Gamma function

$\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt$ for $\operatorname{Re}(z)>0$ , extending factorial to complex $z$ .

The Gamma function is defined for $\operatorname{Re}(z)>0$ by the integral $\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt$ , and it extends by analytic continuation to a function defined on the whole complex plane except at the nonpositive integers, where it has simple poles. Its defining feature is that it interpolates the factorial, $\Gamma(n)=(n-1)!$ for every positive integer $n$ , so $\Gamma$ answers the question of what " $z!$ " should mean for a complex number that is not a nonnegative integer.

Examples

$\Gamma(1)=\int_0^\infty e^{-t}\,dt=1$ , matching $0!=1$ .
$\Gamma(5)=4!=24$ , and $\Gamma(1/2)=\sqrt{\pi}$ .