Gamma function

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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