evaluating integrals, writing macros, and comparing economic alternatives |
What to teach? How to teach? How much to teach? Does everybody need higher math? Opposing theories and answers have led to what is now known as the "math war."
Nicholson Baker's "Wrong Answer: The Case Against Algebra II" appeared in the September 2013 issue of Harper's Magazine. Baker argues that higher math in the form of Algebra II is a major cause of school drop-outs; and that most people, for most professions, don't need it anyway. The article is free to subscribers at the Harper's Magazine website or can be purchased as a single issue here from Amazon.
What do you think? Wrong Answer challenges students respond to Baker's essay, defending a position about whether or not high schools and colleges should require students to study higher math. Included is a list of guide questions and suggestions.
From the New York Times, professor Andrew Hacker's Is Algebra Necessary? argues that for many of us, it isn't.
Paul Lockhart's A Mathematician's Lament (Bellevue Literary Press, 2009) is just 140 pages long but it's a powerful critique of math education as currently practiced. Lockhart writes, "Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions." Instead, math should involve exploration, imagination, and creativity. For teenagers and adults.
Mathematically Sane is a website devoted to promoting the "rational reform of mathematics education" – a topic about which there's a lot of controversy. The site has news, informational articles, research reports, relevant TED talks (including Conrad Wolfram on "Teaching Kids Real Math with Computers"), and more.
By John Allen Paulos, Innumeracy: Mathematical Illiteracy and Its Consequences (Hill and Wang, 2001) is invaluable. Paulos explains, fascinatingly, just why a basic grasp of math – notably a familiarity with probability and statistics – is essential for making reasonable decisions about the world we live in. By the same author, see A Mathematician Reads the Newspaper and Once Upon a Number. For teenagers and adults.
The Best Evidence Encyclopedia, created by the Johns Hopkins University School of Education, rates curricula and teaching approaches in math, reading, and science for elementary, middle, and high school students. Find out what the research says works. Useful for parents and educators. (Best approach for elementary-level math: peer tutoring.)
According to the What Works Clearinghouse (Institute of Education Sciences), here's what works in terms of math curricula. Saxon Math is way down the list. Top of the line is SRA Real Math (McGraw-Hill).
There are hundreds – thousands – of math books, math programs, and math curricula. See below for some of the best.
NOT JUST YOUR ORDINARY BOOKS ABOUT MATH this time plants both seeds, getting two plants and a harvest of four seeds. This time he eats one and plants three – and things rapidly multiply, becoming more and more complicated. For ages 4 and up and upEmily Gravett's The Rabbit Problem (Simon & Schuster, 2010) is a delightful month-by-month take on the Fibonacci series – which is named for the mathematician who first described it in the 13th century, while solving a problem about multiplying rabbits. First there's one lonely rabbit (an invitation stuck to the page reads "Join me"); subsequent months feature baby rabbit record books, rabbit newspapers, carrot recipes, and – by November – wildly overcrowded rabbits. For ages 6-11.
Theoni Pappas is the inventor of Penrose the Mathematical Cat, featured in The Adventures of Penrose the Mathematical Cat (World Wide Publishing/Tetra, 1997) and Fractals, Googols, and Other Mathematical Tales (1993). Each is a collection of mathematical stories in which Penrose explores pancake world, meets a fractal dragon and a Fibonacci rabbit, discovers the golden rectangle and the world of Tangrams, visits the planet Dodeka, and more. Friendly introductions to interesting math concepts for ages 7-11.
Author Cindy Neuschwander introduces kids to geometry through the adventures of gallant Sir Cumference, his wife, Lady Di of Ameter, their son, Prince Radius, and a cast of supporting characters. Titles in the series include Sir Cumference and the First Round Table (Charlesbridge, 1994), Sir Cumference and the Dragon of Pi (1999), and Sir Cumference and the Great Knight of Angleland (2001). For ages 8-12.
Cynthia Zaslavsky's Number Sense and Nonsense (Chicago Review Press, 2001) is subtitled "Building Math Creativity and Confidence Through Number Play" – which it attempts to do by encouraging kids to fool around with number games and puzzles. Chapter titles include "Odds and Evens," "Prime and Not Prime," "Zero – Is It Something? Is It Nothing?" "Money, Measures, and Other Matters," "Counting: Fingers, Words, Sticks, Strings, and Symbols," and "The Calculator and Number Sense." Figure out how many of what arrive over the Twelve Days of Christmas, solve the problem of the King's Chessboard, play a Liberian stone game, and much more. For ages 8 and up.
Johnny Ball's award-winning Go Figure! (Dorling Kindersley, 2005) is subtitled "a totally cool book about numbers," and it is just that. Illustrated with great color photos, charts, and diagrams, the book covers the origins of counting, "magic numbers" (such as Fibonacci numbers, the golden ratio, pi, and Pascal's triangle), geometry (including polyhedra, buckyballs, cones and curves, and symmetry), and "The World of Math" (including probability, chaos theory, and fractals). Challenging puzzles and questions and a great read for ages 8-12.
Simon Basher's Math: A Book You Can Count On (Kingfisher, 2010) – in classic snarky Basher fashion – personifies mathematical concepts as first-person entities, each with its own Japanese-style cartoon character. For example, here's Subtract: "People often think I'm gloomy. Okay, I admit it, I'm the exact opposite of Add, that bubbly ball of smirking positivity." Learn all about Zero, Line, Quadrilateral, Ratio, and X. And more. For ages 9-14.
Math Trek by mathematician Ivars Peterson and Nancy Henderson (John Wiley & Sons, 1999) is a terrific interactive math book in ten short chapters, organized as an "amusement park" of mathematical concepts. Entry into the park – Chapter 1 – is through the Knot Zone; to get in, you have to figure out which of the knots that locks the gate is NOT a KNOT. Kids then experiment with knots (and non-knots) by duplicating patterns with string, find out how to make a trefoil knot (the simplest of mathematical knots) and a Jacob's ladder knot (an impressive-looking non-knot) and learn a good deal about knot theory, its uses, and its history. At the Crazy Roller Coaster – it's a Mobius strip – kids make Mobius strip models, learn about topology, and see some interesting examples of topological artwork. In other chapters, they learn about fractals and make fractal snowflakes, experiment with "weird dice," build a chaos machine, learn to decode a binary secret message, and much more. Included are a glossary and a supplementary reading list. For ages 9 and up.
Glory St. John's How to Count Like a Martian (Random House, 1975) begins with mysterious beeps from Mars – which might just be numbers. The book then covers a range of number systems, among them those of the Egyptians, Babylonians, Mayans, Greeks, Chinese, and Hindus, plus abaci and computers. Out of print; check your local library. For ages 9-12.
Actress Danica McKellar is also a math whiz, and is now known not only for movies and TV, but for educational advocacy, especially when it comes to girls and math. Titles of her informative, friendly, and funny books include Math Doesn't Suck: How to Survive Middle School Math (Plume, 2008), Kiss My Math: Showing Pre-Algebra Who's Boss (Plume, 2009), Hot X: Algebra Exposed (Plume, 2011), and Girls Get Curves: Geometry Takes Shape (Hudson Street Press, 2012). Readers learn math using friendship bracelets, shoes, shopping, pizza, and cute boys. And anyway, who can resist chapter titles like "How to Entertain Yourself While Babysitting a Devil Child" and "Creative Uses for Bubblegum"? For ages 11 and up.
Clifford A. Pickover's The Math Book (Sterling, 2012) is a fascinating chronological history of mathematics "From Pythagoras to the 57th Dimension" in 250 double-page spreads, each illustrated with great color photographs. Actually it starts well before Pythagoras: the first entry, "Ant Odometer," is dated 150 million BC. Other entries include Zeno's Paradox, Archimedes's Spiral, Franklin's Magic Squares, Turing Machines, Rubik's Cube, and Fractals. Something for everybody.
Derrick Niederman's Number Freak (Perigee, 2009) runs from 1 to 200, listing interesting facts and background information about each number. For example, at 23, you find out about the birthday paradox; at 46, you learn that there are 46 peaks in the Adirondack Mountains and that a "46-er" is someone who has climbed them all; and at 85, you find that there are just 85 ways in which to knot a necktie. For teenagers and adults.
By Alexander Humez, Zero to Lazy Eight (Touchstone, 1994) is an information-packed collection of essays, variously on zero, the numbers 1 to 13, and infinity. Readers learn about everything from numerical word origins to the mathematics of ciphers, bell-ringing, and dice games. Find out why we say "three sheets to the wind" and "dressed to the nines." For teenagers and adults.
When it comes to communicating complex concepts, analogies are often the way to go, and Joel Levy's A Bee in a Cathedral and 99 Other Scientific Analogies (Firefly Books, 2011) is crammed with nothing but. The book is divided into seven graphically creative sections, variously covering physics, chemistry, biology, astronomy, earth science, the human body, and technology. For example, if an atom were the size of a cathedral, its nucleus would be the size of a bee. For teenagers and adults.
By author and mathematician Keith Devlin, Devlin's Angle is a collection of monthly columns written for the Mathematical Association of America on math in everyday life and math education. Check them out.
By autistic savant Daniel Tammet (author of Born on a Blue Day), Thinking in Numbers (Little, Brown, 2013) is a collection of 25 essays about seeing the world through numbers, with anecdotes and examples that range from haiku to chess, snowflakes, and Omar Khayyam's calendar. Recommended for both math-loving and totally math-phobic teenagers and adults.
Jennifer Ouellette's The Calculus Diaries (Penguin Books, 2010), subtitled "How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse," is a truly reader-friendly account of applying calculus to everyday life by a self-described math-phobic. Crammed with intriguing anecdotes and examples, from Disneyland's spinning tea cups to speedometers, the Black Death, tulipomania, and the housing bubble. For teenagers and adults.
Larry Gonick's 230+-page The Cartoon Guide to Calculus (William Morrow, 2011) covers all the basics with wonderful little cartoon illustrations and a sense of humor. Delightful, which is something I never thought I'd hear myself say about calculus. Chapter titles include "Speed, Velocity, Change," "Meet the Functions," "Limits," "The Derivative," and "Introducing Integration." For all students of calculus.
THE LURE OF REALLY BIG NUMBERS
Andrew Clements's picture-book A Million Dots (Simon & Schuster Children's Publishing, 2006) indeed contains one million dots, along with a lot of catchy factoids to help readers visualize enormous numerical quantities. Readers learn, for example, that there are 525,600 minutes from one birthday to the next and that when the cow jumped over the moon, she soared upward 238,857 miles. For ages 4-8.
In David Birch's picture book The King's Chessboard (Puffin, 1993), rice is used to teach simultaneous lessons in morals and the mathematics of big numbers. A proud and pushy king insists on giving his counselor, who doesn't want it, a reward; the pestered counselor finally asks for a grain of rice, the amount to be doubled each day for as many days as there are squares (64) on the king's chessboard. The king thinks this is a fine joke and sends one grain, then two, then four – but as the days pass and the doubling continues, soon amounting to humongous quantities of rice, he realizes that he has made a fatal mistake.
In Demi's version of the story, One Grain of Rice: A Mathematical Folktale (Scholastic, 1997), gorgeously illustrated with touches of gold, young Rani outsmarts a selfish raja and saves her hungry village with her rice-and-chessboard request. Here the rice is delivered by animals: birds, leopards, tigers, a goat pulling a cart, and – impressively, on day 30 – a fold-out page of 256 rice-toting elephants.
Big Numbers by Mary and John Gribbin (Wizard Books, 2005), subtitled "A Mind-Expanding Trip to Infinity and Back," traces the history of big numbers and shows how big numbers are used in a range of scientific disciplines, such as astronomy, biology, and geology. For ages 9-12.
The Megapenny Project demonstrates big numbers with stacks of pennies, from a piddling pile of sixteen to a foot-square cube of 50,000 to a towering structure of a million.
Powers of Ten: About the Relative Size of Things in the Universe is a film by Charles and Ray Eames in which viewers journey from the outer limits of the universe to the subatomic quark in 42 ten-fold steps. It's a wonderful progression in color photographs, beginning with two picnickers in a park and moving outward through city, continent, planet, solar system, and galaxy; then inward through skin, cells, DNA, atoms, and subatomic particles. Fascinating for all ages.
A Question of Scale is a clickable illustrated tour of the universe ("from quarks to quasars") in powers of ten.
Cosmic View, based on Kees Boeke's classic 1957 books, travels to the ends of the universe and to the innards of the atom, beginning with a little girl with a cat on her lap. Includes detailed explanations.
The View From the Back of the Envelope is a creative and multifaceted website on big numbers, featuring – among much else – a page displaying a million dots; a big-number Pinocchio estimation game; a guide for scaling the universe to a desktop; explanations of exponential notation; a list of "Powers of Ten" scales; and a demonstration of the scope of big numbers using grains of salt.
The Stan's Café Theatre Company's installation exhibit Of All the People in All the World uses piles of rice to represent a host of human statistics. One person is represented by one grain of rice; the entire population of the world – that is, some six and a half billion grains of rice – by a 104-ton rice mountain. Other piles of rice variously represent the population of the United States, the number of Americans who are millionaires, the number of people worldwide who play the computer game "World of Warcraft," the number of people killed in the Holocaust, the number of people in an average year who go on a pilgrimage to Mecca. (And much more.) A fascinating exercise in statistics, a startling social commentary, and a powerful demonstration of big (and small) numbers.
Though you can't count to infinity, you can learn about it: Welcome to the Hotel Infinity includes a kid-friendly illustrated explanation of the infinity concept along with a clever short-story-cum-puzzle on a hotel with an infinite number of guests and rooms.
From UC Berkeley's Lawrence Hall of Science, Family Math, by Jean Stenmark, Virginia Thompson, and Ruth Cossey (Equals Series), promotes math as an enriching whole-family activity. This 300+-page information and activity collection promotes understanding of basic arithmetic, logical thinking, probability and statistics, geometry, measurement, and calculator math. The book also contains reproducible game boards, hundred charts, graph paper, and a fill-in-the-blank calendar. Great for a range of ages.
Also see the sequel, Family Math II, for ages 5-12.
In the same series, FamilyFamily Math: The Middle School Years (Lawrence Hall of Science, 1998) concentrates on activities intended to inculcate algebraic thinking and number sense. Kids explore simultaneous equations with a game of Flowerpots; study area and perimeter with pentominoes and polyominoes; learn a series of clever tricks for quick mental arithmetic; study fraction/decimal equivalents with a game of Towers; tackle greatest common divisors with the Game of Euclid; and fool around with fraction calculators. Game boards and patterns are included in the text; there is also a list of additional family math resources and a description of the math concepts ordinarily covered in the middle school. For ages 10-14.
By James Overholt and Laurie Kincheloe, Math Wise! (Jossey-Bass, 2010) is a collection of over 100 hands-on activities designed to promote "real math understanding." For example, kids make toothpick storybooks and everyday things number books, experiment with paper plate fractions, and make flexagons, sugar-cube buildings, and paper airplanes. For ages 5-13.
Marilyn Burns's dynamic duo, The I Hate Mathematics Book (Little, Brown, 1975) and Math for Smarty Pants (Little, Brown, 1982) are wonderful 120+-page illustrated collections of math puzzles, games, and experiments designed to show kids that math – rather than a series of rote exercises – is an inventive way of thinking. Determine how close you can get to a pigeon, take a shoelace survey, make a topological map of your house, make sidewalk chalk shapes that can be drawn without lifting the chalk from the sidewalk or retracing any line. Highly recommended for ages 8 and up.
Ann McCallum's The Secret Life of Math (Williamson Books, 2005) is an interactive history of numbers from prehistoric times to the present, illustrated with photographs of artifacts, puzzle and fact boxes, and timelines. In Part I, which describes mankind's first forays into counting, kids make a tally stick with a chicken leg bone, learn how to count like a Zulu or a Roman, hold a native American nature count, and make an Inca quipu. In Part II, which covers the history of numerical symbols, kids make a cuneiform birthday tablet, learn to count in Egyptian hieroglyphs, and learn about zero and Fibonacci numbers. Part III leaps from counting to calculation: kids become "algorithm detectives," tackle lattice multiplication puzzles, and make an abacus and a set of Chinese counting rods. Excellent for ages 9-12.
Amazing Math by Laszlo C. Bardos (Nomad Press, 2010) in the Build It Yourself Series is arranged in four sections – Numbers & Counting; Angles, Curves, and Paths; Shapes; and Patterns – each of which features hands-on projects with instructions and templates, activities, interesting information in text and sidebars, and new word definitions in boxes. Readers learn, for example, about Fibonacci rabbits, four-color maps, and Koch snowflakes, and discover that a potato chip is in the shape of a hyperbolic paraboloid. The projects are cool. For ages 9 and up.
Claudia Zaslavsky's Math Games and Activities from Around the World (Chicago Review Press, 1998) is a 160-page collection of multicultural math games, puzzles, and projects arranged by game category. Chapters include "Three-In-a-Row Games," "Games of Chance," "Puzzles with Numbers," "Geometry All Around Us," and "Repeating Patterns." Kids can play 9 Men's Morris or Mankala, experiment with hexagrams and Magic Squares, make Pennsylvania Dutch love patterns and Japanese Mon-Kiri cut-outs, and much more. Included for each game or project are background information, instructions, and "Things to Think About." For ages 9 and up.
Mark Wahl's A Mathematical Mystery Tour (Prufrock Press, 2008) is an interactive exploration of numbers in nature and art. For example, discover Fibonacci numbers in pinecones, daisies, and pineapples; learn about spiral galaxies and Plato's polyhedra; and study geometry while building a model of the Great Pyramid. For ages 11 and up.
Hands-On Equations is an algebra program that uses fat red and green number cubes (representing positive and negative numbers), colored pawns (positive and negative unknowns), and a balance scale (printed and laminated) to teach kids how to set up and solve algebraic equations. Fun and clever for ages 8 and up.
TOPScience sells multi-lesson modules of coordinated hands-on learning activities for grades 3-10 – and these are extraordinarily clever in that they do a lot with truly simply materials such as pennies, tape, clothespins, and paper clips. Click on "Math and Measurement" for math-oriented units for a range of ages. Highly recommended.
From UC Berkeley's Lawrence Hall of Science, the GEMS (Great Explorations in Math and Science) Teachers' Guides use integrated activities to teach science and math topics. Sample titles include Frog Math, Early Adventures in Algebra, In All Probability, Math Around the World, and Math on the Menu.
Hands-On Math Activities is a collection of printable games and projects, categorized under Numbers and Operations, Geometry, Problem Solving, Data Management and Analysis, and Measurement. For example, kids make and play pentominoes, experiment with geoboard sheets, build a Lego graph, and make and compare the capacities of paper cylinders. For ages 8-12.
Hands On Math is a helpful blog devoted to creative ideas for teaching math. Lots of interesting approaches and activities for a range of ages.
From the Ohio Resources for Early Childhood, For Mathematics Educators is a list of helpful projects and resources including lists of Common Core math standards for grades K-12, large collections of "inquiry-oriented" math problems for a wide range of ages, and an annotated Mathematics Bookshelf.
MathFour is a website devoted to creative approaches to teaching math. For example, kids can make Fibonacci Valentines, whip up a batch of mathematical eggnog, and research invented numbers (eleventeen?).
From Annenberg Learner, Math in Daily Life is an interesting interactive tutorial covering Playing to Win, Savings and Credit, Population Growth, Home Decorating, Cooking by Numbers, and The Universal Language. Also at the Annenberg website check out the extensive list of great math lesson plans.
Patterns in Nature is a collection of cool interactive applets demonstrating math concepts. For example, find out how to compute pi by throwing darts at a dartboard and discover what ants in an anthill have to do with molecular motion.
MATH TOOLS
The National Library of Virtual Manipulatives has a huge list of creative applets for preK-12, categorized under Numbers & Operations, Algebra, Geometry, Measurement, and Data Analysis & Probability. Anything you could possibly want, from pattern blocks and geoboards to fractal generators.
At Mathwire's Math Manipulatives, make your own dominoes and hundred boards. Included are instructions for games, activities, and literature connections.
Learning Resources: Math is a good commercial source for math tools and manipulatives, such as base-ten and pattern blocks, Cuisenaire rods, geoboards, dice and spinners, and fraction games.
From the National Council of Teachers of Mathematics (NCTM), Core Math Tools is a downloadable collection of software tools for high-school-level students for problem-solving in the areas of Algebra and Functions, Geometry and Trigonometry, and Statistics and Probability.
Johnnie's Math Page has an extensive collection of interactive online tools, games, and manipulatives, categorized under Number, Geometry, Multiplication, Fractions, Statistics, Probability, and Measurement. There's also a category called Fun, where visitors can play Alien Addition, tackle the Towers of Hanoi puzzle, and experiment with origami.
