Gamma function

Function

definition $\Gamma: \mathbb C\setminus\{-k\ |\ k\in\mathbb N\}\to \mathbb N$
definition $\Gamma(z) := \begin{cases} \int_0^\infty\ \ t^{z-1}\ \mathrm{e}^{-t}\ \mathrm d t & \mathrm{if}\ \mathrm{Re}(z)>0 \\\\ \frac{1}{z}\Gamma(z+1) & \mathrm{else} \end{cases}$

Discussion

$\Gamma(z)=\Pi(z-1)$

Theorems

$n\in\mathbb N\land n\neq 0 \implies \Gamma(n)=(n-1)! $
$\Gamma(z+1) = z\cdot\Gamma(z) $
$\Gamma(z)\cdot\Gamma(1-z) =\frac{\pi}{\sin(\pi\ z)} $
$\Gamma(z)\cdot\Gamma(z+1/2)=2^{1-2z}\ \pi^{1/2}\ \Gamma(2z) $

Reference

Wikipedia: Gamma function

Context

Function integral on ℝⁿ, Complex exponents with positive real bases

Equivalent to

Pi function

Factorial function