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 ![\[\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}\; dt\]](http://latex.artofproblemsolving.com/1/9/c/19ca723f97e03c22e78c09cc9670957fa7cdcf8c.png)
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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