From Math is Fun, Math Tools and Calculators has many online calculators with which visitors can calculate percentages, convert units, create graphs, experiment with polyhedra, solve quadratic equations, change fractions to decimals, and more.
Wolfram Alpha aims to collect all objective data and to implement every known method to "compute whatever can be computed about anything." Want to know how much paint it would take to cover the moon? Wolfram Alpha can tell you. A spectacular math tool.
MATH LESSONS
Miquon Math is a curriculum for grades 1-3 in six color-coded workbooks, developed in the 1960s by Lore Rasmussen of Pennsylvania's Miquon School. These are designed to be used with Cuisenaire rods and stress investigation, problem-solving skills, and creativity rather than rote drill.
By creative math educator Marilyn Burns, Writing in Math Class (Math Solutions, 1996) has many examples of how writing helps kids of all ages learn math. Many suggestions, among them keeping math journals, writing math autobiographies, and combining math with creative writing. Resources for ages 7 and up.
JUMP Math was designed by mathematician John Mighton (who almost flunked calculus in college), author of The Myth of Ability: Nurturing Mathematical Talent in Every Child (Walker & Company, 2004). JUMP, which stands for Junior Undiscovered Math Prodigies, is a comprehensive program that integrates games, puzzles, magic tricks, hands-on activities, and extensions. Check out the free samples at the website. For grades K-8.
Singapore Math, a comprehensive curriculum for grades K-12, progresses from the concrete to the pictorial to the abstract – that is, it emphasizes translating problems into concrete and/or visual images to help younger learners understand concepts. Excellent reviews.
Stanley Schmidt's Life of Fred series offers a complete math curriculum from soup to nuts – or rather, from simple addition to Calculus, Statistics, and Linear Algebra. The books are set up in chapters, each telling a story about Fred, who teaches at KITTENS University. At the end of each story, kids grab a pencil and tackle a number of questions and challenges related to the story. Frequently these involve additional interesting tidbits and facts. The Fred approach is intended to be multifaceted and thought-provoking – the opposite, in other words, of the drill-and-ill approach so often found in school workbooks. Lightly disguised traditional math.
By Asa Kleiman, David Washington, and Marya Washington Tyler, It's Alive! Math That Makes You Squirm (Prufrock Press, 1996) – written by a pair of young computer geeks and a math teacher – is a hoot, crammed with zany problems based on the kinds of quirky facts and gicky trivia that kids adore. For example, readers calculate the number of earthworms in a football field, the probability of being eaten by a salt-water crocodile, the amount of liquid in a giant squid eyeball, the travel rate of eyelash mites, and the storage capacity (in megabytes) of the human brain. There's a helpful answer key at the back of the book. For ages 9-13.
A sequel, It's Alive and Kicking (Prufrock Press, 1996) – subtitled "Math the Way It Ought to Be – Tough, Fun, and a Little Weird" – continues in the same vein, with problems based on sweat glands, rat litters, cow manure, and the number of rivets holding up the Eiffel Tower.
By Harold R. Jacobs, Mathematics: A Human Endeavor (W.H. Freeman, 1994) is a ray of light in the grim gray field of textbooks. Math, Jacobs-style, is taught through puzzles, games, experiments, and enthralling real-life examples. Chapter 1, "Mathematical Ways of Thinking," for example, plunges students into experiments with the behavior of billiard balls, the notorious four-color map problem, and the invention of the Soma cube puzzle. In later sections, readers learn about number sequences with the hexagrams of I Ching and Francis Bacon's 17th-century diplomatic cipher; are introduced to coordinate graphing with the leaping speed of kangaroos; and learn about logarithms with the electromagnetic spectrum, the frets on a guitar, and the Richter scale. This is real math, and it's great. Highly recommended for ages 13 and up.
The Interactive Mathematics Program (IMP) is an integrated four-year program, intended to replace the traditional math sequence in which kids progress from Algebra I to Geometry, then to Algebra II/ Trigonometry and Pre-calculus. Instead the IMP series teaches algebra, geometry, trigonometry, statistics, and probability in combination, through active investigation of "open-ended situations" – that is, problems without pre-programmed simple answers. In lieu of rote exercises, kids are encouraged to experiment and explore – often with manipulatives, graphing calculators, and computers. For high-school-level students.
A derivative of IMP called Meaningful Math follows a more traditional format and employs graphing calculators.
The Great Courses are a wide range of classes, variously available on video, DVD, audio CD, or audiocassette, for high-school- and college-level students. Among these is The Joy of Thinking (subtitled "The Beauty and Power of Classical Mathematical Ideas"), a 24-lesson lecture series, jointly taught by professors Edward Burger of Williams College and Michael Starbird of the University of Texas at Austin, whose stated goal is to both introduce some of the truly creative and intriguing ideas behind mathematics and to show students how to develop effective thinking strategies. The result is a wide-ranging discussion of counting, geometry, and probability, using clear and easy-to-follow presentations and lots of catchy examples. There are forays, for example, into Fermat's Last Theorem, Fibonacci numbers in pineapples, Mobius bands and Klein bottles, Turing machines and Dragon Curves, coin-flipping, coincidences, and the question of whether monkeys, randomly typing, could eventually produce Hamlet.
Suggested readings for The Joy of Thinking are taken from Burger and Starbird's The Heart of Mathematics: An Invitation to Effective Thinking (Key College Publishing, 2000), a very readable and attractively designed text reminiscent of Harold Jacobs's Mathematics: A Human Endeavor (W.H. Freeman, 1994). (See above.)
Annenberg Learner has some terrific resources for math, among them video courses (many available online for free), lesson plans, and interactives. Video courses include Against All Odds: Inside Statistics, Algebra: In Simplest Terms, and numerous workshops for educators on creative techniques for teaching math. Also at the site are extensive lists of categorized lesson plans (K-2, 3-5, 6-8, 9-12, and college) and a lot of great interactives. For example, kids can experiment with a balance scale, generate graphs, build a number line, manipulate congruent shapes, explore rotational symmetry, and much more.
MathBits has a wealth of resources for math students, including tutorials on Java and C++ programming, projects and worksheets for the Geometer's Sketchpad, instructions for finding your way around a graphing calculator, downloadable graph paper (31 kinds), and many Math Caching games at a range of levels, in which kids must solve problems and submit answers in order to discover the next Internet "box."
Mathcats is a multifaceted site that lets visitors experiment and explore. Try to solve a logic problem involving Sailor Cat, a goat, a wolf, and a cabbage; find out how old you are in seconds; play with architecture blocks; and use the Math Cats Balance to balance everything from electrons to galaxies. There are also dozen of interactive projects (for example, generate fractal snowflakes and geometric spider webs) and math-based crafts. Aimed at open-ended inquiry learning.
Doodling in Math Class is the creation of Vi Hart, mathemusician and employee of Khan Academy. These are a terrific, fun, and irreverent collection of math-and-drawing exercises on such topics as spirals, fractals, Fibonacci numbers, and Sierpinski triangles. I love these. Check them out.
From the Math Forum, Suzanne's Mathematics Lessons are categorized under Numbers & Operations, Algebra, Data Analysis & Probability, Measurement, and Geometry. Most involve hands-on projects and experiments; also included are lists of helpful and creative Internet resources. Primarily for grade levels 6-8.
IXL Math has a complete list of all the (hundreds of) skills required by the public schools at each grade level, with online practice problems and printable worksheets for each.
SOSMath is an online workbook with examples and practice problems for high school and college students, variously covering Algebra, Trigonometry, Calculus, Differential Equations, Complex Variables, and Matrix Algebra.
Free Math – which is free – has detailed lists of all the skills required in public-school math classes, categorized by grade, with associated practice exercises.
For online math classes for grades 3 and up, see Khan Academy. Khan Academy is a non-profit educational website created by Salman Khan (graduate of MIT and Harvard Business School) with the mission of providing a free, world-class education online to anyone, anywhere, anytime. Zillions of exercises, mini-lectures, and tutorials.
EdX provides free online courses from such colleges and universities as Harvard, MIT, and Stanford in a wide range of disciplines (among them, math).
GRAPHS
What do the numbers show? The Real World Data Series from the Heinemann/Raintree Library is a collection of 32-page books that use real-world data – organized in charts, tables, and graphs – to introduce kids to current world issues. Titles include Graphing Food and Nutrition (Isabel Thomas, 2008), Graphing Crime (Barbara Somervill, 2010), Graphing Natural Disasters (Barbara Somervill, 2010), Graphing Water (Sarah Medina, 2008), and Graphing Sports (Casey Rand, 2010). For the complete list, see Real World Data. For ages 8-12.
Edward Tufte's stunning The Visual Display of Quantitative Information (Graphics Press, 2001) – despite its not-very-exciting title – is a classic on the art of presenting mathematical data in graphs, charts, and tables. The book is packed with terrific historical and modern illustrations, demonstrating the best (and worst) in graphics. For teenagers and adults.
At Kids' Zone's Create a Graph, visitors can select from five different types of graphs (line, bar, area, pie, and XY), enter data, label, preview, and print.
Graphing Activities is a collection of 18 projects targeted at elementary-level kids. For example, kids determine preferred car colors by counting cars in a parking lot and graphing the results, or research the most common size of a family or the most disliked vegetable.
Building Brilliant Bar Graphs has several projects in which kids collect data and make bar graphs, among them a pretzel taste test and a pet survey. Included at the site are printable worksheets. For ages 6-9.
eNASCO is a commercial source for activity books and games involving graphing. For example, kids learn coordinate graphing by making picture graphs or geometry quilts.
Also see Lakeshore Learning for commercial hands-on graphing materials. For example, make bar graphs with tiny colored cars.
Teachnology's Graphing Lesson Plans has a long list of activities and projects, plus printable graph paper and graphing worksheets. Sample lessons include All About Me Graphing, the Drawing Bugs Game, and Graphing Equations. For a range of ages.
Carolyn's Unit on Graphing has clear explanations of line graphs, bar graphs, scatter plots, and pie charts, with illustrations and examples.
From the Biology Corner, Measuring Lung Capacity is a hands-on science experiment that involves data collection and graphing. (You'll need a ruler and a round balloon.) Included at the site are worksheets, instructions, and sample data.
PROBABILITY AND STATISTICS
In Edward Einhorn's A Very Improbable Story (Charlesbridge, 2008), Ethan wakes up one morning with a talking cat on his head – who absolutely refuses to move until Ethan wins a game of probability. Ethan then struggles with challenges involving socks, coins, cereal shapes, and marbles, gradually learning how best to judge odds and predict outcomes. (The cat's name, incidentally, is Odds.) For ages 7-10.
By Sheila Dolgowich and colleagues, Chances Are (Libraries Unlimited, 1995) is a 125-page collection of hands-on activities in probability and statistics. For ages 8-13.
Darrell Huff's 144-page How to Lie With Statistics (W.W. Norton, 1993) is a funny, friendly, and informative overview of statistics and the way in which – if we're not on the ball – they can fool us into drawing the wrong conclusions. Learn all about sampling and bias, deceptive averages, "gee-whiz" graphs, and more. Illustrated with vintage-style cartoons. For teenagers and up.
Charles Wheelan's Naked Statistics (W.W. Norton, 2014), subtitled "Stripping the Dread from Data," is an overview of what makes numbers meaningful, dealing – in reader-friendly fashion – with such questions as "How does Netflix know what movies you like?" "What's a batting average?" and "How useful is a GPA?" Various chapters cover correlation, basic probability, the importance of data, the Central Limit Theorem, polling, and regression analysis. For teenagers and adults.
From the New Jersey Mathematics Curriculum Framework, Probability and Statistics is a detailed and useful overview of what kids should know and do at each grade level (K-12), with suggestions for activities and resources.
TeacherVision's Resources for Teachers has a selection of printables and lesson plans for probability and statistics studies. Lesson plan titles include Heads or Tails: Penny Math; Using Scatterplots; Range, Median, and Mode; Baseball Fun; and U.S. Immigration. For ages 7-12.
The BBC's Handling Data has videos, written tutorials, and quizzes on frequency diagrams, mode, median, mean, and range, and probability.
From Cut the Knot, Probability Problems has a detailed tutorial with definitions, explanations, and a long list of challenging problems. For older students.
From the extensive Core Knowledge website, Probability and Statistics is a multi-lesson study with recommended resources and instructions for ages 9-13. Included is a project to collect and analyze data on the "Top Movies of All Time."
From Annenberg Learner, Against All Odds: Inside Statistics, consists of 32 video modules plus coordinated guides. Available online or on DVD. For high-school-level students and up.
SUPER-GOOD GAMES
Set, "the family game of visual perception," is a diabolically clever exercise in mathematical thinking. It's a (deceptively) simple card game, consisting of 81 cards, each printed with one of three basic shapes: a diamond, a lozenge, or a fat squiggle. On each card, the shapes appear in different numbers, colors, and shadings. To play, the dealer lays out 12 cards, face up, and all players attempt to identify three cards that make a set: that is, three cards in which each feature (shape, number, color, shading) is either exactly the same or completely different. When you've managed to do so, you yell "Set!" and remove those three cards from the board; the dealer then adds three new cards and the set-search begins again. Everybody plays at once, which means that nobody has a chance to get bored, and the game is considerably more challenging than it first appears. It is appropriate for persons aged 5 through adult, and adults – believe me – have no advantages over younger players.
Chess might be the ideal teaching tool. It's all about strategy and patterns, lines and angles, spatial analyses, weighing options and making decisions. Research shows it boosts academic achievement, but it's also challenging and fun. Also Harry Potter played it.
Doubtless one reason that it's so successful is that it's self-empowering – players figure a lot of it out on their own – and it provides a range of intellectual benefits without overtly trying to do so. Recommended age for introducing chess to kids is around 8 or 9, but there are no hard and fast rules.
ChessKid has a tutorial on playing chess targeted at kids; young players can also sign up (safely) to play with others online.
Sudoku puzzles are applied logic puzzles, played on a 9×9 grid, with nothing more than a pencil (eraser also highly recommended) and brains. The puzzle grid is subdivided into nine 3×3 blocks or regions; the trick is to enter the numbers 1 through nine (with no repetition) in each horizontal row, vertical column, and block. ("Sudoku" or "su doku" means "numbers singly" in Japanese.) In each puzzle, a few number clues are present on the grid – these cannot be changed and players must work with and around them while solving the puzzle. Sudoku puzzles range in difficulty from the easy to the fiendish; and all are excellent and mind-expanding exercises in the art of logical thinking. (This isn't arithmetic. It's more like chess.)
There are many books of sudoku available, including some specifically for children – see, for example, Alastair Chisholm's The Kids' Book of Sudoku 1 (Simon & Schuster, 2005).
Web Sudoku offers zillions of puzzles – variously classified as easy, medium, hard, and evil – that can be printed or played online. Also see Gamehouse Sudoku, which has online puzzles at five levels of difficulty.
Math Playground has dozens of online math games, variously involving numbers, logic, math manipulatives, and word problems, along with interactive projects, worksheets and flashcards, and more. Click on "Common Core Math" to find grade-by-grade games and challenges aligned to the Common Core.
PBSKids' selection of Math Games includes dozens, among them Juggling George, Send in the Trolls, Star Swiper, Vegetable Planting, the Great Shape Race, and many more. Experiment.
MATH AND FICTION
A collaboration of author Jon Scieszka and artist Lane Smith, Math Curse (Viking, 1995) is clever, funny, and thought-provoking. The curse – laid on the hapless narrator by her math teacher, Mrs. Fibonacci – causes her to think of everything (everything, from getting dressed in the morning to lunchtime pizza to birthday cupcakes) as a math problem. For ages 7 and up.
In Pam Calvert's The Multiplying Menace: The Revenge of Rumpelstiltskin (Charlesbridge, 2006), the crown prince Peter has turned ten and Rumpelstiltskin is back, demanding payment for all that straw he spun into gold. Furthermore, he's armed with a multiplying stick that he uses to awful effect, making things disappear (by multiplying them by fractions) or making them awkwardly big (say, by multiplying noses by six). Luckily Peter solves the problem with a clever math trick. Also see the sequel, The Multiplying Menace Divides. For ages 7-10.
In Edward Eager's Half Magic (Houghton Mifflin Harcourt, 1999), Jane finds a magic talisman that grants just half of every wish. She and her siblings – Mark, Katherine, and Martha – find that this makes for some complications. A great read for ages 8-12.
In Norton Juster's The Phantom Tollbooth (Bulleseye Books, 1988), Milo passes through the Phantom Tollbooth and ends up in a magical country where he sets out on a quest to find the sisters Rhyme and Reason, thus restoring peace to the warring kingdoms of Dictionopolis and Digitopolis. A wonderful cast of characters and a lot of brilliant play on words and numbers. A must-read for ages 8-12.
In The Number Devil by Hans Magnus Enzensberger (Metropolitan Books, 1997), Robert, twelve, loathes Mr. Bockel, his math teacher, who refuses to let him use his calculator and who afflicts him with word problems, such as: "If 2 pretzel makers can make 444 pretzels in 6 hours, how long does it take 5 pretzel makers to make 88 pretzels?" ("How dumb can you get?" said Robert.) Then one night Robert falls asleep and meets the Number Devil, a little bright red man the size of a grasshopper, dressed in knickers and carrying a silver-knobbed walking stick. The Devil, who has his own calculator (it's slimy and green), introduces Robert – night by night – to the many fascinations of mathematics. Among these are the concept of infinity, "prima donna" numbers (those uppity primes that can only be divided by themselves and 1), repeating fractions, square roots, triangular numbers, Fibonacci numbers (and rabbits), factorials, topology, irrational numbers, and more. Humor, memory-sticking mathematical information, and a lot of terrific color illustrations for ages 10 and up.
Twelve-year-old Willow Chance of Holly Goldberg Sloan's Counting by 7s (Dial, 2013) is a scientific genius who loves gardens, books, and the number 7, but doesn't have much luck with her peers. Then her adoptive parents are killed in a car crash and she's left completely on her own – except for new friends Mai and Quang-ha, who live with their mother, Pattie, who has a manicure business, in a garage; her disturbed school counselor Dell Duke, and Jairo Hernandez, a Mexican taxi driver. A great story, interspersed with counting by sevens, for ages 10 and up.
By the fictious Malba Tahan, The Man Who Counted (W.W. Norton & Company, 1993) is the Arabian-Nights-style tale of Beremiz Samir – a.k.a. the Man Who Counted – first encountered sitting on a rock by the side of the road, calling out mysterious and enormous numbers. The book, which purports to be Samir's life story, is actually a series of puzzles: in one story, for example, Samir has to help three quarreling brothers settle their inheritance (35 camels, of which their father has left half to the oldest son, 1/3 to the middle son, and 1/9 to the youngest). In another, he has to determine the eye color of veiled concubines (the blue-eyed ones always lie and the brown-eyed ones always tell the truth). For ages 10 and up.
In Wendy Lichtman's Do the Math: Secrets, Lies, and Algebra (Greenwillow Books, 2008), eighth-grader Tess sees the world in terms of math – in this case including tangles with friends, a school cheating scandal, and a mysterious death. Chapter titles are all math terms, such as "Inequalities," "Graphs," "Tangents," and "The Quadratic Equation." For ages 10-14.
Edwin Abbot's classic Flatland (Dover Publications, 1992), originally written in 1884, is a clever satire set in a two-dimensional world, where the women are lines and the men, polygons. The narrator, a Square, then meets a Sphere and discovers the third dimension. Not only math, but a critique of rigid Victorian society. For teenagers and adults.
At Project Gutenberg, the complete text of Flatland is available online.
For the bookish mathematician, Clifton Fadiman's Fantasia Mathematica (Copernicus, 1997) is a collection of stories, poems, and excerpts all drawn from the "universe of mathematics." Included, for example, are Robert Heinlein's sci-fi short story "And He Built a Crooked House," Martin Gardner's "The Island of Five Colors," George Gamow's "An Infinity of Guests," and poems by Vachel Lindsay, Edna St. Vincent Millay, and Carl Sandburg. For teenagers and adults.
Mathematical Fiction is a long (over a thousand entries) list of books and stories incorporating math and/or mathematicians. For each title, there's a synopsis, examples of math features, and a list of similar titles.
Math Playground's Pattern Blocks has online blocks in an array of geometric shapes with which kids can build patterns of all kinds.
From Mathcats, at the Polygon Playground, visitors can make patterns, tessellations, symmetrical designs, and pictures with a range of colorful geometric shapes in various sizes.
Math and the Art of M.C. Escher is an interactive online book on the mathematics of Escher's work with associated student art projects. Topics covered include symmetry, frieze patterns, tessellations, polygons, fractals, and knot theory. For teenagers and up.
