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Another useful limit for asymptotic approximations is : The derivatives of the gamma function are described in terms of the polygamma function . | The derivatives of the gamma function are described in terms of the polygamma function . |
Another champion for that title might be Both formulas were derived by Bernhard Riemann in his seminal 1859 paper '' Über die Anzahl der Primzahlen unter einer gegebenen Grösse '' ( `` On the Number of Prime Numbers less than a Given Quantity '' ) , one of the milestones in the development of analytic number theory - the branch of mathematics that studies prime numbers using the tools of mathematical analysis . | This was in 1859 paper '' Über die Anzahl der Primzahlen unter einer gegebenen Grösse '' ( `` On the Number of Prime Numbers less than a Given Quantity '' ) |
For example : For positive integer m the derivative of gamma function can be calculated as follows ( here γ is the Euler-Mascheroni constant ) : The n-th derivative of the gamma function is : The gamma function has simple poles at z = − n = 0 , − 1 , − 2 , − 3 , ... The residue there is Moreover , the gamma function has the following Laurent expansion in 1 valid for z − 1 ( 1 . | For example : For positive integer m the derivative of gamma function can be calculated as follows ( here γ is the Euler-Mascheroni constant ) : The n-th derivative of the gamma function is : Some particular values of the gamma function are : Gauss introduced the Pi function . |
That is , if n is a positive integer : The gamma function is defined for all complex numbers except the negative integers and zero . | This means that if n is a positive integer The gamma function is defined for all complex numbers . |
If the real part of the complex number t is positive ( Re ( t ) ) 0 ) , then the integral converges absolutely , and is known as the Euler integral of the second kind ( the Euler integral of the first kind defines the Beta function ) . | It is known as the Euler integral of the second kind . |
denotes the double factorial and , when n = 0 , n !! | denotes the double factorial and , when n = 0 , ( -1 ) !! |
An elegant and deep application of the gamma function is in the study of the Riemann zeta function . | In terms of the gamma function , the Pi function is so that for every non-negative integer n . The gamma function is used to study the Riemann zeta function . |
In mathematics , the gamma function ( represented by the capital Greek letter Γ ) is an extension of the factorial function , with its argument shifted down by 1 , to real and complex numbers . | In mathematics , the gamma function ( Γ ( z ) ) is an extension of the factorial function to positive real and complex numbers . |
Other important functional equations for the gamma function are Euler 's reflection formula which implies and the duplication formula The duplication formula is a special case of the multiplication theorem A simple but useful property , which can be seen from the limit definition , is : Perhaps the best-known value of the gamma function at a non-integer argument is which can be found by setting z = 1\/2 in the reflection or duplication formulas , by using the relation to the beta function given below with x = y = 1\/2 , or simply by making the substitution u = √ t in the integral definition of the gamma function , resulting in a Gaussian integral . | It says : This means The gamma function also has a duplication formula : This formula is a special case of the multiplication theorem A property of the function is that : A famous value of the function at a non-integral point is : This is found if we set z = 1\/2 in the reflection or duplication formulas . |
A fundamental property of the Riemann zeta function is its functional equation : Among other things , this provides an explicit form for the analytic continuation of the zeta function to a meromorphic function in the complex plane and leads to an immediate proof that the zeta function has infinitely many so-called `` trivial '' zeros on the real line . | A property of the Riemann zeta function is its functional equation : Bernhard Riemann found a relation between these two functions . |
