Gamma function

$\Gamma (z):={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt$

Properties

$$\begin{array}{rll}\Gamma (n)& =(n-1)!& \forall n\in {\mathbb{Z}}^{+}\\ \Gamma (z+1)& =z\Gamma (z)& \text{Euler’s reflection formula}\\ \Gamma (z)\Gamma (z+\frac{1}{2})& =2^{1-2x}\sqrt{\pi }\Gamma (2z)& \text{Legendre Duplication Formula}\end{array}$$

where

• $n!$ is the factorial function