In Amy Axelrod's Pigs in the Pantry (Aladdin, 1999), Mrs. Pig is sick in bed, so Mr. Pig and kids decide to make her a tempting pot of Firehouse Chili. Unfortunately measuring mistakes lead to disasters, including the arrival of real firefighters. Included is a recipe so you can see where Mr. Pig went so wrong. For ages 4-8.
Deborah Hopkinson's picture book Fannie in the Kitchen (Aladdin, 2004) – subtitled "The Whole Story from Soup to Nuts of How Fannie Farmer Invented Recipes with Precise Measurements" – is told from the point of view of young Marcia Shaw, who is not exactly pleased when Fannie Farmer comes to cook for her family's Victorian household. Soon, though, she's hooked on Fannie's delicious meals and even has a hand in writing the famous Boston Cooking-School Cook Book. For ages 5-9.
Joan D'Amico and Karen Eich Drummond's The Math Chef (John Wiley & Sons, 1997) teaches math through applesauce, waffles, homemade animal crackers, and banana muffins. The book is divided into four main parts, each devoted to a different math concept: Measuring, Arithmetic, Fractions and Percents, and Geometry. For example, kids learn how to figure out how many grams are in a pound of potatoes, how to triple a sandwich recipe, and how to calculate the area of a brownie, the diameter of a cupcake, and the circumference of a pie. For ages 9-12.
From PBS, Math and Science Gumbo, hosted by the Kitchen Mathematician, uses food and cooking to teach math and science. Math concepts include unit pricing, fractions, estimation, units of measure, and so on. Episodes (among them "Grocery Shop," "Bake Shop," and "Pizza Shop") are available online.
See COOKING for many more books, projects, and resources for curious cooks.
MATH IN THE MOVIES and on TV
Donald in Mathmagic Land (1959) is a clever 27-minute animated film on math in real life – in music, in nature, in games like chess and baseball, and in architecture and art. Nominated for an Oscar.
Simon Singh's The Simpsons and Their Mathematical Secrets (Bloomsbury USA, 2013) shows how the popular (and hilarious) animated series "The Simpsons" is simply loaded with math. Singh uses the episodes as jumping-off points to discuss everything from calculus to baseball statistics. A fun mathematical read for teenagers and adults.
Simpson's Math covers the math in the Simpson's episodes, with episode-by-episode descriptions and associated problems and worksheets.
The TV series, Numb3rs – which ran for six seasons, 2005-2010 – features a pair of crime-fighting brothers in Los Angeles, one an FBI agent, the other a mathematical genius. An exciting pitch for math.
From Cornell University, Numb3rs Math Activities has background info, materials, and projects based on each episode of the series. For advanced math students.
From Wolfram Research, The Math Behind Numb3rs has episode-by-episode descriptions with links to descriptions and explanations of specific math features in each.
Dimensions is a gorgeous film in nine 13-minute "chapters," beginning with Hipparchus, stereographic projections, and maps of the world and proceeding through M.C. Escher, four-dimensional polytopes, complex numbers, "fibration," and mathematical proofs. Free download. For teenagers and adults.
Math in the Movies, aimed at seventh-graders, is subtitled "Motivating Students with the Silver Screen." Included at the site are a list of movies and sample suggestions for associated math projects.
From MathBits, Math in the Movies has a long list of movies that in some way feature math, with summaries and printable worksheets to accompany each. Categorized by grade level (for the math, not the movie). Most worksheets are targeted at middle- and high-school-level students. Among the movies: Alice in Wonderland, Contact, October Sky, and Proof.
The Math in the Movies Page is an opinionated guide to movies (and plays) "with scenes of real mathematics," with brief reviews and ratings both for math presentation and overall performance. A Beautiful Mind (2001), for example, starring Russell Crowe as brilliant mathematician John Nash, gets 3 stars for Math and five stars for Film; Good Will Hunting (1997), the story of a young math genius from South Boston (Matt Damon) and a helpful psychologist (Robin Williams), scores one star for Math and three for Film.
The Mathematical Movie Database is a long (long) alphabetized list of math-containing movies. Included is a separate much shorter list of "must-see" math movies.
A must-read for the mathematically frustrated, Carl Sandburg's poem Arithmetic begins "Arithmetic is where numbers fly like pigeons in and out of your head."
Selected by Lee Bennett Hopkins, Marvelous Math (Simon & Schuster, 2001) is an illustrated collection of poems about math by a range of poets – among them "Counting Birds" by Felice Holman, "Pythagoras" by Madeleine Comora, and "Nature Knows Its Math" by Joan Bransfield Graham. For ages 5-8.
Theoni Pappas's Math Talk (Wide World Publishing, 1993) is a collection of 25 mathematical poems for two voices, covering everything from circles, fractals, and zero to Mobius strips, tessellations, googols, and infinity. For ages 7 and upFor much more math poetry (or science poetry, history poetry, and geography poetry), see POETRY II.
FAMOUS MATHEMATICIANS
Deborah Heiligman's The Boy Who Loved Math (Roaring Brook Press, 2013) is a delightful picture-book biography of Hungarian mathematician Paul Erdos who loved numbers from the time he was a toddler. (Tell him your birthday and he could tell you how many seconds you'd been alive.) For ages 4-8.
Jennifer Berne's On a Beam of Light (Chronicle Books, 2013) is a picture-book biography of Albert Einstein, charmingly illustrated in pen-and-ink and watercolor. Kids learn about Einstein's early fascination with a compass ("Suddenly he knew there were mysteries in the world…") and how – one day while riding his bicycle – he wondered what it would be like to ride on a beam of light. Eventually he grew up to theorize about atoms, mass, and energy, and to devise his famous Theory of Relativity. For ages 6-9.
Joseph D'Agnese's Blockhead (Henry Holt and Company, 2010) is a charmingly illustrated picture-book biography of Leonardo Fibonacci – the daydreaming medieval "blockhead" (and famous mathematician) whose astute observations of numbers in nature led to the discovery of the "Fibonacci series." Pictures show Fibonacci happily counting pomegranate and sunflower seeds, flower petals, and seashell chambers; text includes a beautifully clear description of his signature number pattern. For ages 6-10.
For more on the Fibonacci sequence for the same age group, see Sarah Campbell's Growing Patterns (Boyds Mill Press, 2010), illustrated with gorgeous (and countable) color photographs; and Ann McCallum's Rabbits, Rabbits Everywhere (Charlesbridge Publishing, 2007), a tale of a wizard, the Pied Piper, a lot of rabbits, and a clever little girl named Amanda. Also see Emily Gravett's The Rabbit Problem (above). (For Fibonacci rabbit lesson plans, see
Julie Glass's A Fly on the Ceiling (Random House, 1998) is a Step-Into-Reading book about French mathematician Rene Descartes and his discovery of the Cartesian system of coordinates. For ages 7-9.
By Julie Ellis, What's Your Angle, Pythagoras? (Charlesbridge, 2004) is a fictionalized picture-book account of the famous Greek mathematician. Here Pythagoras, a curious young boy, travels to Egypt with his father, learns about right triangles, and comes up with the Pythagorean theorem. For ages 7-10
This YouTube video shows how to make a rope triangle of the sort used to solve problems in What's Your Angle, Pythagoras?
Also see the sequel, Pythagoras and the Ratios (Charlesbridge, 2010), in which Pythagoras and his cousins want to win a music contest, but their pipes and lyres sound awful. Pythagoras saves the day by elucidating the mathematical ratio that creates harmony. For ages 7-10.
By Luetta Reimer and Wilbert Reimer, Mathematicians Are People Too! (Dale Seymour Publications, 1994) is a collection of short friendly biographical stories about fifteen famous mathematicians, among them Thales ("Pyramids, Olives, and Donkeys"), Archimedes ("The Man Who Concentrated Too Hard"), Blaise Pascal ("Count on Pascal"), Sophie Germain ("Mathematics at Midnight"), and Srinivasa Ramanujan ("Numbers Were His Greatest Treasure"). For ages 7-12.
The Ohio Resource Center's Mathematics Bookshelf has chapter-by-chapter suggestions and printable worksheets to accompany both volumes of Mathematicians Are People Too!
MATH AND SPORTS
Titles in Ian F. Mahaney's Sports Math series (PowerKids Press, 2011) include The Math of Baseball, The Math of Basketball, The Math of Soccer, The Math of Football, and The Math of Hockey. Each has an overview of the featured sport, measurements of the relevant playing field or court, and information on scoring or statistics. "Figure It Out" sidebars challenge readers to solve problems. Illustrated with photos, charts, and diagrams. For ages 7-12.
The Math & Movement program, developed by math educator Suzy Koontz, is tailor-made for non-sitters. Koontz describes the program as a "kinesthetic multisensory" approach to math that involves physical exercise (jumping, hopping, bending), dance, and yoga, plus an array of "visually pleasing floor mats" to teach and reinforce basic math concepts. Kids dance, wiggle, and leap their way through counting, skip counting, addition and subtraction facts, the multiplication tables, positive and negative numbers, and more. The Math & Movement Training Manual, which describes the program in detail, is available in paperback or eBook formats; the floor mats – clearly intended for schools – are pricey, but creative families can get around that. There's always sidewalk chalk, paint, and duct tape.
How many? How big? How far? How long? And when should kids know what? From PBS, the Child Development Tracker has descriptions of what kids generally know and do, year by year, from ages 1 to 9, in the fields of Creative Arts, Language, Literacy, Mathematics, Physical Health, Science, and Social and Emotional Growth.
There are – literally – hundreds of books aimed at introducing just-beginners to numbers; check out some good resources below.
There are several picture-book versions of the loved-by-everybody song/nursery rhyme "Ten in the Bed:" "There were 10 in the bed and the little one said/"Roll over! Roll over!'/So they all rolled over and 1 fell out…" David Ellwand's Ten in the Bed (Chronicle Books, 2001) is illustrated with enchanting photographs of ten teddy bears (including one in a striped night cap and one in wire-rimmed spectacles). For ages 1-4.
In Donald Crews's rhyming Ten Black Dots (Greenwillow, 1994), various numbers of black dots (from 1 to 10) can be anything from a sun and a moon, to the eyes of a fox, the face of a snowman, or beads "for stringing on a lace." Illustrated with big bright graphics for ages 1-5.
Math Literature Connections: Number Sense has activities and downloadable cards, worksheets and charts to accompany Donald Crews's Ten Black Dots, Theo LeSieg's Ten Apples Up on Top, and Jerrie Oughton's How the Stars Fell Into the Sky.
Lois Ehlert's Fish Eyes ("A Book You Can Count On") (Houghton Mifflin Harcourt, 1992), illustrated with gorgeous bright-colored fish, makes for a great interactive read, with many fish and fish eyes to count, plus shapes and colors to identify. For ages 2-5.
By Mitsumasa Anno, Anno's Counting Book (Crowell, 1997) is an enchanting picture book that teaches the numbers 0 to 12 as a small village grows through the months of the year. The book opens with an empty snow scene (0); by 1, we have one house, one snowy pine tree, one bridge over the river, one snowman, and one skier; by 7, there are seven buildings, seven pine trees, seven spotted cows, a clothesline hung with seven sheets and, in the sky, a seven-colored rainbow. Delightful for ages 2-6.
In Rick Walton's rhyming So Many Bunnies (HarperFestival, 2000) – an ABC and counting book – Old Mother Rabbit, who lives in a shoe, is putting her 26 alphabetical offspring to bed, counting them one by one, from (1) Abel (who sleeps on a table) to (26) Zed, who sleeps in a shed. For ages 2-6.
By Maurice Sendak, One Was Johnny (HarperCollins, 1991) begins with Johnny, who lives alone, happily reading by himself. Then a rat leaps in, followed by a cat, a dog, a turtle, and so on until an annoyed Johnny cleverly counts backwards, getting rid of his uninvited guests and restoring peace and quiet. For ages 2-7.
For dinosaur lovers, How Do Dinosaurs Count to Ten? by Jane Yolen and Marc Teague (Blue Sky Press, 2004) features enormous dinosaurs perched on kid-sized beds and playing with kid-sized toys. Readers count to 10 beginning with 1 tattered teddy bear. One of a series for ages 3-5.
Cynthia Cotton's At the Edge of the Woods (Henry Holt and Company, 2002) is a rhyming counting book of woodland animals, beginning with "At the edge of the woods, the grass grows tall/The daisies dance and the blackbirds call/One chipmunk lives in the old stone wall/At the edge of the deep, dark woods." An evocative numerical read for ages 3-6.
In Louise Yates's Dog Loves Counting (Knopf Books for Young Readers, 2013), Dog has tried counting sheep, but still can't get to sleep – so off he goes to find other animals to count. He begins with one baby dodo, and together the two of them set off in search of number three – a three-toed sloth, followed by a four-legged camel, a five-lined skink, and so on up to ten. At the end of the book, all ten animals end up counting stars. Other books featuring Dog include Dog Loves Books (2010) and Dog Loves Drawing (2012). For ages 3-6.
Richard Scarry's Best Counting Book Ever (Sterling, 2010) counts by ones to twenty, then by tens to one hundred – all with Scarry's busy little pictures in which there's a lot to study and count. For ages 3-6.
Paul Giganti's How Many Snails? (Greenwillow, 1994) is a clever counting book that introduces kids to the idea of sets and subsets. (How many clouds? How many clouds are big and fluffy? How many clouds are big and fluffy and gray?) The School Library Journal trashed it for ambiguity (What constitutes a truck? Will kids know that fire trucks are trucks?) – but I think that's a plus. Discuss and debate. That's what books are for. For ages 3-7.
In Stioshi Kitamura's When Sheep Cannot Sleep (Square Fish, 1988), Woolly, a pop-eyed little sheep in blue-and-white striped pajamas, can't get to sleep – so off he goes for a walk, counting along the way, from one butterfly to two ladybugs, three owls, and four bats, up to 20 stars. Back in bed again, he thinks about his family – 21 relatives, all sheep – and so, finally, counting sheep, he falls asleep. Great watercolor illustrations. For ages 3-7.
We're all primates! Anthony Browne's One Gorilla (Candlewick, 2013) is a counting book of primates, from 1 gorilla to 2 orangutans, 3 chimpanzees, and so on, through gibbons, macaques, and mandrills to 10 ring-tailed lemurs. The book ends with 20 portraits of people ("All primates/All one family"). Illustrated with wonderful detailed paintings. For ages 3-7.
Alison Jay's 1 2 3 (Dutton Juvenile Books, 2007) is a charmer, beginning with one sleeping little girl who is carried away on the back of a (golden-egg-laying) goose to an enchanting fairy-tale world, populated with three pigs, four frog princes, seven magic beans, and so on, up to ten and back again. Each wonderful illustration is filled with numbers and references to fairy tales. (Figure out which one.) For ages 4-7.
In Philemon Sturges's Ten Flashing Fireflies (NorthSouth, 1997), a pair of children capture – one by one – ten fireflies in a jar, and then, as the lights begin to blink out, let them go (and glow) again, counting back down from 10 to 1. The illustrations are soft summer night scenes in pastels, with luminous balls of glowing fireflies. For ages 4-8.
Firefly Activities include making a wax-paper-winged fireflies, ice-cream-spoon fireflies, and a firefly keepsake jar. (Count them!)
Alice Melvin's Counting Birds (Tate, 2010), written in rhyming couplets, counts birds (1-20) over the course of a day, beginning at dawn with one cockerel, then two love birds in a cage, then three ducks. Readers learn 21 different birds (the book ends at evening, with one nocturnal barn owl.) For ages 4-8.
By April Pulley Sayre, One is a Snail, Ten is a Crab (Candlewick, 2008) is a counting book of feet, beginning with the one-footed snail – then 2 (people), 4 (dog), 6 (insect), 8 (spider), and 10 (crab). Odd numbers are represented by an even-footed animal plus one snail. The numbers 10 to 100 are then represented by various combinations of animals – 80, for example, can be eight crabs or ten spiders. Cheerful cartoon illustrations. For ages 4-8.
Rosemary Wells's Emily's First 100 Days of School (Disney-Hyperion, 2005) covers the numbers 1 to 100, with Emily's daily number journal. Crammed with creative number ideas. (Make a number journal of your own!) Great project possibilities for ages 4 and up.
Lola M. Schaeffer's Lifetime: The Amazing Numbers in Animal Lives (Chronicle Books, 2013), is a mix of biology and math, as kids learn numbers and cool animal facts from 1 to (with skips) 1000. For example, in a single lifetime, a spider will spin one egg sac, a caribou will shed ten sets of antlers, a woodpecker will drill 30 nesting holes in trees, a rattlesnake will add 40 beads to its rattle, and a pair of seahorses will produce 1000 baby seahorses. For ages 4-8.
In Loreen Leedy's Missing Math (Two Lions, 2008), all the numbers in town have simply disappeared – leaving behind a mess: clocks and calendars don't work, money has no value, sports competitions and elections can't be resolved, and nobody knows how old or tall they are. The culprit is finally caught: a number thief with a powerful vacuum, trying to make a number large enough to reach infinity. For ages 4-8.
Christina Dobson's Pizza Counting (Charlesbridge, 2003) covers counting, addition, large numbers, and fractions, all through the medium of creative and yummy-looking pizzas. Pizza toppings not only demonstrate the numbers 1-20, but are combined to make pictures, such as a pizza face, a pizza cat, a pizza clock. A pizza tricked out with 100 topping pieces is duplicated 10 times (to demonstrate 1000) and then 100 times (10,000); millions and biliions are discussed in terms of numbers of pizzas necessary to circle the globe or reach to the moon. Try pairing this one with making your own numerical paper or baked-in-the-oven pizzas. For ages 5-8.
In David Birch's The King's Chessboard (Puffin, 1993), the king insists on giving his wise counselor a reward. Finally the counselor asks for a single grain of rice, the quantity to be doubled each day for as many days as there are squares on the king's chessboard. The king soon realizes that he has made a dreadful mathematical mistake. For ages 6-10.
Demi's One Grain of Rice (Scholastic, 1997) is a gorgeously illustrated version of the same tale, set in India; Helena Clare Pittman's A Grain of Rice (Yearling, 1995) is a Chinese version of the story, in which a mathematically clever farmer's son wins the hand of a princess.
Andrew Clements's picture-book A Million Dots (Simon & Schuster Children's Publishing, 2006) contains one million dots, along with a lot of catchy factoids to help readers visualize crucial numerical quantities along the way. Kids learn, for example, that there are 525,600 minutes from one birthday to the next and that when the cow jumped over the moon, she soared upward 238,857 miles. For ages 4-8.
In David Schwartz's How Much Is a Million? (HarperCollins, 2004), kids learn about millions, billions, and trillions, with the help of Marvelosissimo the Mathematical Magician and a lot of clever analogies. Readers discover, for example, that it would take 23 days to count to a million, that a goldfish bowl big enough for a million goldfish could hold a blue whale, and that a stack of a million kids, standing on each other's shoulders, would reach all the way to the moon. For ages 4-8.
In David Schwartz's On Beyond a Million (Dragonfly Books, 2001), Professor X and Dog Y (both in sweater vests) show kids how to count exponentially (by powers of ten). The book is appealingly designed, with conversation in cartoon bubbles and a lot of fascinating "Did you know?" side bars filled with numerical facts. For example, readers learn that one colony of weaver ants contains 500,000 ants, that there are 40,000 characters in Chinese, and that Americans eat 500,000,000 pounds of popcorn each year. Readers learn about the enormous googol (a 1 with a hundred zeroes after it) and the even more enormous googolplex (a googol raised to the power of a googol). However, they find that it's impossible to count to infinity, and the book ends with: "No matter what number you have, there is always one bigger."For ages 5-8.
Robert E. Wells's Is A Blue Whale the Biggest Thing There Is? (Albert Whitman & Company, 1993), for ages 6-9, is a cleverly illustrated exercise in big numbers and relative sizes: For example, it takes about 12 minutes to count to a thousand, but a good three weeks to count to a million, and a lifetime to count to a billion; and yes, a blue whale is big, but it's tiny in comparison to massive Mount Everest, which is tiny in comparison to planet Earth, which is dwarfed by the Sun, which is puny compared to the red supergiant Antares. For ages 6-11In Kate Hosford's Infinity and Me (Carolrhoda Books, 2012), young Uma – gazing at the star-filled night sky – grapples with the difficult-to-grasp concept of infinity. Family and friends all offer different takes on infinity, and eventually Uma comes to terms with it, realizing that her love for her grandma is "as big as infinity." With gorgeous illustrations by Gabi Swiatkowska. For ages 5-8.
MATH SERIES BOOKS
The Math Counts Series (Children's Press) by Henry Pluckrose is a collection of 32-page books, each with a simple text and illustrated with attractive color photos, introducing a range of math topics. Titles include Numbers, Counting, Sorting, Shape, Patterns, Size, Length, Capacity, and Weight. For ages 3-6.