There are a number of depictions of this parable , the most famous being The Importunate Neighbour ( 1895 ) by William Holman Hunt , held in the National Gallery of Victoria , Australia . | The most famous depiction of this parable is The Importunate Neighbour ( 1895 ) by William Holman Hunt , kept at the National Gallery of Victoria , Australia . |
As with verses 9-13 , the parable is therefore an incentive to pray . | The parable is therefore an incentive to pray . |
The Parable of the Friend at Night ( or of the Importunate Neighbour ) , is a parable of Jesus , which appears in only one of the Canonical gospels of the New Testament . | The Parable of the Friend at Night ( Importunate Neighbour ) , is a parable of Jesus written in the New Testament in Luke 11:5 -8 . |
This parable appears in the Gospel of Luke immediately after Jesus teaches the Lord 's Prayer , and can therefore be viewed as a continuation of Jesus teaching his disciples how to pray , while the verses which follow help to explain the meaning of the parable : Joel B. Green suggests that the question that opens the parable ( `` Which of you who has a friend ... ? '' | Jesus told this parable after he taught his disciples how to pray the Lord 's Prayer . |
This is the best study area in the world for beaked whales . | This is the best study area for beaked whales in the world . |
It lies along the western coast of France from Brest south to the Spanish border , and the northern coast of Spain west to Cape Ortegal . | It is along the western coast of France from Brest south to the Spanish border , and the northern coast of Spain west to Cape Ortegal . |
Most importantly , it is one of the few places where the beaked whales , such as the Cuvier 's beaked whale , have been observed relatively frequently . | It is one of the few places where the beaked whales such as the Cuvier 's beaked whale have been seen frequently . |
The Bay of Biscay is home to some of the Atlantic Ocean 's fiercest weather . | The Bay of Biscay is home to some of the Atlantic Ocean 's baddest weather . |
Large storms occur in the bay , especially during the winter months . | Large storms happen in the bay , especially during the winter months . |
The average depth is 1,744 metres ( 5,722 ft ) and the greatest depth is 4,735 metres ( 15,535 ft ) . | The average depth is 1,744 metres ( 5,722 ft ) and maximum depth is 2,789 metres ( 9,150 ft ) . |
The best areas to see the larger cetaceans lie in the deep waters beyond the continental shelf , particularly over the Santander Canyon and Torrelavega Canyon in the south of the Bay . | The best areas to see the larger cetaceans are in the deep waters beyond the continental shelf . |
Many species of whales and dolphins can be seen in this area . | The main rivers that empty into the Bay of Biscay are : The main cities on the shores of the Bay of Biscay are : Many species of whales and dolphins can be seen in this area . |
Currently , there are five Famitsū magazines : Shūkan Famitsū , Famitsū PS3 + PSP , Famitsū Xbox 360 , Famitsū DS+W ii , and Famitsū Wave DVD . | Currently , there are five Famitsū magazines : Shūkan Famitsū , Famitsū PS3 + PSP , Famitsū Xbox 360 , Famitsū Wii + DS , and Famitsū Wave DVD . |
, formerly Famicom Tsūshin ) is a line of Japanese video game magazines published by Enterbrain , Inc. and Tokuma . | , formerly Famicom Tsūshin ) is a Japanese language video game magazine . |
Offa died in 796 ; his son , Ecgfrith , succeeded him , but reigned for less than five months before Coenwulf of Mercia became king . | Offa died in 796 and was succeeded by his son , Ecgfrith , who reigned for less than five months before Coenwulf of Mercia became king . |
Offa reigned as King of Mercia from 757 until his death in July 796 . | Offa was the King of Mercia from 757 until his death in July 796 . |
He was succeeded by his son , Ecgfrith , but according to the Anglo-Saxon Chronicle Ecgfrith died after a reign of only 141 days . | His son , Ecgfrith , lasted only 141 days as king . |
Offa died on 29 July 796 , and was buried in Bedford . | Offa died in 796 . |
Native to China and Korea , there are two subspecies : the Chinese water deer ( Hydropotes inermis inermis ) and the Korean water deer ( Hydropotes inermis argyropus ) . | There are two subspecies : the Chinese Water Deer ( Hydropotes inermis inermis ) and the Korean Water Deer ( Hydropotes inermis argyropus ) . |
