# Difference between revisions of "Gamma function"

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− | The '''Gamma function''' is a generalization of the notion of a [[factorial]] to [[complex number]] | + | The '''Gamma function''' is a generalization of the notion of a [[factorial]] to [[complex number|complex numbers]]. |

== Definition == | == Definition == | ||

− | For <math>\Re(z)>0</math>, we define < | + | For <math>\Re(z)>0</math>, we define <cmath>\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dz</cmath> |

+ | It is easy to check with [[integration by parts]] that <math>\Gamma(z+1)=z\Gamma(z)</math>. This is almost the same as the factorial identity <math>(n+1)!=(n+1)n!</math>, but it is off by one. Since <math>\Gamma(1)=1</math>, we therefore have <math>\Gamma(n+1)=n!</math> for nonnegative integers <math>n</math>. But with the integral, we can define the <math>\Gamma</math> function for other complex numbers. We can then use the identity to extend the Gamma function to a [[meromorphic]] function on the full [[complex plane]], with simple poles at the nonpositive integers. | ||

{{stub}} | {{stub}} | ||

+ | [[Category:Definition]] |

## Revision as of 21:43, 21 June 2009

The **Gamma function** is a generalization of the notion of a factorial to complex numbers.

## Definition

For , we define It is easy to check with integration by parts that . This is almost the same as the factorial identity , but it is off by one. Since , we therefore have for nonnegative integers . But with the integral, we can define the function for other complex numbers. We can then use the identity to extend the Gamma function to a meromorphic function on the full complex plane, with simple poles at the nonpositive integers.

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