Brian Cleary's Math is Categorical series (Lerner Publishing) includes such titles as The Action of Subtraction, The Mission of Addition, and Windows, Rings, and Grapes – a Look at Different Shapes. (See complete list at the website.) All are simple introductions to math concepts, with friendly examples, a rhyming text, and a lot of bright zany animal illustrations. For ages 4 and up.
Stuart J. Murphy's extensive MathStarts series is categorized by age group: Level 1 (ages 3 and up), Level 2 (ages 6 and up), and Level 3 (ages 7 and up). See the website for the complete list, with descriptions of math concepts covered.
The Math Matters series (Kane Press) by various authors is a series of picture-book stories, each related to a specific math concept and variously targeted at ages 5-7 or 6-8. For example, Gail Herman's Bad Luck Brad covers probability; Jennifer Dussling's Fair is Fair introduces readers to bar graphs; and Linda Williams Aber's Grandma's Button Box is all about sorting. See the complete list of titles at the website.
The Mouse Math series (Kane Press), variously by Eleanor May, Daphne Skinner, and Laura Driscoll, are picture-book introductions to simple math concepts for preschoolers, starring a pair of adorable mice, Albert and his big sister Wanda. Albert Is Not Scared, for example, covers direction words; Albert's Amazing Snail emphasizes position words; and Albert the Muffin-Maker introduces ordinal numbers. See all the titles at the website. Cute and funny.
MATH CONCEPTS
Suzanne Aker's What Comes in 2s, 3s, and 4s (Aladdin, 1992) in a picture-book introduction to sets – starting with your own two eyes, two ears, two arms, and two legs. For ages 2-5.
In Margarette S. Reid's The Button Box (Puffin, 1995), a little boy gets out his grandmother's enormous button box and begins to play, sorting the buttons into rows and piles – all the flower-painted china ones, all the sparkly jewel-like ones, and so on. There's not much to it, but it would be great paired with an actual button box. (Got one?) For ages 3-6.
Eve Merriam's 12 Ways to Get to 11 (Aladdin, 1996) is a clever twist on the counting book, showing 12 different combinations of things that all add up to 11: 9 pine cones and 2 acorns, for example; or 4 flags + 5 rabbits + 1 pitcher of water + 1 bouquet of flowers, all pulled from a magician's hat. For ages 3-7.
In Kathryn Cristaldi's Even Steven and Odd Todd (Cartwheel, 1996), Todd is definitely odd, in that he insists everything come out even, from his breakfast pancakes to the fish in his goldfish bowl. Then cousin Odd Todd arrives, who prefers his numbers odd. Eventually all works out – and the book ends with a handful of questions and simple activities on even and odd numbers. For ages 4-8.
Michael Dahl's Eggs and Legs (Nonfiction Picture Books, 2005) is a clever exercise in learning to count by twos, as a hen watches pairs of legs emerge from hatching eggs. Also see Dahl's Lots of Ladybugs: Counting by Fives and Toasty Toes: Counting by Tens. For ages 4-8.
In Lily Toy Hong's Chinese folktale Two of Everything (Albert Whitman & Company, 1993), Mr. Haktak unearths an ancient pot in the garden that turns out, miraculously, to double anything placed inside it. He and Mrs. Haktak happily double their money (again and again), but then Mrs. Haktak herself falls into the pot. And doubles. For ages 4-8.
In Stuart J. Murphy's Double the Ducks (HarperCollins, 2002), a pint-sized cowboy is caring for his flock of five ducks. Then each duck brings home a friend, which means twice as much food, twice as much bedding, and twice as much work. For ages 4-8.
Cynthia DeFelice's One Potato, Two Potato (Farrar, Straus & Giroux, 2006) is an Irish version of the doubling story, in which Mr. and Mrs. O'Grady are so ragged and poor that they have only one of everything – one potato for dinner, one blanket on their bed, one chair to sit in, and one winter coat. Until, that is, Mr. O'Grady finds a magic pot, that doubles everything put inside. For ages 4-8.
In Pat Hutchins's The Doorbell Rang (Greenwillow, 1989), Sam and Victoria have just divided one dozen of their mother's freshly baked cookies, when the doorbell starts ringing and more and more friends arrive. With each new guest, the dozen cookies must be divided all over again. An exercise in beginning division (and sharing) for ages 4-8.
Paul Giganti's Each Orange Had Eight Slices (Greenwillow, 1999) is a simple picture-book introduction to counting, addition and, by extension, multiplication. ("On my way to the zoo I saw 3 waddling ducks. Each duck had 4 baby ducks trailing behing, Each duck said, "QUACK, QUACK, QUACK." So: how many ducks, how many baby ducks, how many quacks? For ages 4-8.
In Elinor J. Pinczes's One Hundred Hungry Ants (Houghton Mifflin Harcourt, 1999), a tale of division, one hundred ants are headed toward a picnic when they are halted by one mathematically minded ant, who suggests that they will get food more efficiently if they split up into ranks. Obediently the ants rearrange themselves in groups of 50, 25, 10, and so on – only to discover by the time they've finished that the picnickers have packed up and left. For ages 4-8.
Also by Pinczes is A Remainder of One (Houghton Mifflin Harcourt, 2002), in which the ants struggle to form even ranks to march in the big parade. For ages 4-8.
Margaret Mahy's rhyming 17 Kings and 42 Elephants (Dial, 1987) features a royal procession through the jungle in which 17 kings and 42 elephants meet a tongue-twisting array of animals. A fun romp with potential for problem-solving. (How to divide 42 elephants among 17 kings?) For ages 4-8. so plants both seeds, getting two plants and a harvest of four seeds. This time he eats one and plants three – and things rapidly multiply, becoming more and more complicated. For ages 4-8.
Laura Overdeck's Bedtime Math (Falwel & Friends, 2013) is a cute idea – if bedtime stories, why not bedtime math? Each chapter starts by with a kid-friendly topic – Lego bricks, dog-walking, cookies, sticky ketchup bottles – and then goes on to pose three math problems at increasing levels of difficulty. The reviews have been very positive. I, however, was disappointed – there's not much in the way of math-interesting detail in the lead-ins, and the problems, though catchily worded, are workbook-type arithmetic problems. ("If you squirt 2 cups of ketchup and each cup used 14 tomatoes, how many tomatoes' worth of ketchup did you just squirt?") For ages 3-7.
Greg Tang is a master of math riddles, and his books – written in catchy rhyme – encourage kids to identify patterns and combinations and to devise effective problem-solving strategies. Titles include The Grapes of Math (Scholastic, 2004), Math for All Seasons, and Math Potatoes. For ages 4-8.
See Greg Tang Math for online versions of the books and many brain-boosting math games and puzzles-8.
Amanda Bean, main character of Cindy Neuschwander's Amanda Bean's Amazing Dream (Scholastic, 1998), loves to count, but she's not at all interested in learning her multiplication facts. Until, that is, she has a dream in which eight sheep on bicycles each buy five balls of yarn, and the resultant counting confusion reveals the usefulness of learning how to multiply. The book's cartoon-style illustrations are crammed with things to count (and multiply), from lollipops to windowpanes to puffy bushes in the park. For ages 5-8.
In Marilyn Burns's The Greedy Triangle (Scholastic, 1998), the greedy triangle wants more than just three sides and three angles. With the help of the local shapeshifter, he acquires more and more, becoming in turn a quadrilateral, pentagon, hexagon, heptagon, and octagon before finally deciding that life as a triangle was really the best of all. For ages 4-8.
In Ann Tompert's Grandfather Tang's Story (Dragonfly, 1997), a Chinese grandfather tells his little granddaughter a story about a pair of magical shape-changing foxes, illustrating the story with geometrical tangram puzzle pieces. The book includes a reproducible tangram template for making a set of your own. For ages 4-8.
From ABCYa.com. Tangrams for Kids has tangram puzzles to solve online. Click and drag to rearrange the shapes. Also see Tangram Game from PBS Kids.
From the Museum of Play, Tangrams has both an online game and a set of colorful printable tangrams.
In Duncan Birmingham's Look Twice (Tarquin, 1993), readers use an enclosed mirror card to turn a pair of identical objects into a pair of opposites. A fun study in symmetry for ages 4-8.
Also see Birmingham's M is for Mirror (Tarquin, 1988).
Bruce Goldstone's That's a Possibility! (Henry Holt and Company, 2013) is an introduction to probability, using an interactive question-and-answer format and bright color photographs to discuss concepts of possible, probable, improbable, and certain. For example, a teddy bear has ten shirts and ten pairs of pants, which combine to make 100 different outfits – so it's unlikely (100 to 1) that anyone can correctly guess what outfit he's going to wear. For ages 5-9.
In Lauren Leedy's The Great Graph Contest (Holiday House, 2006), Chester (a snail) is monitoring a contest between friends Beezy (a lizard) and Gonk (a toad) over who can make the best graph. In the process, the friend explore data collection processes and many different kinds of graphs, among them bar graphs, pie graphs, pictographs, and Venn diagrams. For ages 5-8.
Ann Whitehead Nagda's Polar Bear Math (Square Fish, 2007) is a real-life exercise in fractions based on data from two polar bear cubs born at the Denver Zoo. Each double-page spread includes a page of data – how to mix polar-bear formula, for example – while the facing page tells the story of the bears, illustrated with photographs. For ages 6-9.
Also by Ann Whitehead Nagda, in Cheetah Math (Henry Holt and Company, 2007) kids learn division with real-life data from a pair of cheetah cubs; Tiger Math (Square Fish, 2002) in which kids learn to graph by tracking the growth of a tiger cub; and Chimp Math (2002), in which readers learn to keep time records.
By Yelena McManaman and Maria Droujkova, Moebius Noodles (Delta Stream Media, 2014) – subtitled "Adventurous Math for the Playground Crowd" – is a 80+-page collection of games and investigations for kids, plus helpful hints for parents hoping to provide a mind-expanding math environment. The book is divided into four sections: Symmetry, Quantity, Function, and Grid. Kids learn real math terms – say, transitive property – through play. Delightful, substantive, and sensible. For ages 1 and up.
The Mother Goose Programs, developed by the Vermont Center for the Book, pair math- and science-related pictures book with open-ended investigative experiments and hands-on activities. Excellent for ages 3-5.
Associated with the Mother Goose Programs is the What's the BIG Idea? workbook series, a collection of six creatively interactive books designed to get kids excited about and involved in science and math. The books – crammed with hands-on activities and games – are illustrated with a mix of big bright-colored drawings and photo collage, and each comes with a companion CD featuring an appropriately themed picture book, printable activity cards and manipulatives, and a resource list. The books also include complete parent/teacher instructions, lots of extension suggestions, and an answer key. Titles are Counting (with Rick Walton's How Many, How Many, How Many), Measuring (with Susan Hightower's Twelve Snails to One Lizard), Shapes (with Dayle Ann Dodds's The Shape of Things), Patterns (with Trudy Harris's Pattern Fish), Sorting (with W. Nikola-Lisa's Bein' with You This Way), and Maps (with Pat Hutchins's Rosie's Walk).
FamilyMargaret McNamara's How Many Seeds in a Pumpkin? (Schwartz and Wade, 2007) turns into a mathematical guessing game as the kids in Mr. Tiffin's class try to figure out how many seeds are in large, small, and middle-sized pumpkins (A million? 500? 22?) Finally they cut the pumpkins open, scoop out the seeds, and count them, which is (1) messy and (2) the most straightforward way to find out. For ages 4-8.
There are dozens of sources for commercial math manipulatives and hands-on kits. A good starting point is Learning Resources, which sells dozens, including plastic counters, pattern blocks, tangrams, magnetic numbers, base-ten blocks, balances, and more.
In Yuyi Morales's Mexican-themed Just a Minute (Chronicle Books, 2003), a skeleton arrives at Grandma Beetle's door, demanding that she "come along." Grandma, however, cleverly puts him off with a series of (countable) chores: she has one house to sweep, two pots of tea to brew, three pounds of corn to make into tortillas, and nine grandchildren to invite to her birthday party. Children plus skeleton – guest number ten – have such a wonderful time that the skeleton decides that Grandma doesn't need to come along after all. Readers learn to count to ten in both English and Spanish. For ages 4-7.
In Lezlie Evans's Can You Count Ten Toes? (Houghton Mifflin Harcourt, 2004), readers learn to count to ten in ten different languages: Spanish, French, Japanese, Chinese, Korean, Tagalog, Russian, Hindi, Hebrew, and Zulu. Included are phonetic pronunciations for each number word and a map showing where the featured languages are spoken. For ages 4-8.
By Muriel Feelings, Moja Means One (Puffin, 1992) is a Swahili counting book, in which kids learn numbers 1-10 in Swahili as well as interesting facts about the land and culture of East Africa. The book begins with one impressive Mount Kilimanjaro, and continues through two kids playing a game of Mankala, three coffee trees, and so on, culminating in a group of ten children listening to a traditional storyteller. With lovely earth-toned illustrations by Tom Feelings. For ages 4-8.
In Andrea Cheng's Grandfather Counts (Lee & Low, 2003), Helen's grandfather, newly arrived in America from China, speaks no English and Helen and her siblings speak no Chinese. Gradually, though, as they watch passing trains together, her grandfather begins to teach Helen to count in Chinese, while she teaches him to count in English. A lovely story of an intergenerational relationship (with counting). For ages 4-8.
From Math Is Fun, Roman Numerals has an explanation of the symbols and their combinations, rules for forming numbers, how to write really big numbers (up to a million), and a couple of handy mnemonics for remembering what's what.
MATH AND ART
From the San Antonio Museum of Art, 123 Si! (Trinity University Press, 2011) is a counting book illustrated with color photos of art works from the Museum, among them Mexican puppets, Olmec clay statuettes, and Korean pen-and-ink tigers. For ages 3-6.
In Lucy Mickelthwait's I Spy Two Eyes: Numbers in Art (HarperTeen, 1993) readers search for objects in classical works of art, from 1 fly and 2 eyes to 12 squirrels, 17 birds, and 20 angels. For ages 4-7.
MATH ANXIETY
In Mark Pett's The Girl Who Never Made Mistakes (Sourcebook Jabberwocky, 2011), nine-year-old Beatrice never ever makes a mistake (unlike little brother Carl, who eats crayons). In fact, Beatrice is absolutely perfect, until the day of the annual talent show, when she makes a colossal and very public mistake. And discovers that it's not the end of the world. For ages 4-8.
In Joan Horton's rhyming Math Attack! (Farrar, Straus & Giroux, 2009), a little girl – at her wit's end when her teacher asks for the answer to seven times ten – has a math attack: numbers EXPLODE out of her head and wreak havoc all over town, disrupting everything from the prices in the supermarket to the helicopters of the National Guard. Finally she gets the answer, and all goes back to normal. For ages 5-9.
In Danny Schnitzlein's The Monster Who Did My Math (Peachtree Publishers, 2012), a math-hating kid is struggling with his impossible multiplication homework when a monster arrives and offers to take care of it for him – all he has to do is sign the contract on the dotted line. All is well until the teacher sends him to the blackboard, and he discovers the contract's fine print ("In paragraph seven of clause ninety-three/If you don't learn anything, do not blame me!"). And then, as in all Faustian bargains, he has to come up with the pay-off. Which involves some math. For ages 6-8.
Barbara Esham's Last to Finish (Mainstream Connections Publishing, 2008), one of the Adventures of Everyday Geniuses series, features third-grader Max who has always liked math – but falls apart when his teacher starts giving the class timed tests. Max is miserable. Eventually, however, the teacher discovers that Max has been working problems from his older brother's algebra book (for fun), and Max ends up on the school math team. A nice reminder that different kids learn in very different ways. For ages 6-9.
JUST FOR FUN
Amy Krouse Rosenthal's Wumbers (Chronicle Books, 2012), with bright cartoonish illustrations by Tom Lichtenheld, is a picture book for the text-messaging generation. Wumbers are words spelled with sound-alike numbers, familiar to anyone who has ever texted "gr8!" For example, try these: At a tea party (attended by a teddy bear and two little girls in purple): "Would you like some honey 2 swee10 your tea?" "Yes, that would be 1derful." At a family picnic: "We have the 2na salad and the pl8s. What have we 4gotten?" (Dismay!)"The 4ks!" Fun creative word puzzles for beginning readers ages 5-8.
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Zapped Zs, Alphabet Cities, dozens of cool and unusual alphabets, a lot of great books and projects, and alphabet flashcards for future nerds! (And it's not all just for little kids.)
MOSTLY FOR BEGINNERS
Artist Lisa DeJohn's colorful Alphabet Animals Flash Cards are printed in bright colors on heavy cardboard. Each has a capital alphabet letter, an animal word in lower-case print, and a great animal illustration, from Ant, Blackbird, and Caterpillar, through Mouse, Octopus and Zebra. For ages 1-4.
In Lisa Campbell Ernst's The Letters Are Lost (Puffin, 1999), the letters of the alphabet – each represented as an old-fashioned alphabet block – have been scattered: A flew off in an Airplane, B tumbled into the Bath, C joined a family of Cows. By the end, they're finally all back in order in their box again – but where will they end up next? (Invent your own lost-letter scenarios.) For ages 2-6.
In The Human Alphabet (Roaring Brook Press, 2005) by John Kane and the Philobolus Dance Company, dancers in bright-colored leotards take on the shapes of the alphabet letters. For ages 2-6.
Steve Martin's The Alphabet from A to Y with Bonus Letter Z! (Flying Dolphin Press, 2007) begins with "Amiable Amy, Alice, and Andie/Ate all the anchovy sandwiches handy." The illustrations, by brilliant cartoonist Roz Chast, are crammed with extra alphabetical goodies: under B, for example, readers can find everything from boomerangs, bears, and buckets to balloons, a ballerina, and a bowling ball. A great vocabulary builder for ages 2-6.
Sandra Boynton's A is for Angry: An Animal and Adjective Alphabet (Workman, 1997) runs from Angry Aardvark (deprived of ants) to Bashful Bear, Frightened Fox, and Zany Zebra (grinning, in pointy yellow party hat). Readers learn the alphabet, a host of animal names, emotion words, and the meaning of "adjective." For ages 3-5.
Mary Elting's Q is for Duck (Houghton Mifflin, 1985) is an alphabetical guessing game of animal sounds: Q is for duck because ducks quack. (Now try B is for Dog.) For ages 3-5.
In Sara Pinto's interactive The Alphabet Room (Bloomsbury USA, 2003), A is predictably for Apple and Z for Zebra – but each letter is accompanied by a revealing lift-the-flap door, behind which increasing numbers of labeled objects are continually shuffled and rearranged. (The Cat and Dog play with the Fish; the little Lamb eats Ivy; and the Moustache pops up everywhere.) For ages 3-6.
In Maira Kalman's What Pete Ate From A to Z (Puffin, 2003) – subtitled "Where We Explore the English Alphabet (in its entirety) In Which a Certain Dog Devours a Myriad of Items Which He Should Not" – Pete chows down on an astonishing array of alphabetical stuff, beginning with Uncle Rocky's Accordion. All with explanatory asides from his frustrated, but loving, owner. Funny and clever for ages 3-7.
In Leslie Tryon's Albert's Alphabet (Aladdin, 1994), Albert – the school carpenter and a very creative duck – builds all the letters of the alphabet. For ages 3-7.
In H.A. Rey's Curious George Learns the Alphabet (Houghton Mifflin Harcourt, 1973), everyone's favorite little monkey learns the upper- and lower-case letters of the alphabet, with help from the Man in the Yellow Hat. The trick is picture mnemonics: upper-case A, for example, looks like an alligator's open mouth and lower-case a like a slice of apple; H looks like a house and h like a horse. For ages 3-7.
Maurice Sendak's Alligators All Around (HarperCollins, 1991) is a delightful alphabet romp with alligators, in which a family of three variously bursts balloons, catches cold, entertains elephants, makes macaroni, and pushes people. For ages 3-7.
From ReadWriteThink, Alliteration All Around is a five-part lesson plan in which kids make their own alliterative alphabet books and write alliterative poetry. (Targeted at grades 3-5.)
In Lesa Cline-Ransome's Quilt Alphabet (Holiday House, 2002), each letter of the alphabet – framed in a quilt square – is paired with an alphabetical riddle poem and a folk-art painting. Answer are country-cosy: APPLE, COW, KETTLE, PIE, SCARECROW. For ages 3-6.
In Tana Hoban's 26 Letters and 99 Cents (Greenwillow Books, 1995), colorful photos of plastic letters are paired with photos of objects – D with a toy dinosaur, F with a goldfish, J with a handful of jellybeans. Flip the book over and it becomes a counting book in the same format. For ages 3-8.