The water deer ( Hydropotes inermis ) is a small deer superficially more similar to a musk deer than a true deer . | The Water Deer ( Hydropotes inermis ) is an even-toed ungulate mammal of the deer family . |
The first three were largely because of the all-star play of Ken Dryden . | The first three were mostly because of the all-star play of Ken Dryden . |
During that era , the trophy was given to the principal goalies on the team allowing the fewest goals in the regular season . | During that time , the trophy was given to the main goaltenders on the team allowing the fewest goals in the regular season . |
Michel Raymond `` Bunny '' Larocque ( April 6 , 1952 - July 29 , 1992 ) was a Canadian professional ice hockey goaltender who played for the Montreal Canadiens , Toronto Maple Leafs , Philadelphia Flyers and St. Louis Blues in the National Hockey League . | Michel ` Bunny ' Larocque ( April 6 , 1952 in Hull , Quebec , Canada - July 29 , 1992 ) was a Canadian professional ice hockey goaltender who played for the Montreal Canadiens , Toronto Maple Leafs , Philadelphia Flyers , and St. Louis Blues in the National Hockey League . |
Larocque would be co-winner of four Vezina Trophies . | Larocque would help win four Vezina Trophies . |
His fourth and final Vezina Trophy was shared with Denis Herron and Richard Sevigny in 1981 . | The final trophy was with Denis Herron and Richard Sevigny in 1981 . |
He won a total of four Stanley Cups with Montreal in 1976 , 1977 , 1978 and 1979 . | He would win four Stanley Cups with Montreal in 1976 , 1977 , 1978 , and 1979 . |
From 1438 to 1533 , the Incas used a variety of methods , from conquest to peaceful assimilation , to incorporate a large portion of western South America , centered on the Andean mountain ranges , including , besides Peru , large parts of modern Ecuador , western and south central Bolivia , northwest Argentina , north and central Chile , and a small part of southern Colombia into a state comparable to the historical empires of Eurasia . | From 1438 to 1533 , the Incas used conquest and non-violent assimilation to gain a large portion of western South America , that centered on the Andean mountain ranges , and included large parts of what is now Ecuador , Peru , Bolivia , Argentina and Chile . |
The administrative , political and military center of the empire was located in Cusco in modern-day Peru . | This was the administrative , political and military center of the empire . |
The Incas considered their King , the Sapa Inca , to be the `` child of the sun . '' | The Inca were ruled by an Emperor known as the Sapa Inca . |
The line of succession to the Japanese throne is the list of all people who may become Emperor of Japan . | The line of succession to the Japanese throne is the list of people who may become Emperor of Japan . |
In October 2009 , he told Wine Spectator magazine that he had been diagnosed with a heart condition . | In October 2009 , he told Wine Spectator magazine that he has been diagnosed with a heart condition . |
( \ / ˈʃɔːn ˈkɒnəri \ / ; 25 August 1930 ) is a Scottish actor and producer who has won an Academy Award , two BAFTA Awards ( one of them being a BAFTA Academy Fellowship Award ) and three Golden Globes ( including the Cecil B. DeMille Award and a Henrietta Award ) . | He has won an Academy Award , two BAFTA Awards ( one of them a BAFTA Academy Fellowship Award ) and three Golden Globes ( including the Cecil B. DeMille Award and a Henrietta Award ) . |
Thomas Sean Connery , named Thomas after his grandfather , was born in Fountainbridge , Edinburgh , Scotland . | He was born in Fountainbridge , Edinburgh . |
The becquerel ( symbol Bq ) ( pronounced : ` be-kə-rel ) is the SI derived unit of radioactivity . | The becquerel ( symbol Bq ) is the SI derived unit of radioactivity , defined as the activity of a quantity of radioactive material in which one nucleus decays per second . |
This SI unit is named after Henri Becquerel . | The unit is named in honor of Henri Becquerel . |
By the late 1950s , Frederik Pohl was helping Gold with most aspects of the magazine 's production . | By the late 1950s , Frederik Pohl was helping Gold with the magazine . |
Under Pohl Galaxy had continued success , regularly publishing fiction by writers such as Cordwainer Smith , Jack Vance , Harlan Ellison , and Robert Silverberg . | Pohl often published fiction by important writers such as Cordwainer Smith , Jack Vance , Harlan Ellison , and Robert Silverberg . |
