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]]s.
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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 <math>\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dz</math>. 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.
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For <math>\Re(z)>0</math>, we define <cmath>\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dz</cmath>  
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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.
  
 
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[[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 $\Re(z)>0$, we define \[\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dz\] It is easy to check with integration by parts that $\Gamma(z+1)=z\Gamma(z)$. This is almost the same as the factorial identity $(n+1)!=(n+1)n!$, but it is off by one. Since $\Gamma(1)=1$, we therefore have $\Gamma(n+1)=n!$ for nonnegative integers $n$. But with the integral, we can define the $\Gamma$ 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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