In Alethea Kontis's AlphaOops! (Candlewick, 2012), put-upon Z ("Zebra and I are SICK of this last-in-line stuff!") creates havoc in the alphabet, until A manages to pull things back together. A delightful read for ages 3-8.
Arnold Lobel's On Market Street (Greenwillow Books, 2006) chronicles in alphabetical order the list of objects a small sailor-suited child buys on Market Street. The illustrations – from apples, books, and clocks to lollipops, playing cards, quilts, and wigs – are wonderful Arcimboldo-type paintings of people made entirely from their wares. For ages 3-8.
In David Pelletier's The Graphic Alphabet (Scholastic, 1996), a Caldecott Honor book, each page is essentially a concrete poem. A, for example, is an A-shaped mountain, crumbling at the top with a tumbling avalanche. For ages 6 and up.
In Leo Lionni's The Alphabet Tree (Alfred A. Knopf, 1990), each letter has a favorite leaf on the alphabet tree – until a gale-force wind swoops in and blows them all over the place. The solution is cooperation, as the letters band together to form words. For ages 3-7.
The Alphabet Tree has multidisciplinary extension activities to accompany the book, among them learning about seasons, creating story sequence cards, making a word tree poster, and studying tree growth and planting seeds.
Al Pha, main character of Amy Krouse Rosenthal's Al Pha's Bet (Putnam Juvenile, 2011), lived "back when all sorts of things were being invented" – among them, the alphabet. Al takes on the challenge of putting all the letters in proper order. For ages 3-5.
Tony DiTerlizzi's G is for One Gzonk (Simon & Schuster, 2006) is an outrageously zany "alpha-number-bet book" in which readers learn letters and numbers through such imaginary creatures as the Angry Ack, Dinkalicious Dinky, and Ravenous Rotoid. Lots of clever vocabulary and witty asides. For ages 4-7.
In Kelly Bingham's Z is for Moose (Greenwillow Books, 2012 ), Zebra is directing the line-up of the alphabet, a task continually disrupted by the over-eager Moose, who keeps bursting onto the scene, demanding "Is it my turn now?" "NOW?" Devastatingly, when M finally comes along, the letter goes to Mouse – but Zebra saves the day at Z, when Z stands for "Zebra's friend Moose." A great (and funny) read for ages 4-8.
In Neil Gaiman's The Dangerous Alphabet (HarperCollins, 2010), two kids and their pet gazelle launch themselves into a spooky underground in search of treasure. The story, rife with pirates, monsters, and trolls, is told in rhyming alphabetical (slightly scrambled) couplets. With Victorian-style illustrations by Gris Grimly. For ages 6-9.
In James Thurber's The Wonderful O (Simon & Schuster, 1957), a pirate named Black in search of buried treasure takes over the island of Ooroo and proceeds to ban the letter O. As the pirates forcibly remove everything with an O in its name, the islanders, led by a poet named Andreus, vow that four O words will not be lost: hope, valor, love, and freedom. This short chapter book is appropriate for ages 8 or so and up – probably not much younger; the word play is so clever that kids need well-developed reading and vocabulary skills to fully appreciate it.
Ella Minnow Pea. Say it once or twice, fast, and you'll see what it has to do with the alphabet. Ella is the protagonist of Ella Minnow Pea by Mark Dunn (Anchor, 2002), set on the fictional island of Nollop off the coast of South Carolina. The island is named for its founder, Nevin Nollop, inventor of the famous pangram (that is, a sentence using all 26 letters of the alphabet) "The quick brown fox jumps over the lazy dog." This pangram is set in tiles on the base of Nollop's memorial monument and when the tiles start falling off, the Nollopian governmental committee attributes it not to failing glue but to a sign from the beyond. The Z is the first to fall, and it is promptly decreed that the letter Z be expunged from the Nollopian alphabet. This is a problem for Nollopians named Zeke or Zachary, and a disaster for the island beekeeper (the bees, which make ZZZ sounds all the time, have to be eliminated), but most people manage to get by. As more and more letters fall, however, life becomes increasingly difficult; and the island takes on aspects of a fascist state. For teenagers and adults.
Visit Pangrams to learn all about these slippery alphabetical sentences and have a try at inventing one of your own.
In fantasy author Patricia McKillip's Alphabet of Thorn (Ace Trade, 2005), Nepenthe, a foundling with an unusual talent for language and translation, is raised by the librarians of the Royal Library of Raine, where she leads a secluded ivory-tower existence, devoted to books. Then a student mage brings her a new book written in a strange thorn-like alphabet that only she can read – and that appears to have strange magical powers. For teenagers and adults.
ALL ABOUT THE ALPHABET
James Rumford's There's a Monster in the Alphabet (Houghton Mifflin Harcourt, 2002) is the story of the pictorial beginnings of our modern alphabet, supposedly first brought to ancient Greece by the Phoenecian hero Cadmus. An appended chart compares English, Phoenecian, Greek, Hebrew, and Arabic alphabets.
Don Robb's Ox, House, Stick: The History of Our Alphabet (Charlesbridge, 2007) is a 48-page picture-book history of the alphabet targeted at ages 8-12.
David Sacks's Letter Perfect (Broadway Books, 2004) is the "marvelous history of our alphabet" from the Phoenecians to the present day. Included are general information, a family tree of world alphabets, many alphabetic charts, photographs of artifacts, and 26 informative chapters, each devoted to a different letter of the alphabet. Find out how letters got their shapes, why some letters have multiple sounds, and why X marks the spot. For teenagers and adults.
Roy Blount's Alphabet Juice (Sarah Crichton Books, 2009), arranged A to Z alphabet-style, is an info- and anecdote-filled overview of words and letters. The enormous subtitle gives you a sense of the content: "The Energies, Gists, and Spirits of Letters, Words, and Combinations Thereof; Their Roots, Bones, Innards, Piths, Pips, and Secret Parts, Tinctures, Tonics, and Essences; With Examples of Their Usage Foul and Savory." For teenagers and adults.
POSITIVELY WICKED ALPHABETS
Chris Van Allburg's The Z was Zapped (Houghton Mifflin Harcourt, 1987) is a clever play in 26 acts, in which each letter – appearing in black-and-white on a curtained stage – has something (generally awful) happen to it. A, for example, is caught in an Avalanche, B is Badly Bitten, K is Kidnapped, Y is Yanked offstage with a crook. And you can see by the title what happened to Z. A creative read for ages 4 and up.
By Far the Best Alphabet Book Ever is a lesson plan in which kids create their own "alphabet riddles" based on The Z Was Zapped. Included is a printable page of a curtained stage.
By Shel Silverstein, Uncle Shelby's ABZ Book (Touchstone, 1985) is a wickedly funny alphabet book supposedly for adults only. ("Meet Ernie, the giant who lives in the ceiling. Ernie likes eggs. Catch, Ernie, catch!") My kids found it hilarious. For all ages, depending on sense of humor.
In Edward Gorey's rhyming The Gashlycrumb Tinies (Houghton Mifflin Harcourt, 1997), a succession of Victorian children come to sad, bad ends, from Amy (who fell down the stairs) and Basil (assaulted by bears) to Zilla (who drank too much gin). My macabre children adored and memorized it. For a wide range of appropriately twisted ages. In our house, it was found hysterical by age 7.
Also see Gorey's Thoughtful Alphabets (Pomegranate, 2012), a pair of grimly hilarious 26-phrase stories ("The Just Dessert" and "The Deadly Blotter"), both running from A to Z. ("Apologize. Bewail complications.")
By Jory John and Avery Monsen, K is for Knifeball (Chronicle Books, 2012) is a rhyming A to Z collection of truly terrible advice, supposedly directed at (but not really for) kids. B is for Blender. F is for Fire. You can see where this is going.
In Roz Chast's What I Hate From A to Z (Bloomsbury USA, 2011), a cartoon compendium of miseries, B is for Balloon ("imminent explosion"), C is for Carnival, G for General Anaesthesia, and S for Spontaneous Human Combustion. For teenagers and adults. Make one of your own. Think therapy.
ALPHABETS IN THE REAL WORLD
By architectural photographer Elliott Kaufman, Alphabet Everywhere (Abbeville Kids, 2012) shows how the letters of the alphabet appear in all sorts of unexpected ways in the world around us, from bridge supports to sidewalk shadows to branches, leaves, and ocean waves. (Would make a great family project.) For ages 3 and up.
Stephen Johnson's Alphabet City (Puffin, 1999), a Caldecott Honor book, is a wordless tour of the alphabet, finding letters A to Z in construction sites, fire escapes, traffic lights, lamp posts, and church windows. For ages 3 and up.
By Krystina Castella and Brian Boyd, Discovering Nature's Alphabet (Heyday, 2006) is a fascinating collecting of color photographs of alphabet images from nature, found in everything from branches, vines, and rocks to seaweed on the beach. (Try taking an alphabet nature walk.) For ages 5 and up.
Karl Blossfeldt's The Alphabet of Plants (Schimmer/Mosel, 2007) is not an alphabet, but rather a collection of stunning black-and-white photographs of plant patterns in nature. All ages.
HISTORY, ALPHABET-STYLE
Martin Jarrie's ABC USA (Sterling, 2005) is an alphabetical overview of American history and culture (B is for Baseball, F is for Flag, I is for Immigrant), with charming folk-art-style illustrations. For ages 3-8.
Lynne Cheney's America: A Patriotic Primer (Simon & Schuster, 2002) is an alphabet book of American history and culture, with multifaceted cartoon-style illustrations by Robin Preiss Glasser. Lots to look at and discuss. For ages 5-10.
Lynne Cheney's A is for Abigail (Simon & Schuster, 2003) is an alphabet of famous American women, beginning with the indomitable Abigail Adams. Clever cartoon-style illustrations by Robin Preiss Glasser are crammed with extra information. Many pages are composites, such as E (for Educators) and W (for Writers). For ages 5-10.
James Rumford's Sequoyah: The Cherokee Man Who Gave His People Writing (Houghton Mifflin Harcourt, 2004) is a picture-book biography of the inventor of the Cherokee syllabary. The text appears in both English and Cherokee; included is a Cherokee alphabet chart. For ages 5-8.
By artist/historian Eric Sloane, the ABC Book of Early Americana (Dover Publications, 2012) is a beautifully illustrated compendium of American inventions and antiquities, from Axe, Almanack, Bathtub, and Conestoga wagon to Zig-zag fence. Included is a section on "The Alphabet in Early America." For all ages.
AROUND THE WORLD WITH ALPHABETS
From Sleeping Bear Press, the Discover the World series consists of alphabet books on various countries of the world, among them America, England, Italy, China, and India. See the link for a complete list, plus accompanying recipes, games, and maps. For ages 6-8.
Margaret Musgrove's Ashanti to Zulu (Puffin, 1992) is an alphabet of African tribes and traditions, with an appended map showing where each of the featured tribes lives. The gorgeous illustrations by Leo and Diane Dillon won this book a Caldecott Medal. For ages 3-8.
By Maya Ajmera and Anna Rhesa Versola, Children from Australia to Zimbabwe (Charlesbridge Publishing, 2001) is an alphabetical and photographic journey around the world. For each country are included a colorful map, helpful background information, the word "hello" in the dominant language, and a lot of terrific photos. For ages 7-12.
Rather than a phonetic alphabet, some languages – like Chinese – are written with pictographic characters. Peggy Goldman's Hu is a Tiger (Scholastic, 1996) is a simple introduction to Chinese characters for kids.
A survey of multicultural and alternative alphabets can be a fascinating project for all ages. See Omniglot for background information on the history of writing and an immense and fascinating list of writing systems. Visitors can view the Cyrillic, Etruscan, Runic, and Greek alphabets, and many many more. The site also includes a list of "alternative alphabets," including Braille, Morse code, and the Shavian alphabet – inspired by George Bernard Shaw, who touted a phonetic alphabet designed to simplify English spelling.
SCIENCE, MATH, AND ALPHABETS
Nerdy Baby ABC Flashcards are not your ordinary A-is-for-Apple flashcards. In these 26 laminated, illustrated cards, aimed at future geeks and scientists, A is for Atom, C for Cell membrane, M for Mandelbrot set, and N for Neuron.
Lois Ehlert's Eating the Alphabet runs the gamut from Apricot, Apple Avocado, and Asparagus to Zucchini. A brightly illustrated compendium of multicultural fruits and veggies, including such not-so-common selections as Jicama, Kiwi, Yam, and Xigua. For ages 2-5.
Anita Lobel's Alison's Zinnia (Greenwillow, 1996) is a lovely interlinking alphabet of girls' names, flower names, and verbs, from "Alison acquired an Amaryllis for Beryl" to the neatly tied up "Zena zeroed in on a Zinnia for Alison." Illustrated with beautiful and botanically accurate flower paintings. For ages 4-8.
In the same format, see Azarian's A Farmer's Alphabet (David R. Godine, 2009). (APPLE, LAMB, PUMPKIN, ZINNIA.) For ages 4-8.
By David McLimans, Gone Wild (Walker Children's Books, 2006) – a Caldecott Honor book – is an alphabet of endangered animals from Chinese Alligator to Grevy's Zebra. Black-and-white letters are cleverly transmogrified into animals, complete with horns, eyes, tongues, and wings. For ages 4-6.
Name a topic and Jerry Pallotta has almost certainly written an alphabet book about it. For a complete list – everything from Airplanes, Beetles, and Birds to Vegetables and Yucky Reptiles – see here.
Particularly fascinating for young scientists is Jerry Pallotta's The Skull Alphabet Book (Charlesbridge, 2002) which pictures the skulls of 26 different animals (anteater to zebra). The skulls aren't labeled; readers have to figure out the source for themselves from clues in the text. For ages 5-8.
In Kjell Sandved's The Butterfly Alphabet (Scholastic, 1999), readers find the letters of the alphabet in the patterns on butterfly wings – that is, real butterfly wings. The author, a nature photographer, decided to create the book when he found a perfect letter F on the wing of a tropical moth that he was studying under the microscope. Double-page spreads show the whole butterfly or moth with its scientific name, paired with a close-up of the wing showing an alphabet letter pattern. For all ages.
Available from Butterfly Alphabet, Inc., is a Butterfly Alphabet poster. (There's also an option to write your name in butterfly wings.)
By David M. Schwartz, G is for Google (Tricycle Press, 1998) is a math alphabet book, running from A is for Abacus to Z is for Zillion. (In between, Binary, Exponent, Fibonacci, and X-axis.) Each entry is accompanied by catchy cartoon-style illustrations and two to three pages of reader-friendly explanation. For ages 9-12.
In David M. Schwartz's multidisciplinary Q is for Quark (Tricycle Press, 2009) – a science alphabet book – A is for Atom, B for Black Hole, C for Clone, and X for Xylem. Each entry comes with appealing cartoon illustrations and two to three pages of background information and explanation. For ages 9-12.
My First Physics Poster is a great A to Z infographic poster in which a is for acceleration, c for speed of light in a vacuum, f for frequency, and h for Planck's constant.
ALPHABETICAL ART
In Denise Fleming's Alphabet Under Construction (Square Fish, 2006), artistic Mouse is busily creating an alphabet, using a different creative technique from each letter – for example, Air-brushing the A, Buttoning the B, Carving the C, and Dyeing the D. For ages 3-6.
From the Metropolitan Museum of Art, Museum ABC (Little, Brown Books, 2002) is a tour of the alphabet through dozens of works of art from the Museum's collection. A beautiful book for ages 3 and up.
By Cynthia Weill, ABeCeDarios (Cinco Puntos, 2007) is an alphabet book of Mexican folk art animals, in which animal names are listed in both English and Spanish. The animals are carved and brightly painted sculptures. Grab some modeling clay and make some of your own. For ages 2-4.
Lucy Mickelthwait's I Spy: An Alphabet in Art (Greenwillow, 1998) is a collection of 26 famous paintings, among them works by Rousseau, Goya, Chagall, Picasso, Renoir, and Matisse. Each is chosen to illustrate a letter of the alphabet, which often involves a bit of a hunt. The book begins with Rene Magritte's Son of Man, with its prominent green Apple. For ages 3-8.
In Stephen Johnson's A is for Art (Simon & Schuster, 2008), there's more than initially meets the eye. The book consists of 26 original works of abstract art, each containing concealed alphabet letters. For ages 6 and up.
Some of the most gorgeous alphabets ever are surely the illuminated letters of medieval manuscripts. Kids can learn about the process of 15th-century book-making in Bruce Robertson's Marguerite Makes a Book (J. Paul Getty Museum Publications, 1999) in which young Marguerite, when her artist father is injured, takes over and finishes his beautiful hand-written and painted book. Fold-out pages explain the technicalities of the process, including how paints were mixed and gold leaf applied. For ages 7-12.
This Illuminated Letter project has background information, color photos of examples, and instructions.
Theodore Menten's The Illuminated Alphabet (Dover Publications, 1971) is an inexpensive coloring book with 50 detailed black-line medieval letters to color. For ages 8 and up.
Tony Seddon's Draw Your Own Alphabets (Princeton Architectural Press, 2013) is a workbook with which users learn to draw thirty different creative fonts (and invent some of your own). For ages 10 and up.
Jane Bayer's A, My Name is Alice (Puffin, 1992) is an alphabetical picture-book version of the traditional jump rope rhyme, with illustrations by Steven Kellogg. (Add a jump rope and give it a try.) For ages 4-8.
Laura Rankin's The Handmade Alphabet (Puffin, 1996) teaches American Sign Language with clever letter-related visual cues. For each letter, a hand demonstrates the finger positions of the ASL alphabet, along with an alphabetical extra: the G hand, for example, wears a glove; I points to an Icicle; the T hand sports three thimbles; the V holds a paper valentine. For ages 6 and up.
Chris L. Demarest's Alpha Bravo Charlie (Margaret K. McElderry, 2005) is a picture-book introduction to the military or International Communications Alphabet (ICA), along with a chart of the U.S. Navy's alphabetical signal flags. For ages 6-9.
Tobi Tobias's A World of Words (Lothrop Lee & Shepard, 1998) is a beautiful illustrated alphabet of quotations by such authors as Emily Dickinson, e.e. cummings, J.R.R. Tolkien, and Lewis Carroll. (Interested older kids might enjoy making alphabetic quotation books of their own.) For all ages.
ALPHABETICAL POETRY
Katrina Vandenberg's The Alphabet Not Unlike the World (Milkweed Editions, 2012) is a collection of poems named for the Phoenician letters of the alphabet. A compelling collection for teenagers and adults.
Edward Lear's Alphabet Poem runs from "A tumbled down and hurt his arm" to "Z said, 'Here is a box of Zinc!'"
By Lee Bennett Hopkins, Alphathoughts (Wordsong, 2003) is an illusrated collection of 26 poems, each representing a letter of the alphabet. B is for Books, J for Jelly, L for Library, P for Pencil. For ages 6-8.
Richard Wilbur's picture book The Disappearing Alphabet (Houghton Mifflin Harcourt, 2001) demonstrates in clever rhymes what would happen if each letter of the alphabet should vanish: "What if the letter S were missing?/Cobras would have no way of hissing/And all their kin would have to take/The name of ERPENT or of NAKE." Terrific for all ages.
Jeanne Steig's Alpha Beta Chowder (HarperTrophy, 1994) is a collection of hysterical alliterative alphabet rhymes. (T, for example, features Tactless Toby who teases Tina with tadpoles in her tapioca.) For ages 7 and up.
Sara Midda's How to Build an A (Artisan, 2008) is a simple alphabet book (A for Apple, B for Boy) that comes with eleven plastic puzzle pieces with which kids can build all the upper- and lower-case letters of the alphabet. For ages 2-5.
Judy Press's Alphabet Art (Williamson Publishing, 1997) is a collection of poems, songs, projects, games, and fingerplays for teaching the letters of the alphabet. For example, kids make upper- and lower-case Bs from bubblewrap (templates can be traced from the book), assemble a paper Butterfly, and read Eric Carle's The Very Hungry Caterpillar (which sounds like a C book, but there's a gorgeous and enormous butterfly on the last page).For ages 2-6.
From Enchanted Learning, Alphabet Book has instructions for putting together a simple version of an alphabet book for early learners. You'll need construction paper and a lot of old magazines.
For older students, ABC Books Aren't for Babies! has creative alphabet book activities for grades K-12. Included is an A to Z list of suggestions: students can make, for example, an Ancient Civilizations Alphabet Book, a Biology Alphabet Book, a Mathematics Alphabet Book, or a Technology Alphabet Book.
Decorate with the alphabet! At Alphabet Around the Room, find instructions for making a cool wrap-around alphabet and word display.
From No Time for Flash Cards, 25 Alphabet Activities for Kids include making a magnetic alphabet garden, a letter pizza, a recycled alphabet, and a set of alphabet peg dolls.
