# Difference between revisions of "Gamma function"

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(According to Wolfram Mathworld, the integral is with respect to t NOT z. http://mathworld.wolfram.com/GammaFunction.html) |
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== Definition == | == Definition == | ||

− | For <math>\Re(z)>0</math>, we define <cmath>\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; | + | For <math>\Re(z)>0</math>, we define <cmath>\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dt</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. | 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]] | [[Category:Definition]] |

## Latest revision as of 22:26, 22 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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