Gamma function
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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