The snake is almost always the bad guy. Voldemort's Nagini in the Harry Potter series, Asmodeus Poisonteeth in Redwall, Cleopatra's asp, and Satan in Milton's "Paradise Lost" are all – well, pretty much evil. Kaa, the ancient python in Kipling's Jungle Book, helps rescue Mowgli (the "man-cub") from a band of hostile monkeys – but he also tries (several times) to eat him.
Still, to be fair, snakes can be adorable and cool. And they're always interesting.
SNAKE STORIES
Colors! With a snake! In Alexander Wilensky's The Splendid Spotted Snake (Workman Publishing, 2011), the cheerful snake – which, in the book, is made from a sturdy ribbon – is born with red spots, but as he grows, he gains more and more spots in sequentially different colors. For ages 2-5.
Keith Baker's Hide and Snake (Houghton Mifflin Harcourt, 1995) is a gorgeously illustrated interactive picture book, in which readers try to find a colorfully patterned snake as it slithers from page to page, entangling itself in yarn, hats, baskets, cats, and shoelaces. For ages 2-6.
In Ellen Stoll Walsh's Mouse Count (Houghton Mifflin Harcourt, 1995), a hungry snake decides to fill a jar with ten (charming) mice to take home for dinner. He counts them as he adds them, one by one, to the jar – but the mice cleverly manage to un-count themselves again and escape. An exercise in counting forward and backward for ages 2-6.
Salina Yoon's Opposnakes (Little Simon, 2009) is an appealing lift-the-flap book about opposites, with snakes – which are variously clean and dirty, quiet and loud, hot and cold, straight and tangled, and more. For ages 3-6.
Joan Heilbroner's A Pet Named Sneaker (Random House, 2013) is a delightful Beginner Book starring Sneaker, a pet-store snake who wants only to be adopted. Finally a little boy named Pete takes him home, and Sneaker proves to be a wonderful pet (and even a hero). For ages 4-7.
In Patricia Reilly Giff's Watch Out! Man-Eating Snake! (Yearling, 1988) – one of the New Kids of Polk Street School series – it's Stacy's first day in kindergarten and she tries to make friends, but instead ends up terrifying everybody with her stories about her man-eating snake. (It's really stuffed.) Luckily big sister Emily has some helpful advice about friendship. For ages 4-7.
Tomi Ungerer's Crictor (HarperCollins, 1983) is the story of a perfectly delightful boa constrictor sent as a birthday present to Madame Bodot by her son who studies reptiles in Africa. Soon Crictor charms the entire village. For ages 4-8.
Trinka Hakes Noble's The Day Jimmy's Boa Ate the Wash (Puffin, 1992) is the riotous story of the disaster-laden day when Jimmy brings his pet boa constrictor on a class trip. With illustrations by Steven Kellogg. For ages 4-8.
In Julia Donaldson's The Gruffalo (Puffin, 2006), clever Mouse manages to scare off a hungry fox, owl, and snake by inventing a fearsome gruffalo (with terrible claws, terrible tusks, and terrible jaws). Mouse isn't worried, because there's no such thing as a gruffalo – until, to his horror, he actually encounters one. But it turns out that inventive Mouse has a story ready. For ages 4-8.
In Janell Cannon's Verdi (Houghton Mifflin Harcourt, 1997), Verdi is a bright-yellow little python who is determined never to grow up to be "lazy, boring, or green!" When, to his horror, he discovers that he's developing a green stripe, he does his very best to get rid of it, which nearly leads to disaster. Eventually Verdi comes to terms with adulthood, discovering that – even though he's green – "I'm still me!" For ages 4-8.
In Randy Siegel's My Snake Blake (Roaring Brook Press, 2012), the protagonist gets a super-long, bright green snake for his birthday as a present from his dad. ("I think your father is nuts," said my mom, as she walked in, frowning. "And proud of it," answered dad. "Now let the snake out.") Luckily Blake is a very talented snake, capable of spelling words with his coils. Like "RELAX." Funny and terrific for ages 4 and on up.
"Rikki-Tikki-Tavi," which first appeared in Rudyard Kipling's 1894 classic The Jungle Book, is the story of a brave little mongoose who saves a British family in India from Nag and Nagaina, a pair of deadly cobras. For a picture-book version of the tale, see Rikki-Tikki-Tavi (Morrow Junior Books, 1997), with illustrations by Jerry Pinkney. For ages 5 and up.
David Adler's Cam Jansen: The Scary Snake Mystery (Puffin, 2005) stars young detective Cam Jansen, whose nickname (Cam is short for "Camera") refers to her photographic memory. This one involves a loose snake and a stolen video camera. For ages 5-8.
In Alexander McCall Smith's Akimbo and the Snakes (Bloomsbury, 2006) – one of a series – ten-year-old Akimbo's father is head ranger on a game preserve in Africa. In this book, Akimbo goes to visit his uncle's snake farm, where he becomes involved in a hunt for a rare (and dangerous) green mamba. For ages 7-9.
David Almond's Mouse Bird Snake Wolf (Candlewick, 2013) is an original creation tale in which the gods – now back in the clouds, having naps and tea – have left gaps in the world. These are filled in by three imaginative children who conjure up a mouse, bird, snake, and finally a wolf. With the wolf, things get dangerous. A thought-provoking read for ages 7-10.
Joy Cowley's Snake and Lizard (Kane/Miller Books, 2008) covers – in many short chapters – the adventures of two unlikely friends, laid-back Snake and excitable Lizard. For ages 7-10.
The narrator of Patrick Jenning's We Can't All Be Rattlesnakes (HarperCollins, 2011) is Crusher, a gopher snake, captured by "an oily, filthy, fleshy human child" named Gunnar. ("Gunnar thinks I'll be his adoring pet. He's wrong.") Snarky Crusher decides to pretend to be tame while plotting to escape, but soon finds herself feeling sorry for clueless Gunnar. Clever, funny, and a great animal voice. For ages 8-12.
In The Reptile Room (HarperCollins, 2007) – Book 2 of Lemony Snicket's A Series of Unfortunate Events series – the luckless Baudelaire orphans are living with their herpetologist Uncle Monty, owner of the Incredibly Deadly Viper. Hilariously miserable for ages 8-12.
In Kurtis Scaletta's Mamba Point (Yearling, 2011), 12-year-old Linus moves to Liberia where his father has a job at the American embassy. Linus is terrified of deadly black mambas – but somehow mambas seem to be drawn to Linus. It turns out that Linus is a kaseng – a person with a mysterious connection to certain animals, in his case, mambas. Soon he has adopted a mamba as a pet – and with the help of the snake, he eventually becomes what he wants to be: "a whole new Linus." For ages 8-12.
Antoine de Saint-Exupery's The Little Prince (Houghton Mifflin Harcourt, 2013), originally published in 1943, is the poetic and philosophical tale of an aviator who, as a child, drew a picture of an elephant that had been swallowed by a boa constrictor – but which all the adults around him said was a hat. Now an adult, the aviator is stranded in the desert, attempting to repair his plane, when he meets the mysterious little prince – here from his tiny distant home planet. For starts, the little prince knows the picture is of an elephant inside a snake. A wonderful story. Readers will find a lot to talk about. For ages 10 and up.
The 1974 film version of The Little Prince, in which Bob Fosse plays a superb Snake, is rated G.
In Pat O'Shea's The Hounds of the Morrigan (HarperTeen, 1999), ten-year-old Pidge and his little sister Brigit find a book in a second-hand bookshop that turns out to be an ancient prison for the powerful and evil serpent Olc-Glas. The Morrigan – goddess of death and destruction – wants to use the serpent's power to take over the world, and the children soon find that they're involved in a great battle between the forces of good and evil. This is a wonderful fantasy, filled with characters and images from Celtic mythology. It's available through libraries and from used-book dealers. For fans of Tolkien and C.S. Lewis, this one is well worth tracking down. Highly recommended for ages 11 and up.
Rumer Godden's The River (Trans-Atlantic Books, 2004) is the story of Harriet, who lives with her British family in India. There Harriet, a would-be poet, struggles with change and relationships – with her older sister, Bea; the wounded soldier, Captain John; and her little brother, who is fascinated with the cobra in the garden. A thoughtful and beautiful exploration of life and death, love, loss, and growing up. For ages 12 and up.
In the Disney version of Robin Hood (1973), all the characters are animated – including Sir Hiss, wicked Prince John's wicked snake sidekick. (Robin and Maid Marian are foxes.) Rated G.
SNAKE MYTHS AND LEGENDS
By Sheila MacGill-Callahan, The Last Snake in Ireland (Holiday House, 1999) is the story of (not-yet-saint) Patrick ousting the snakes from Ireland with a magic bell – all but one particularly ornery snake who persistently dogs Patrick's heels until he finally manages to banish it to the depths of Scotland's Loch Ness. When he returns, years later, to check on it, it's still there. And much bigger. For ages 4-8.
Sean Taylor's The Great Snake: Stories from the Amazon (Frances Lincoln Children's Books, 2008) is a collection of nine pourquoi-style folktales collected in the course of a boat trip along the Amazon, among them "The Great Snake." Illustrations are colored woodcuts. For ages 8 and up.
Aaron Shepard's lavishly illustrated Lady White Snake (Pan Asian Publications, 2001) is a tale from Chinese opera in which the beautiful Lady White – a snake in human form – falls in love with a young man and marries. A monk, who discovers Lady White's true nature, finally succeeds – after a battle – in driving the lovers apart and imprisoning Lady White under a pagoda. (Though Lady White still has warrior friends who come to her aid.) Readers can also learn a handful of Chinese characters, including the one for "snake." For ages 8-12.
Geraldine McCaughrean's Perseus (Cricket Books, 2005) is a 160-page retelling of the Greek myth of Perseus, who slays the snake-headed gorgon Medusa and rescues the beautiful princess Andromeda. For ages 10 and up.
Edited by Gregory McNamee, The Serpent's Tale (University of Georgia Press, 2000) is a collection of snakes stories and tales from around the world, including folktales from Germany, the American Southwest, China, Yugoslavia, Iceland, and India. For ages 13 and up.
SNAKE SCIENCE
Patricia Lauber's Snakes Are Hunters (HarperCollins, 2002) – one of the Let's-Read-and-Find-Out Science series – is a nicely presented picture-book overview of the natural history of snakes, variously covering anatomy, senses, predators, food, and egg-laying. For ages 4-8.
By Janet Halfman, Garter Snake at Willow Creek Lane (First Edition, 2011) – one of the Smithsonian's Backyard series – is the story of a young garter snake, on her own since she was a two-day-old snakeling, learning how to survive and, as winter approaches, searching for a safe place to spend the coming cold months. For ages 4-8.
Sarah L. Thomson's Amazing Snakes (HarperCollins, 2006) – one of the I Can Read series – is a nicely done introduction to snakes for beginning readers. ("There are more than 2000 different kinds of snakes. Some are shorter than a pencil. Some are almost as long as a school bus.") For nonfiction lovers ages 5-7.
Amanda O'Neill's I Wonder Why Snakes Shed Their Skin (Kingfisher, 2011) is an engaging overview of reptiles, organized in question-and-answer format for a great interactive read. Which is the biggest reptile? Which is the biggest snake? Why do snakes stare? Illustrated with color photos and cartoons. For ages 5-9.
By Diane Burns, Snakes, Salamanders, and Lizards (Cooper Square Publishing, 1995) is one of the Take Along Guides, with background information, identification helps, note pages, and three simple craft projects (make a dried-bean picture of a snake, for example). For ages 6-9.
In Kate Jackson's Katie of the Sonoran Desert (Arizona-Sonoran Desert Museum Press, 2009), Katie is a diamondback rattlesnake. Told from the point of view of Katie as she hunts for food, fights predators, and protects her young, this is a good introduction to snake science for ages 7 and up. In both English and Spanish.
By Marianne Taylor, What If Humans Were Like Animals? (Readers Digest, 2013), in icky science mode, details "The Amazing and Disgusting Life You'd Lead as a Snake, Bird, Fish or Worm." For ages 9-12.
By Sy Montgomery with photographer Nic Bishop, The Snake Scientist (Houghton Mifflin Harcourt, 2001) – one of the Scientists in the Field series – follows the research of herpetologist Robert Mason on the red-sided garter snake. Find out what snake scientists do. For ages 10 and up.
David Badger's 144-page Snakes (Crestline Books, 2011) has information on the history of humans and snakes, nonfiction accounts of various aspects of snake biology, and over 100 spectacular color photographs of snakes. For snake-lovers of all ages.
Also see the similarly authoritative Snakes (Firefly Books, 2012) by David Gower, Katherine Garrett, and Peter Stafford.
From the California Academy of Science, Snakes and Lizards: Length and Movement is a hands-on activity in which kids measure and research a range of snakes and lizards. Find out what a squamate is. For ages 5-11.
From the Smithsonian, read about the 40-foot-long Titanboa, the largest snake ever.
Snakes and Ladders (or Chutes and Ladders) is a classic board game that originated in ancient India. Play a round online here. (The aim: maneuver your game piece across the board, while being helped by ladders and hindered by snakes.) Snakes and Ladders games, including the classic board game, a pirate-themed version, a snakeless Chutes and Ladders version, and an Android app are available from Amazon.com.
Jan Sovak's Snakes of the World Coloring Book (Dover Publications, 1995) has brief information and black-line, ready-to-color portraits of some 40 different snakes, among them the anaconda, king cobra, cottonmouth, puff adder, and garter snake.
AND DON'T LEAVE OUT…
Ouroboros, the ancient symbol of a snake swallowing its own tail, represents the cycle of life.
The famous Gadsden flag – named for its designer Christopher Gadsden – was created during the American Revolution and pictures a coiled rattlesnake with the motto "Don't Tread on Me." This article has interesting information on historical snake flags and snake symbolism.
If into each life some rain must fall, we might as well have some fun with it. Try making a cloud in a bottle or baking a thunder cake. See below for fiction and nonfiction books, poems, projects, experiments, recipes, and arts and crafts, all having to do with clouds and rain.
CLOUD AND RAIN STORIES
Robert Kalan's Rain (Greenwillow Books, 1991) is as much about colors as rain, beginning with "Blue sky," "Yellow sun," and "White clouds." Then the sky turns gray and rain falls – and finally there's a wonderful multicolored rainbow. For ages 2-5.
In Eric Carle's Little Cloud (Philomel, 1996), Little Cloud changes itself into a handful of different shapes – a sheep, a tree, a bunny, an airplane – before joining in with the other clouds to make a rainstorm. For ages 2-6.
In Manya Stojic's Rain (Dragonfly Books, 2009), rain finally comes to the hot dry African savanna. The porcupine smells it first, and runs to tell the zebras, who see distant lightning. They rush to tell the baboons, who hear thunder; then the rhinoceros feels the first falling drops. Both a rain story and an exploration of the five senses for ages 3-7.
In Charlotte Zolotow's The Storm Book (HarperCollins, 1989) – a Caldecott Honor book – it's a hot summer day in the country when a storm sweeps in, and then retreats, leaving behind a beautiful rainbow. For ages 3-7.
Charles Shaw's It Looked Like Spilt Milk (HarperCollins, 1988) is a collection of splotchy white shapes on a dark blue background, with an attention-grabbing refrain: "Sometimes it looked like a Rabbit. But it wasn't a Rabbit./Sometimes it looked like a Bird. But it wasn't a Bird.") On the last page, readers find out just was it is: a floating white cloud. For ages 3-8.
In David Shannon's The Rain Came Down (Blue Sky Press), the rain makes everybody cross. The chickens squawk, the cat yowls, the dog barks, people yell, and in no time the entire neighborhood is squabbling – all to the refrain of "the rain came down." Then (!) the rain stops, the sun comes out, and soon all problems are magically resolved. For ages 3-8.
Linda Ashman's picture book Rain (Houghton Mifflin Harcourt, 2013) combines two very different takes on the weather – that of a disgruntled old man ("Nasty galoshes!" "Dang puddles!") and that of an exuberant little kid pretending to be a frog. A charmer for ages 4-7.
In David Wiesner's Sector 7 (Clarion Books, 1999) – a Caldecott Honor Book – a little boy on a visit to the Empire State Building befriends a cloud and is carried off to the Cloud Dispatch Center in the sky, responsible for Sector 7 which encompasses New York City. There he discovers that the clouds are unhappy with their strictly regulated shapes and sizes, and so sets out to remedy the matter, turning them into a marvelous variety of fantastic shapes. For ages 4-8.
In "Clouds" – one of the short clever stories in Arnold Lobel's Mouse Tales (HarperCollins, 1978) – a little mouse enjoys watching the changing shapes of clouds until, to his horror, a cloud takes the shape of an enormous cat. For ages 4-8.
Uri Shulevitch's Rain Rain Rivers (Square Fish, 2006) is a lyrical celebration of rain, beginning with a little girl sitting in her attic bedroom listening to rain on the roof. For ages 4-8.
In Rob Scotton's The Rain is a Pain (HarperCollins, 2012), Splat the Cat is happily trying out his new purple rollerskates when a determined and annoying cloud moves in. For ages 4-8.
In Tom Lichtenheld's Cloudette (Henry Holt and Company, 2011), the title character is a very small and adorable cloud. Being small has many advantages, but Cloudette sees how bigger clouds behave, watering crops and filling waterfalls and rivers, and she wants to make a difference too. And finally she does, for one small unhappy frog. For ages 4-8.
In James Stevenson's We Hate Rain! (Greenwillow Books, 1988), Louie and Mary Ann are fretting because it has rained for two days straight, so Grandpa tells a tale from his youth when he, his brother Wainey, and family were deluged in a truly spectacular rain that filled their Victorian house to the roof. Like all Stevenson books, it's clever and hilarious. It's also infuriatingly out of print; check your local library. It's also available from used-book suppliers. For ages 4-8.
In Judi Barrett's Cloudy with a Chance of Meatballs (Atheneum, 1978), the village of Chewandswallow gets its food three times a day from the weather – it rains, snows, and blows orange juice, mashed potatoes, and hamburgers. Then, suddenly, the food-bearing weather turns vicious. For ages 4-8.
From Library Lessons, Cloudy with a Chance of Meatballs is a multidisciplinary lesson plan for grades 2-5. (Among the projects: create a Chewandswallow newspaper reporting on the weather disaster.)
In Dr. Seuss's Bartholomew and the Oobleck (Random House, 1949), King Derwin is bored with rain, snow, and fog, and so demands that the royal magicians ("mystic men who eat boiled owls") create something new and different to fall from the sky. They produce a disastrous storm of gooey green oobleck, and it's up to the king's commonsensical page boy, Bartholomew, to solve the problem. For ages 4-9.
In Michael Catchpool's The Cloud Spinner (Knopf Books for Young Readers, 2012), a young boy can weave beautiful cloth from clouds: "gold in the morning with the rising sun, white in the afternoon, and crimson in the evening" and "soft as a mouse's touch and warm as roasted chestnuts," He's always careful, though, never to weave too much, having been taught by his mother that "enough is enough, and not one stitch more." Then the king discovers the wonderful cloth and demands more and more of it – until the kingdom is at risk of losing its clouds, with awful consequences for all. Luckily the wise young princess intervenes. A lovely ecological tale for ages 5-8.
In David Wisniewski's Rain Player (Houghton Mifflin Harcourt, 1995), the land is threatened with a drought, so Pik, a young Mayan boy, challenges Chac, the rain god, to a game of ball. With wonderful Mayan-style cut-paper illustrations. For ages 5-9.
THE SCIENCE AND HISTORY OF CLOUDS AND RAIN
Franklyn M. Branley's Down Comes the Rain (HarperCollins, 1997), one of the Let's-Read-and-Find-Out Science series, is an appealing picture-book overview of the water cycle. Readers learn all about evaporation, condensation, cloud formation, and precipitation. For ages 4-8.
Anne Rockwell's Clouds (HarperCollins, 2008), one of the Let's-Read-and-Find-Out Science series, is a simple introduction to the different kinds of clouds and how they help us predict the weather. Included are instructions for making a cloud in a jar. For ages 4-8.
Tomie dePaola's The Cloud Book (Holiday House, 1984) covers ten different kinds of clouds ("Cumulus clouds are puffy and look like cauliflowers"), cloud mythology and traditional sayings, and ends with a short and silly cloud story. The illustrations are delightful. For ages 4-7.
Scholastic's The Cloud Book Teaching Plan has several science activities to accompany de Paola's The Cloud Book, among them making a cloud in a jar and a model water cycle, collecting and graphing rainfall data, and measuring the size of raindrops.
Lawrence Lowery's Cloud, Rain, Clouds Again (NSTA Press, 2013), one of the I Wonder Why series, is a picture-book introduction to the water cycle with an included activity handbook. For ages 5-8.