Galaxy Science Fiction was an American digest-size science fiction magazine , published from 1950 to 1980 . | Galaxy Science Fiction was an American science fiction magazine . |
However , Pohl never won the annual Hugo Award for his stewardship of Galaxy , winning three Hugos instead for its sister magazine , If . | However , Pohl never won the annual Hugo Award for his work at Galaxy . |
In 1969 Guinn sold Galaxy to Universal Publishing and Distribution Corporation ( UPD ) and Pohl resigned , to be replaced by Ejler Jakobsson . | In 1969 , Galaxy was sold to Universal Publishing and Distribution Corporation ( UPD ) and Pohl quit . |
Gold published many notable stories during his tenure , including Ray Bradbury 's `` The Fireman '' , later expanded as Fahrenheit 451 ; Robert A. Heinlein 's The Puppet Masters ; and Alfred Bester 's The Demolished Man . | Gold published many well-known and important stories , including : Robert A. Heinlein 's The Puppet Masters and Alfred Bester 's The Demolished Man . |
Galaxy was quickly established as one of the three leading science fiction magazines , along with Campbell 's Astounding and The Magazine of Fantasy and Science Fiction ( usually abbreviated to F&SF ) . | He made Galaxy the leading science fiction magazine of its time . |
H. L. Gold , Galaxy 's first editor , had worked at Standard Magazines in the early 1940s as an assistant editor , reading for Standard 's three science fiction pulps : Startling Stories , Thrilling Wonder , and Captain Future . | H. L. Gold was the first editor . |
The person who decides the amount of the mixing parts is called the murage . | The man who decides the amount of the mixing parts is called the murage . |
The word tama means `` round and precious '' , like a gem . | The word tama means `` round and precious '' , like a gem or jewel . |
The clay tub is heated to a high temperature , about 1000 ºC ( 1832 ºF ) . | The clay tub is heated to a high temperature , about 1000 ºC ( 1.832 ºF ) . |
Tamahagane is used to make Japanese swords , knives , and other kinds of tools . | Tamahagane steel is used to make Japanese swords , knives , and other kinds of tools . |
Tamahagane ( 玉鋼 ) is a high quality metal ( steel ) made in the Japanese tradition . | Tamahagane { tah-mah-ha-ga-nay } is a high quality metal ( steel ) made in the Japanese tradition . |
Tamahagane is made of iron sand . | Tamahagane is made of iron sand ( called satetsu in Japanese ) . |
The iron sand is put in a tatara , a clay tub furnace . | The iron sand is put in a clay tub ( furnace ) called a tatara . |
U.S. House of Representatives Joint Resolution 71 was approved by a vote of 407-0 on October 25 , 2001 . | The U.S. House of Representatives Joint Resolution 71 was approved by a vote of 407-0 . |
President George W. Bush signed the resolution into law on December 18 , 2001 ( as Pub . | President George W. Bush signed the resolution into law on December 18 , 2001 ( as Public Law 107-89 ) . |
On September 4 , 2002 , President Bush used his authority created by the resolution and proclaimed September 11 , 2002 as Patriot Day . | On September 4 , 2002 , President Bush proclaimed September 11 , 2002 as Patriot Day . |
It requested that the President designate September 11 of each year as `` Patriot Day '' . | The Resolution asked that the President would designate September 11 of each year as `` Patriot Day . '' |
Additionally , a moment of silence is observed to correspond with the attacks , beginning at 8:46 A.M. ( Eastern Daylight Time ) , the time the first plane struck the North Tower of the World Trade Center on September 11 , 2001 . | This is the time the first plane struck the North Tower of the World Trade Center on September 11 , 2001 . |
By presidential decree , the American flag is flown at half-staff at the White House and on all U.S. government buildings and establishments throughout the world ; flags are also encouraged to be displayed on individual American homes . | On this day , the American flag is to be flown at half-staff at individual American homes , at the White House , and on all U.S. government buildings , home and abroad . |
