Gamma function

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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Gamma function

Definition