Melvin Berger and Gilda Berger's Can It Rain Cats and Dogs? (Scholastic, 1999), written in interactive question-and-answer format, is an overview of weather divided into three main sections: Sun, Air, and Wind; Rain, Snow, and Hail; and Wild Weather. An interesting interactive read for ages 5-9.
Seymour Simon's Weather (HarperCollins, 2006), illustrated with gorgeous full-page color photographs, is an overview of the causes and effects of the world's weather for ages 6-12.
Seymour Simon has several other excellent weather-related books in the same format, among them Storms, Hurricanes, Tornadoes, and Lightning.
Laura Lee's Blame It on the Rain: How the Weather Has Changed History (William Morrow, 2006) is a fascinating and reader-friendly overview of the historical impact of weather, with such chapters as "Greenland's Vikings," "Gee, It's Cold in Russia," "Washington and Weather," and "Rain Ruins Robespierre." For ages 12 and up.
Richard Hamblyn's The Invention of Clouds (Picador, 2002) is the story of Luke Howard, the early-19th-century amateur meteorologist who came up with the cloud classification and naming system that we still use today. For teenagers and adults.
APPRECIATING CLOUDS/CLASSIFYING CLOUDS
By John A. Day and Vincent J. Schaefer, the 128-page Peterson First Guide to Clouds and Weather (Houghton Mifflin, 1991) includes basic weather info and 116 helpful color photos for cloud spotters. For ages 6 and up.
Other field guides for weather watchers include David Ludlum's National Audubon Society Field Guide to North American Weather (Knopf, 1991).
Gavin Pretor-Pinney's The Cloudspotter's Guide (Perigee, 2007) is a 330+-page account of the science, history and culture of clouds, filled with fascinating facts and helpful illustrated cloud-spotting charts. Also by Pretor-Pinney, see The Cloud Collector's Handbook (Chronicle Books, 2011) which is part cloud identification manual, part journal for recording your cloud sightings. For ages 12 and up.
By Louis D. Rubin and Jim Duncan, The Weather Wizard's Cloud Book (Algonquin Books, 1989) describes a "unique way to predict the weather" by reading the clouds. An appendix explains how to set up a home weather station. For teenagers and adults.
Nephelococcygia is the practice of cloud-watching. This cloud-watching lesson plan has art, writing, and math activities for early-elementary students.
From Plymouth State University, Cloud Boutique is a photo-illustrated overview of cloud classification.
Cool Clouds is a great collection of photos of clouds that (more or less) look like things. Included is a gallery of clouds for viewers to make their own guesses as to what they look like.
From SCOOL, Cloud Types is a straightforward video tutorial with helpful diagrams and photos.
The Clouds 365 Project aims to take a cloud photo every day of the year. (Try it on your own!)
PROJECTS AND ACTIVITIES
Weather Dance Lesson Plan from Arizona State University is a creative (and active) collection of activities in which elementary-level kids learn about clouds and cloud formation and invent representative weather-related dances. Included are resource lists, music recommendations, and teaching suggestions.
From Science@home, Keeping a Weather Diary has suggestions and downloadable fill-in-the-data Easy and Advanced Diaries.
From NeoK12, Water Cycle has online quizzes and puzzles and a series of short educational videos. (One of these shows how to make your own water cycle in a box.)
See The Big Freeze for Make Your Own Cloud, a multi-part lesson plan in which kids learn about clouds and the water cycle and make a cloud in a jar; and Build Your Own Weather Station, in which kids learn about weather instruments and build a barometer, rain gauge, wind vane, and anemometer.
From Steve Spangler Science, the Cloud in a Bottle Experiment has detailed photo-illustrated instructions and an explanation of the results.
From HoodaMath, Cloud Wars is a strategy game, playable on several different levels, in which players attempt to capture clouds and take over the sky.
Studying acid rain? Find out how to make some here, along with suggestions for science projects. Are Plants Affected by Acid Rain? also has detailed instructions for an acid-rain experiment.
DIY Rain Gauge has instructions for building one, using a two-liter plastic bottle.
Weather Science Projects has background information and instructions for making a model water cycle and a cloud in a jar.
The Weather WizKids site has kid-friendly info on weather features (among them Clouds, Rain & Floods, Wind, Temperature, Lightning, Hurricanes, and more), weather experiments, weather games, a list of weather instruments, and a photo gallery.
Scholastic's WeatherWatch has a collection of great interactive projects and activities. Kids can identify and track clouds, gather data using weather instruments, become "Weather Detectives" and research causes of weather, take a try at forecasting the weather, research extreme weather, and check out "Nature in the News."
From the Franklin Institute, Franklin's Forecast has information (and an experiment) on El Niño, instructions for building your own weather station, a tutorial on radar, weather satellite history, and a fun list of weather activities, including a hyperlinked list of Musical Meteorology.
From NASA, ClimateKids has an animated list of the "Big Questions" about Weather and Climate, Air, Ocean, Water, Carbon, Energy, Plants & Animals, and Technology. Also included are a Climate Time Machine, instructions for hands-on projects ("Make Stuff"), and great resources for teachers.
Web Weather for Kids has interactive overviews of Clouds, Hurricanes, Blizzards, and Thunderstorms/Tornadoes, along with hands-on projects, a Cloud Matching game, and step-by-step instructions for reading weather maps and forecasting the weather. Projects include making fog in a jar, modeling convection currents, a tornado, and rain, and making a hot-air balloon.
The Weather Dude has basic info on weather topics, statistics on world weather, daily weather stories, weather maps, and a lot of weather songs (available on CD or as downloads).
From NOVALabs, Cloud Lab Guide has a great collection of educational science videos and links to weather-related NOVA programs (among them "Earth from Space" and "Inside a Megastorm").
By Craig F. Bohren, Clouds in a Glass of Beer (Dover Publications, 1987) is a collection of "simple experiments in atmospheric physics," among them not only "Clouds in a Glass of Beer," but "Mixing Clouds," "Black Clouds," "Indoor Rainbows," and more. Very thorough explanations for teenagers and adults.
STORM STORIES
Nancy Tafuri's The Big Storm (Simon & Schuster, 2009) is a "Very Soggy Counting Book" from 1 to 10 as more and more animals take shelter from the storm in a cave. For ages 2-5.
In Laura Vaccaro Seeger's Walter Was Worried (Square Fish, 2006), the sky turns dark and a storm rolls in, with arouses a whole range of emotions: Walter was worried; Priscilla was puzzled; Shirley, shocked; and Frederick, frightened. Their feelings are literally spelled out in letters on their face, which makes for a fun interactive read. (Walter's eyebrows, for example, are the r's in "worried.") For ages 4-8.
In Patricia Polacco's Thunder Cake (Puffin, 1997), a little girl is frightened by an approaching thunderstorm, and her grandmother reassures her ("This is Thunder Cake baking weather, all right."), by baking a very special cake. For ages 4-8.
From Patricia Polacco's website, Thunder Cake has printables, discussion questions, and activities to accompany the book.
Arthur Geisert's brilliantly illustrated Thunderstorm (Enchanted Lion Books, 2013) is a timeline – the text just a list of sequential times of day – of a thunderstorm, escalating to a tornado, moving in on a small Midwestern farm. For ages 4-8.
The heroine of Jerdine Nolan's tall tale Thunder Rose (Houghton Mifflin Harcourt, 2007) was born on a stormy night and grew up to be a most unusual girl, capable of lifting a cow over her head, trouncing rustlers, squeezing rain out of clouds, and facing down tornados. For ages 5-8.
In Mary Stolz's Storm in the Night (HarperCollins, 1990), a frightening storm has knocked out the power, so a grandfather tells his young grandson a story from when he was a boy in a storm as they sit together in the dark. Wonderful storm imagery and themes of intergenerational connection and overcoming fear. For ages 5-9.
In Peter and Connie Roop's Keep the Lights Burning, Abbie (Carolrhoda Books, 1987), set in 1856 in Maine, young Abbie is left in charge of the lighthouse, her sick mother, and three younger sisters when her father, the lighthouse keeper, goes to the mainland for medicine. When a fierce storm blows up, Abbie is on her own for weeks, keeping the lights burning and caring for her family. For ages 6-9.
For another version of Abbie's story, see Marcia Vaughn's Abbie Against the Storm (Aladdin, 1999).
Bruce Hiscock's The Big Storm (Boyds Mills Press, 2008) is the picture-book story of a landmark storm that swept across the United States in 1982, creating avalanches, tornadoes, and blizzards as it went. Readers learn about warm and cold fronts and air pressure. Illustrated with paintings and diagrams. For ages 6-10.
In Mary Pope Osborne's Twister on Tuesday (Random House, 2001) – one of the popular Magic Tree House series – Jack and Annie are sent to a one-room schoolhouse on the Kansas prairie in the 1870s, and must save their classmates when a tornado moves in. For ages 6-9.
In Jennifer Smith's The Storm Makers (Little, Brown, 2013) twins Ruby and Simon are having a strange summer, bedeviled with weird weather – which, it turns out, is all Simon's fault. Simon is a Storm Maker, one of a group of powerful people capable of controlling the world's weather. Soon opposing forces in the Makers of Storm Society, good and bad, are competing to control him, since Simon may be the most powerful Storm Maker of all time. For ages 8-12.
In Roland Smith's Storm Runners (Scholastic, 2012), Chase Masters and his father John spend their time traveling the country in pursuit of violent storms. In this, the first of a storm-filled adventure series, they encounter horrific Hurricane Emily. For ages 8-12.
In Ivy Ruckman's Night of the Twisters (HarperCollins, 2003), twelve-year-old Dan, his best friend Arthur, and baby brother Ryan are on their own when a fearsome tornado rips through their Nebraska town. Fictionalized, but based on a real event. For ages 8-12.
STORM SCIENCE
In the Let's-Read-and-Find-Out Science series, Franklyn M. Branley's Flash, Crash, Rumble, and Roll (HarperCollins, 1999) is a delightfully illustrated introduction to thunderstorms – with great diagrams – for ages 4-8.
Other weather books in this series include Anne Rockwell's Clouds, Lynda DeWitt's What Will the Weather Be?, Arthur Dorros's Feel the Wind, and – both by Franklyn M. Branley – Down Comes the Rain and Tornado Alert! An entire weather library for ages 4-8.
Myron Uhlberg's A Storm Called Katrina (Peachtree Publishers, 2011) is the harrowing story of the destruction of New Orleans by Hurricane Katrina, seen through the eyes of ten-year-old Louis Daniel – who wants to be a horn player like Louis Armstrong and manages to save only his brass cornet from the wreckage of his family's home. For ages 4-9.
By Simon Basher and Dan Green, Weather (Kingfisher Books, 2012) is terrific, with funny and informative first-person characterizations of important weather features – among them the Sun, the Atmosphere, Hail, Sleet, Hurricane, and El Nino. (Monsoon – a huge water drop – announces "Boy, am I a big crybaby! Every year I change from bright and sunny to sullen and sulky. I turn on the tears, instantly bringing cloudbursts of my favorite play pal, Rain.") For ages 8-13.
Lee Sandlin's Storm Kings (Pantheon, 2013) is a fascinating history of tornados and tornado chasers, beginning with the "Electricians" – stage magicians who performed tricks with static electricity – who inspired Benjamin Franklin to embark on his famous key-and-kite experiment with lightning. An absorbing and exciting read for teenagers and adults.
From Steve Spangler Science, use the Tornado Tube and a couple of one-liter plastic soda bottles to create your own tornado. Tornado in a Bottle has instructions for tornado-tube experiments and an explanation of how the tube works.
Rosalyn Schanzer's How Ben Franklin Stole the Lightning (HarperCollins, 2002) is an upbeat picture-book account of Ben Franklin's inventions and innovations, with emphasis on his interest in electricity and his investigations into the nature of lightning. For ages 6-10.
By poet Elena Roo, The Rain Train (Candlewick, 2011) is an onomatopoetic journey by train in the rain. For ages 3-6.
By Bill Martin, Jr., and John Archambault, Listen to the Rain (Henry Holt and Company, 1988) is an irresistible poem that echoes the sound of rain: "Listen to the rain/the whisper of the rain/the slow soft sprinkle/the drip-drop tinkle/the first wet whisper of the rain." For ages 3-7.
A lover of rain is called a pluvophile. If you are one, visit Rainy Mood to listen to the rain anytime.
By Joan Bransfield Graham, Splish Splash (Houghton Mifflin Harcourt, 2001) is a great collection of concrete poems about all things water, including one titled "Clouds." (Kids will want to invent some of their own.) For ages 4-8.
In Verna Aardema's rhyming Bringing the Rain to Kapiti Plain (Puffin, 1992), there's a drought in Kenya ("These are the cows, all hungry and dry/Who mooed for the rain to fall from the sky") – which Ki-pat the herdsman ends when he fires an arrow far into the air. Wonderful illustrations of African animals. For ages 4-8.
The title poem of Jack Prelutsky's poetry collection It's Raining Pigs and Noodles (Greenwillow Books, 2005) is a celebration of silly and wonderful rains. ("It's raining pigs and noodles/it's pouring frogs and hats/chrysanthemums and poodles/bananas, brooms, and cats.") For ages 5-10.
In Nancy Willard's wonderful and evocative poetry collection A Visit to William Blake's Inn (Houghton Mifflin Harcourt, 1982), with illustrations by Alice and Martin Provensen, see "The Wise Cow Enjoys a Cloud." Highly recommended for all ages.
What if rain dripped in your head and flowed into your brain? Read Shel Silverstein's Rain.
From Mother Goose Caboose, Rain Poems is a great list, including selections from Robert Louis Stevenson, Henry Wadsworth Longfellow, Langston Hughes, and Elizabeth Coatsworth.
Read Rain by Don Paterson, from the poetry collection of the same title.
"Into each life some rain must fall." So says Henry Wadsworth Longfellow in The Rainy Day.
RAIN? OR REIGN?
Fred Gwynne's The King Who Rained (Aladdin, 1998) is a picture book of homophones and idioms, as a puzzled little girl misinterprets forks in the road, fairy tails, boars to dinner, foot prince in the snow, and the king who rained for forty years. For ages 5-8.
By Will Moses, Raining Cats and Dogs (Philomel, 2008) is a collection of "irresistible idioms and illustrations to tickle the funny bones of young people," illustrated with Moses's signature folk art. Lots of fun wordplay for ages 6-11.
CLOUDS, RAIN, AND ART
Wordle is a cool toy for generating "word clouds" from text. See samples here and create word clouds of your own.
Make a cloud collage. You'll need several different kinds of blue paper and some fluffy cotton.
From DLTK's Crafts, Weather Activities is a collection of projects for preschoolers and elementary-level kids. For example, kids make a cloud wind puppet, paper-cut and salt-crystal snowflakes, a handprint sun, a windsock, and a pinwheel.
Little Cloud is an art lesson plan from Kinderart in which kids make stuffed clouds and raindrop pictures.
From Red Ted Art, Weather Get Crafty is a selection of particularly gorgeous weather-based crafts, among them suncatchers, sundials, windspinners, wind chimes, and rain mobiles. There's even a recipe for yummy rainbow jelly (topped with a cloud).
From Holly's Arts and Crafts Corner, the Cloud Jars look like great fun: you'll need jars of water, shaving cream, and food coloring.
From AllKids Network, the Raining Cloud Craft is a great mobile with translucent tissue paper raindrops.
Who doesn't love the birds? And think about all the great birds in literature: Stuart Little's Margalo, Harry Potter's Hedwig, Mo Willems's Pigeon. Edgar Allan Poe's Raven. The doleful Dodo in Alice in Wonderland. And all those piratical parrots.
See below for bird stories, bird science, birds in art, bird food recipes, mathematical birds, famous birds, and the best birds in movies.
BIRD STORIES
Jane Yolen's Owl Moon (Philomel, 1987) is a magical picture-book story about a walk through the winter woods at night to go owling. For ages 3-8.
In Jennifer Sattler's Sylvie (Random House, 2009), Sylvie – a little flamingo – asks her mother why flamingos are pink and discovers that it's because of the pink shrimp that they eat. Sylvie promptly sets out to experiment, snacking on grapes (which turn her purple), chocolate (brown), a red kite (scarlet), and even a paisley bathing suit (paisley-patterned). Finally, however, she discovers that she'd prefer to be her own pink self. For ages 3-7. (Pair this one with Leo Lionni's A Color of His Own (Dragonfly, 1997).)
In I.C. Springman's More! (Houghton Mifflin Harcourt, 2012), an acquisitive magpie learns about the perils of too much stuff with the help of some friendly mice. When the book begins, the magpie has nothing, until a mouse offers him a marble – but soon, obsessively collecting, he passes from "plenty" to "much too much." A nicely done lesson on materialism (with a bird). For ages 4-8.
In Germano Zullo's charming Little Bird, a bright red truck stops by a cliff and the driver – an egg-shaped man in overalls – gets out, opens the back door, and releases a flock of birds. Just one little blackbird is left behind, and the man does his best to encourage him to go with the flock, by flapping his arms to imitate flying. Finally (after sharing a sandwich) the bird leaves – only to return with the entire flock to carry the man up into the sky. For ages 4-8.
Adam Rubin's hilarious and delightful Those Darn Squirrels! (Sandpiper, 2011) features the unspeakably grumpy Old Man Fookwire, who hates pies and puppies – but loves birds. He paints bird portraits and fills his yard with beautiful bird feeders, in hopes of persuading his beloved birds stay with him through the winter. The feeders promptly attract a gang of particularly persistent and innovative (they're good with pulleys and catapults) squirrels. When the birds do fly south, leaving Old Man Fookwire alone in his house mournfully eating cottage cheese, the squirrels decide to do him a good turn in payment for all the goodies they've nabbed. For ages 4-8.
Inspired by Old Man Fookwire? From Deep Space Sparkle Art Lessons for Kids, see How to Draw a Bird for a great bird drawing, painting, and decorating project. Make beautiful bird portraits of your own.
For more on squirrels, including famous squirrels, a purple squirrels, and a robotic squirrel, see SQUIRRELS.
In Jennifer Yerkes's A Funny Little Bird (Sourcebooks Jabberwocky, 2013), the little bird is essentially invisible – so he sets about decorating himself with flowers, leaves, and discarded feathers. The new plumage backfires, however, when it catches the attention of predators, and the little bird decides that it's far better to stay as he is and use his camouflage talents to help his friends. For ages 4-6.
In Dr. Seuss's Horton Hatches the Egg (Random House, 2004), Horton, the kind and patient elephant, determinedly cares for the egg left behind by lazy bird Maysie, who has taken off for Palm Springs. ("I said what I meant and I meant what I said/An elephant's faithful, one hundred percent!") And at last, when the egg hatches, Horton gets a wonderful reward. For ages 4-8.
Janell Cannon's Stellaluna (Houghton Mifflin Harcourt, 1993) is the story of a little fruit bat who, attacked by an owl, falls and lands in a nest of birds. Her new siblings teach her about life as a bird – and she, in turn, shows them what life is like for bats. It's a lovely story about friendship, despite differences. (Stellaluna, at the end, is reunited with her mother and discovers that she's supposed to eat mangoes, not bugs.) For ages 4-8.
In Don Freeman's Will's Quill, or How a Goose Saved Shakespeare (Viking Juvenile Books, 2004), Willoughby, a country goose, heads for London to see the sights. There he has a hard time until befriended by playwright Will Shakespeare – and ultimately, by providing feathers for quill pens, he does Will a great service in return. For ages 5-8.
In Cybele Young's Ten Birds (Kids Can Press, 2011), ten birds – with such names as Brilliant, Extraordinary, and Shows Great Promise – are trying to figure out how to cross a river. Each comes up with an imaginative solution – stilts, a water bicycle, a parachute, a kite – until it's the turn of the tenth bird, known as Needs Improvement. Who comes up with the simplest and cleverest solution of all. For ages 5-10.
In Dick King-Smith's chapter book Harry's Mad (Yearling, 1997), Mad is a bird – a highly intelligent and creative talking parrot named Madison, left to Harry by his eccentric uncle. Trouble strikes when Mad is parrot-napped. For ages 6-10.
In Joan Aiken's Arabel's Raven (Houghton Mifflin Harcourt, 2007) – with illustrations by the incomparable Quentin Blake – Arabel's father, a taxi driver, brings home an injured bird. Subsequently named Mortimer, the raven – who insists on answering the telephone by squawking "Nevermore!" – wreaks havoc. He's a sort of avian Paddington Bear. Arabel loves him and so do I. For ages 7-10.
In E.B. White's The Trumpet of the Swan (HarperCollins, 2001), originally published in 1970, eleven-year-old Sam discovers a family of trumpeter swans while on a camping trip – the youngest of whom, a cygnet named Louis, is mute. Louis's father steals a brass trumpet from a music store to give his son a voice. A wonderful book for ages 8-12.