In the United States , Patriot Day ( known in full as Patriot Day and National Day of Service and Remembrance ) occurs on September 11 of each year , designated in memory of the 2,977 killed in the September 11 , 2001 terrorist attacks . | Patriot Day is a day of remembrance in the United States . |
Impressed with the group 's material , he brought the trio to the attention of Lorne Michaels , the producer of Saturday Night Live , who agreed to see them . | Impressed with their material , he brought them to the attention of Lorne Michaels , the producer of Saturday Night Live , who agreed to see them . |
He was called back for a second audition a few weeks later and was ultimately invited to be a cast member ; Schaffer and Taccone were invited to join the writing staff based on their work in the trio . | He was the only member of TLI who had also been doing stand-up comedy in his free time He was called back for a second audition a few weeks later and was invited to be a cast member ; Schaffer and Taccone were invited to join the writing staff based on their work in the trio . |
Originally from Berkeley , California , the group is currently based in New York City . | They are from Berkeley , California , but are now based in New York City . |
The Lonely Island is an American comedy group formed by Akiva Schaffer , Andy Samberg , and Jorma Taccone , best known for their comedic music . | The people are Andy Samberg , Jorma Taccone , and Akiva Schaffer . |
Since their arrival , the group has written some popular bits in recent SNL history , most notably , the SNL Digital Shorts ( collectively ) and MacGruber ( Taccone , with John Solomon and cast member Will Forte ) . | Since their arrival on SNL , the group has written some of the most popular bits in recent SNL history , most notably , the SNL Digital Shorts . |
Following the success of `` Lazy Sunday '' , the group produced `` Dick in a Box '' , `` Jizz in My Pants '' , `` Like a Boss '' and `` I 'm on a Boat '' , which have subsequently had huge success both on SNL and on the internet , and prompted the release of the 2009 album , Incredibad . | They are known for songs such as `` I 'm On A Boat '' , `` Like a Boss '' , `` Jizz in My Pants '' , `` Dick in a Box '' , and `` Lazy Sunday '' . |
The Proud Family was created by Bruce W. Smith and was produced by Jambalaya Studios . | The Proud Family was created by Bruce W. Smith . |
Many of the later episodes of The Proud Family were produced using Adobe Flash . | Some of the episodes in later seasons were produced using Adobe Flash . |
The Proud Family is an American animated television sitcom that ran on the Disney Channel from September 15 , 2001 to December 31 , 2005 . | It was on the Disney Channel from September 15 , 2001 to August 19 , 2005 . |
The series is the first Disney children 's cartoon which did not premiere on network\/over-the-air television , which had been done since 1984 when the Television Animation unit was started . | This had been done since 1985 when the Television Animation unit was started . |
Originally piloted for Nickelodeon , it was eventually picked up by the Disney Channel and started airing in September 2001 . | However , Disney Channel took it and began airing in September 2001 . |
The production also overran in its shooting schedule and had to be out of the Fox Studios in Sydney to make way for Star Wars Episode II : Attack of the Clones ( in which McGregor also starred ) . | The production also overran in its shooting schedule and had to be out of the Fox Studios in Sydney to make way for Star Wars Episode II : Attack of the Clones ( in which Ewan McGregor also starred ) . |
Filming generally went smoothly , with the only major problem occurring when Kidman injured her ribs while filming one of the more complicated dance sequences . | Filming generally went smoothly , with the only major problem being when Nicole Kidman injured her knee while filming one of the more complicated dance sequences . |
Production began in November 1999 and was completed in May 2000 , with a budget of $ 52.5 million . | Production on the movie began in November 1999 and was finished in May 2000 , with a budget of just over $ 50 million . |
It tells the story of a young English poet\/writer , Christian ( Ewan McGregor ) , who falls in love with the terminally-ill star of the Moulin Rouge , cabaret actress and courtesan Satine ( Nicole Kidman ) . | It tells the story of a young British writer called Christian , who falls in love with an actress called Satine , the leading lady of the famous nightclub , the Moulin Rouge . |
São Paulo is the capital of the state of São Paulo , Brazil 's most populous state . | São Paulo ( Portuguese for Saint Paul ) is the capital of the state of São Paulo , Brazil . |
Obesity increases the risk of many physical and mental conditions . | Obesity increases the chance of getting heart disease . |
Some research shows that obese people are less likely to be hired for a job and are less likely to be promoted . | But obese people with heart disease are less likely to die in a 7 year period . |
People are considered obese when their body mass index ( BMI ) , a measurement obtained by dividing a person 's weight by the square of the person 's height , exceeds 30 kg\/m2 . | To know if a person is overweight , the body mass index ( BMI ) is calculated , by dividing the person 's weight ( in kilograms ) , by their height ( in meters ) squared ( multiplied by itself ) . |
Yoshi has a variety of abilities that make him stand out amongst other characters in the Mario series . | Yoshi is one of the more famous characters in the Mario series . |
Yoshis come in a variety of colors , including green , red , blue , yellow , cyan , pink , purple , black , white , orange , and brown . | These other colors include green , red , light blue , dark blue , pink , yellow , black , and white . |
Chimpanzees have been kept as pets for centuries in a few African villages , especially in the Democratic Republic of Congo . | The bonobo lives in the rain forests of the Democratic Republic of the Congo . |
In the late 19th century August Mann 's concerts were exploring works by well-known composers : Brahms , Liszt , Mendelssohn , Schubert , Schumann , Smetana and Wagner . | Later in the 19th century the programs included more music by famous composers such as Brahms , Liszt , Mendelssohn , Schubert , Schumann , Smetana and Wagner . |
By 1895 the time was ripe for Henry Wood to start his series of promenade concerts which continue today as the BBC Proms . | In 1895 Henry Wood started his series of promenade concerts . |
His orchestra were often joined by the bands of the Royal Artillery or drummers from the French Garde Nationale . | His orchestra were even joined by the bands of the Royal Artillery or drummers from the French National Guard . |
Choruses from Handel 's oratorios were often included . | Sometimes choirs sang choruses from oratorios by Handel . |
Although the term Promenade Concert is normally associated today with the series of concerts founded in 1895 by Robert Newman and the conductor Henry Wood - a festival known today as the BBC Proms - the term originally referred to concerts in the pleasure gardens of London where the audience could stroll about while listening to the music ( French se promener = to walk ) . | These concerts took place in the leisure gardens ( parks ) where people could walk about while listening to the music ( French se promener = to walk ) . |
The songs were often on pastoral subjects , or drinking songs , hunting songs or even songs on morbid subjects . | The music was often long songs ( ballads ) , or drinking songs . |
Other entertainments would have included fireworks , masquerades and acrobatics . | There were lots of entertainments there , e.g. fireworks , dancing , acrobatics and music . |
Soon there was also a series of '' Concerts d'hiver '' under Louis Antoine Jullien ( 1812-1860 ) . | The conductor Louis Antoine Jullien conducted many concerts . |
Another notable conductor was August Manns ( 1825-1907 ) who is associated with the Saturday concerts at London 's Crystal Palace , the enormous glass building which housed the Great Exhibition in 1851 . | Concerts were also given in Crystal Palace , the huge glass building which was built for the Great Exhibition in 1851 . |
However , gestures can have different meanings according to the country in which they are expressed . | Gestures have a cultural significance ( meaning ) . |
Gesture processing takes place in areas of the brain such as Broca 's and Wernicke 's areas , which are used by speech and sign language . | In the neocortex , Broca 's and Wernicke 's areas , are used by speech and sign language . |
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