In Kathleen O'Dell's The Aviary (Yearling, 2012), eleven-year-old Clara and her mother live in a crumbling mansion with old Mrs. Glendoveer. Clara, said to have a weak heart, is forbidden to run, play, or go to school – but nonetheless manages to make a friend and to solve the mystery of the Glendoveers' past. It's a spooky and addictive story that involves vanished children and the birds caged in the great aviary behind the house. For ages 8-12.
In Farley Mowat's Owls in the Family, Billy – growing up on the plains of Canada – adopts two personality-laden pet owls, Wol and Weeps, who promptly turn the family and the neighborhood upside-down. For ages 8-12.
The main character in Jean Craighead George's My Side of the Mountain (Puffin Books, 2004) is 12-year-old Sam Gribley who runs away from home to live on his own in a hollow tree in the Catskills. There he learns to survive, and adopts and tames a peregrine falcon chick, which he names Frightful. A wonderful read for any kid who has ever dreamed of life in the woods – and luckily there are several sequels. For ages 8-12.
Holling C. Holling's Seabird (Houghton Mifflin Harcourt, 1978) is the story of a carved ivory seagull who travels across oceans and through time with four generations of seafarers, from a Nantucket whaling ship to a clipper, a steamship, and an airplane. The carving is made by young Ezra Brown, based on the seagull he saw in a snowstorm from the crow's-nest of the whaling ship. It's a wonderful book, illustrated both with colorful paintings and detailed marginal drawings, diagrams, and maps. For ages 9-12.
FOLK TALES, FAIRY TALES, AND FANTASY
Gerald McDermott's Raven (Houghton Mifflin Harcourt, 2001) is a trickster tale from the Pacific Northwest in which clever Raven feels sorry for the people living in the cold and dark, and so sets out to steal light and warmth from the Sky Chief. Illustrated with colorful native-American-style drawings. For ages 4-8.
In James Mayhew's Ella Bella Ballerina and Swan Lake (Barron's Educational Series, 2011), Ella's ballet class is preparing to dance Tchaikovsky's Swan Lake – and Ella, as she listens to the music, is magically transported into the world of Swan Lake, where she meets Odette, the swan princess, and the evil sorcerer who turned her into a bird. For ages 4-8.
There are now many versions and editions of Hans Christian Andersen's story of "The Ugly Duckling," the homely and heckled little duck who grew up to be a beautiful swan. See Jerry Pinkney's Caldecott Honor book The Ugly Duckling (HarperCollins, 1999).
Jane Ray's The Emperor's Nightingale and Other Feathery Tales (Boxer Books, 2013) is a collection of 12 stories and poems from around the world, all featuring birds. Included, along with Hans Christian Andersen's "The Emperor's Nightingale," are Oscar Wilde's "The Happy Prince," "Jorinda and Joringel" from the Brothers Grimm, and Edward Lear's "The Owl and the Pussycat." For ages 5 and up.
In Katherine Paterson's The Tale of the Mandarin Ducks (Puffin, 1995), set in medieval Japan, a greedy lord captures and cages a beautiful mandarin duck, who pines miserably for freedom and his mate. Yasuko, the little kitchen maid, releases the bird, and she and her friend, the one-eyed ex-warrior Shozo, are sentenced to death by their angry master – only to be saved by a pair of mysterious messengers. For ages 5 and up.
By Ingri and Edgar Parin D'Aulaire, The Terrible Troll-bird (New York Review Children's Collection, 2007), based on Norwegian folklore, is the story of a giant rooster and some even more threatening trolls, all soundly defeated by four brave children, Ola and his sisters Lina, Sina, and Trina. Wonderful folk-art-style illustrations. For ages 5-9.
By Gennady Spirin, The Tale of the Firebird (Philomel, 2002) is a gorgeously illustrated picture-book version of the Russian folktale about the Tsar's youngest son and his quest for the Firebird. Danger, adventures, a helpful wolf, the frightening Baba Yaga who lives in a cottage with chicken feet, a beautiful princess, and a wonderful bird. For ages 6-10.
By R. L. LeFevers, Flight of the Phoenix (Houghton Mifflin Harcourt, 2010) stars ten-year-old Nathaniel Fludd, sent to live with his aunt after his parents are declared lost at sea – where he sets out to learn the family business of beastology. In this, the first of an extensive series (all crammed with mythological creatures), Nate and Aunt Phil travel to Arabia to witness the hatching of a phoenix egg. For ages 7-10.
In Edward Ormondroyd's David and the Phoenix (Purple House Press, 2001) – originally published in the 1950s – David explores the mountains behind his new North Carolina home and there discovers the Phoenix. The Phoenix is being pursued by a Scientist and had been studying Spanish, in preparation for fleeing to South America – but he decides to stay put for a while, and to take David's education in hand. There follows a series of hilarious and often near-disastrous adventures, involving fauns, leprechauns, witches, griffins, and a Sea Monster – and ultimately a painful, but hopeful ending. I've loved this book for years, as it waffles in and out of print. At the moment, it's in. It's also available in used editions and inexpensively (even for free) on Kindle. For ages 8-12.
Kathryn Lasky's Guardians of Ga'hoole series is a gripping battle between good and evil, with owls. In Book 1 of the series, The Capture (Scholastic, 2003), a young owl named Soren has been living happily with his family, raised on tales of the Guardians of Ga'hoole, legendary owls famed for their noble deeds. Then he is knocked out of the nest and captured by evil owls from the Academy of St. Aegolius. There Soren and his new friend Gylfie struggle to survive, resist their captors, and secretly learn to fly. Many exciting sequels. For ages 8-12.
By James Riordan, The Seven Voyages of Sinbad the Sailor (Frances Lincoln Children's Books, 2008) – from the classic A Thousand and One Arabian Nights – is a 64-page illustrated account of astounding adventures with (among others) an island that turns out to be a whale, a sea monster, ogres, and a gigantic bird called a rookh. For ages 9-12.
Clem Martini's The Mob (Kids Can Press, 2005) is the story of the Kinaars, a crow clan, now come together for their annual meeting at the Gathering Tree. Kyp, a headstrong young crow, is ostracized from the Flock for calling down a mob on an encroaching cat; when a blizzard hits, however, Kyp and friends – though they've flouted crow tradition – save the day. It's a great story, and many of the behaviors of the crow clan are based upon those of real crows in the wild. Reminiscent of Watership Down. There are two sequels. For ages 10 and up.
BIRD POEMS
Susan Stockdale's Bring on the Birds (Peachtree Publishers, 2011) is a gorgeously illustrated rhyming account of the many different kinds of birds ("Swooping birds/Whooping birds/Birds with puffy chests/Dancing birds/Diving birds/Birds with fluffy crests"). An illustrated appendix explains just what each bird is. For ages 4-8.
Douglas Florian's on the wing (Houghton Mifflin Harcourt, 2000) is a beautifully illustrated collection of 21 bird poems, each dedicated to a different bird – among them "The Egret," "Magnificent Frigate Birds," "The Quetzal," "The Emperor Penguins," and "The Common Crow." For ages 5-10.
Paul Fleischman's I Am Phoenix (HarperCollins, 1989) is a wonderful collection of "Poems for Two Voices," all about birds. For ages 8-12.
Edited by Billy Collins and illustrated by nature artist David Allen Sibley, Bright Wings (Columbia University Press, 2012) is a wide-ranging anthology of poems about birds, beginning with Stephen Vincent Benet's "John James Audubon." Also included are poems by Seamus Heaney, Marianne Moore, Mary Oliver, Walt Whitman, Sylvia Plath, Delmore Schwartz, Henry David Thoreau, Robert Browning, and many more. For teenagers and adults.
Kevin Henkes's Birds (Greenwillow Books, 2009) is a delightful introduction to birds that conveys the magic of bird-watching through stylized acrylic paintings and an appealing text in the voice of a child narrator. ("Once I saw seven birds on a telephone wire. They didn't move and they didn't move and they didn't move. I looked away for just a second…and then they were gone.") A charmer for ages 3-7.
In Priscilla Belz Jenkins's A Nest Full of Eggs (HarperCollins, 1995) – one of the Let's-Read-and-Find-Out Science series – a pair of children watch as robins build a nest, lay a clutch of eggs, and raise chicks. Finally, the babies grown, the robins leave in the fall to fly south – though the kids look forward to them returning again the next spring. For ages 4-8.
May Garelick's What Makes a Bird a Bird? (Mondo Publishing, 1995) in a thought-provoking exploration of just that. Is it a bird because it flies? But bees, butterflies, bats, and flying fish all fly – and some birds, like ostriches and penguins, can't. The book proceeds in this fashion, question by question, until readers finally discover the defining characteristic of birds: feathers. For ages 4-8.
By Roma Gans, How Do Birds Find their Way? (HarperCollins, 1996) in the Let's-Read-and-Find-Out Science series is an explanation of the hows and whys of bird migration. (Arctic terns travel from northern Maine to the South Pole. Why don't they get lost?) For ages 4-9.
By Bernadette Gervais and Francesco Pittau, Birds of a Feather (Chronicle Books, 2012) is crammed with creative graphics, interactive features – including flaps, pop-ups, and puzzles, and a lot of fascinating facts about birds. (Did you know that flamingos are gray when they're first hatched?) For ages 4-9.
By Sneed B. Collard III, Beaks! (Charlesbridge, 2002) is an exploration of the many kinds and uses of bird beaks, illustrated with impressive 3-D cut-paper sculptures by Robin Brickman. In the same format, see Collard's Wings! (2008). For ages 4-9.
Irene Kelly's Even an Ostrich Needs a Nest (Holiday House, 2009) discusses how different species of birds from all over the world build their nests (plus four species who don't build nests at all). Costa's Hummingbird, for example, builds a nest the size of half a ping-pong ball, while the Bald Eagle constructs a two-ton nest the size of a car. For ages 5-8.
In Melissa Stewart's lovely scrapbook-style picture book Feathers: Not Just for Flying (Charlesbridge, 2014), readers are introduced to sixteen different birds and the many surprising uses of feathers. (For example, they can "warm like a blanket" or "shade out sun like an umbrella;" and the feathers on the willow ptarmigan's feet act like snowshoes.) For ages 5-9.
By Steve Jenkins and Robin Page, How to Clean a Hippopotamus (Houghton Mifflin Harcourt, 2013) is a picture book about unusual animal partnerships, several involving birds. Find out why ravens and wolves, plovers and crocodiles, and egrets and antelopes stick together. For ages 6-9.
Birds live everywhere. Barbara Bash's Urban Roosts (Little, Brown, 1992) shows how 13 different species of birds – from pigeons to peregrine falcons – have adapted to life in the city. For ages 7-11.
In David Burnie's Bird (Dorling Kindersley, 2008) in the Eyewitness Series, each double-page spread covers a different aspect of bird anatomy, physiology, or behavior. (Topics include Feathers, Courtship, Beaks, Making a Nest, Extraordinary Eggs, and more.) Illustrated with wonderful photographs and diagrams. For ages 8 and up.
Where did birds come from anyway? Are they really…dinosaurs? Check out Christopher Sloan's How Dinosaurs Took Flight (National Geographic Children's Books, 2005) for ages 10 and up.
Colin Tudge's The Bird: A Natural History of Who Birds Are, Where They Came From, and How They Live (Broadway Books, 2010) is an excellent overview of all things bird for teenagers and adults.
By Jennifer Price, Flight Maps (Basic Books, 2000) – subtitled "Adventures with Nature in America" – includes terrific essays on the extinction of the passenger pigeon, the trends for birds on women's hats that led to the founding of the Audubon Society, and the history of the pink flamingo lawn ornament. For teenagers and adults.
From PBS, David Attenborough's Life of Birds is a fascinating and beautifully done documentary, variously covering bird brains, evolution, champions, parenthood, bird song, and more.
Attenborough's Life of Birds, the complete ten-part series, is available from Amazon.com as an Instant Video.
The Cornell Lab of Ornithology is an excellent source for all forms of bird information. Included on the website are bird citizen science projects (participate in Project Feeder Watch or join in the Great Backyard Bird Count), online courses (among these a superb home study course in Bird Biology), bird identification guides, bird cams, and much more.
Audubon.org has information on citizen science and bird conservation projects, bird FAQs, an online bird ID guide, and reports on birds in the news.
From UC Berkeley, Introduction to the Aves has detailed information on bird fossils, life history and ecology, systematics, and morphology.
WHICH BIRD?
In Lois Ehlert's rhyming Feathers for Lunch (Houghton Mifflin Harcourt, 1996), a black cat – safely equipped with collar and jingling bell – escapes from the house and encounters twelve common backyard birds, among them a cardinal, blue jay, goldfinch, robin, and hummingbird. Kids learn beginning bird identification and the cat ends up with nothing but feathers for lunch. The painted paper illustrations are wonderful. For ages 3-7.
By Mel Boring, Birds, Nests, and Eggs (Cooper Square Publishing, 2008) is a helpful "Take Along Guide" to help kids identify fifteen different birds, along with a handful of activities (make a bird bath, a blind for bird watching, and a suet feeder) and scrapbook pages for sketches and notes. For ages 5-10.
Bird Bingo is an illustrated bingo game featuring 64 different species of birds from around the world, from the emu and kookaburra to the puffin, robin, and mandarin duck. Play and learn your birds! For ages 6 and up.
Ana Gerhard's picture book Listen to the Birds: An Introduction to Classical Music (Secret Mountain, 2013) explains how many classical composers have been inspired by bird song, among them Mozart, Tchaikovsky, Prokofiev, and Vivaldi. The book includes short biographies of each composer and information the featured birds. An accompanying CD has excerpts of 20 different bird-based musical compositions, among them The Goldfinch, Hens and Roosters, The Cuckoo and the Nightingale, and Dance of the Firebird. For ages 7 and up.
FEEDING THE BIRDS
By the Editors of Birds & Blooms, For the Birds (Readers Digest, 2010) is a collection of 50 easy-to-make recipes for bird food. For all ages.
From CanTeach, A Variety of Bird Feeders has instructions for making five simple feeders, variously using plastic bottles, milk cartons, pine cones, plastic lids (plus a doughnut), and potato chip cans.
From Artists Helping Children, Easy Birdfeeders, House, and Perches has instructions and patterns for several different kinds of bird feeders and bird snacks, among them pinecone, soda bottle, and milk carton feeders. Also included: a recipe for bird biscuits. Squirrels, of course, like these too.
On YouTube, listen to Julie Andrews sing Feed the Birds from Mary Poppins.
SAVING THE BIRDS
By Olivia Bouler, Olivia's Birds: Saving the Gulf (Sterling, 2011) is the story of an 11-year-old girl's campaign to save the Gulf Coast birds after the devastating oil spill of 2010. For ages 3-9.
Kathryn Lasky's She's Wearing a Dead Bird on Her Head! (Disney-Hyperion, 1997) is the picture-book story of Harriet Hemenway and Minna Hall, who founded the Massachusetts Audubon Society. For ages 5-9.
In Meindert DeJong's 1955 Newbery winner, The Wheel on the School (HarperCollins, 1972), young Lina wonders why there are no more storks – birds that are said to bring good luck – in her village. Soon she has co-opted the entire community into luring the storks back home by proving rooftop wheels where they can build their nests. For ages 8-12.
In Gill Lewis's Wild Wings (Atheneum Books for Young Readers, 2012), Iona and Callum in Scotland (an unlikely pair) join forces to protect an endangered osprey – a story that eventually links to people around the world. For ages 8-12.
In Carl Hiaasen's Hoot (Yearling, 2005), Roy Eberhardt, recently moved from Montana to Florida, joins forces with the intimidating Beatrice and her brother Mullet Fingers to save a colony of tiny burrowing owls from Mother Paula's All-American Pancake House. Funny, brave, and wonderful. For ages 10 and up.
Stephen Kress's Project Puffin (Tilbury House Publishers, 2003) is the story of how Kress and his team of "Puffineers" restored the puffin population of Egg Rock, an island off the coast of Maine. For ages 10 and up.
By Pete Salmansohn and Stephen W. Kress, Saving Birds: Heroes Around the World (Tilbury House Press, 2005) has six dramatic stories of people around the world fighting to save wild birds. For ages 10 and up.
STATE BIRDS
Hudson Talbot's United Tweets of America (Putnam Juvenile Books, 2008) is the humorously illustrated story of all 50 state birds, in alphabetical order by state. For each is included information about the bird, a map of the state, and basic state information, including other state symbols, the state capital, famous people, and more. For ages 7-10.
Annika Bernhard's State Birds and Flowers Coloring Book (Dover Publications,1990) has black-line versions of them all, ready for crayons or colored pencils.
MATHEMATICAL BIRDS
In Frank Mazzola's Counting Is for the Birds (Charlesbridge, 1997), birds, two by two, gather at a backyard feeder, until they're scattered by a squirrel. Kids count to 20 and back again, and learn a bit about birds from thumbnail sketches. For ages 3-7.
Alice Melvin's Counting Birds (Tate, 2010) is a lovely counting book with a rhyming text. Kids count to twenty, beginning with one cockerel, two lovebirds, and three flying ducks. For ages 3-7.
Stuart J. Murphy's Double the Ducks (HarperCollins, 2002) – a MathStart book – introduces kids to concepts of addition and multiplication when five little ducks each bring home a friend. For ages 4-8.
Shirley Raye Redmond's Pigeon Hero (Simon Spotlight, 2003) in the Ready to Read series is the true story of G.I. Joe, a World War II homing pigeon, who saved an Italian town by carrying crucial messages through enemy lines. (He was awarded a medal for bravery.) For ages 5-7.
Leo Politi's Song of the Swallows (J. Paul Getty Museum, 2009) is a picture-book story of the famous annual return of the swallows to the Mission San Juan Capistrano in California. For ages 5-9.
For more information and activity suggestions, see Swallows on a Mission. Included at the site are resources, maps, a swallow-sighting report form, and an in-depth lesson on "How Birds Fly."
By Stephanie Spinner, Alex the Parrot: No Ordinary Bird (Knopf Books for Young Readers, 2012) is the true story of scientist Irene Pepperberg and the amazingly intelligent Alex, an African gray parrot, who could count, name colors, and had a vocabulary of hundreds of words. For ages 8-12.
Philip Hoose's award-winning Moonbird (Farrar, Straus & Giroux, 2012) is the true story of the phenomenal travels of a little shorebird known to scientists as B95 – in his lifetime, a distance of over 325,000 miles, enough to have taken him to the moon and halfway back. Illustrated with photographs and maps. For ages 10 and up.
By artist Charley Harper, the Charley Harper Coloring Book of Birds (Ammo Books, 2010) is an attractive collection of 32 stylized black-line drawings for ages 4 and up. (Check out Harper's art at the Charley Harper Gallery.)
Jacqueline Davies's The Boy Who Drew Birds (Houghton Mifflin Harcourt, 2004) – with wonderful illustrations by Melissa Sweet – is the picture-book story of John James Audubon, perhaps the world's best-known painter of birds. For ages 5-9.
Audubon's Birds of America Coloring Book (Dover Publications, 1974) has black-line versions of 44 of Audubon's famous bird paintings.
Inspired by a painting by Peter Breugel, Stepanie Girel's A Bird in Winter (Prestel Publishing, 2011) is the story of Mayken, an eight-year-old peasant girl, who – while ice-skating with friends – finds an injured bird and nurses it back to health. (Included is a beautiful reproduction of Breugel's "The Hunters in the Snow.") For ages 4-8.
Geraldine Elschner's The Cat and the Bird (Prestel Publishing, 2012) – inspired by and illustrated in the style of artist Paul Klee – is the tale of a little cat who, despite a lovely home filled with toys, envies the freedom of the bird. Then one day the bird manages to set the cat free, and at the end the cat is dancing joyfully on the roof in the moonlight. For ages 5-8. |
Master medical calculations in an engaging environment! In these fun and practical lessons, you(TM)ll gain the medical math skills you need for anything from calculating dosages to using scientific formulas. Whatever medical field you(TM)re in, the hands-on activities in this course will help you perform day-to-day math tasks quickly and easily.
First, you'll brush up your basic math skills. You'll begin with a review of fractions, decimals, and percentages, and then dive into measurement systems and conversions used in the medical field.
Next, youll do dosage calculations for oral, parenteral, and intravenous medications. You'll explore three different methods you can use for dosage calculations: proportions, dimensional analysis, and the formula method. You'll also learn an easy formula that you can apply to many dosage calculations.
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Whether you're new to the field of medicine or want to enhance your skills, this is the course for you. By the time you finish these lessons, you'll have a solid grounding in basic medical math, and you'll be ready to tackle any calculation confidently. |
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