Gamma Function

Definition

$\Gamma (\alpha ):={\int }_0^{\infty }{y}^{\alpha -1}{e}^{-y}dy$ where $\alpha \in {\mathbb{R}}^{+}$

$\Gamma (s):={\int }_0^1(-\log u{)}^{s-1}du$ where $s\in {\mathbb{R}}^{+}$

Gamma function is used as an extension of the factorial function

Properties

$\forall 1<\alpha \in \mathbb{R},\Gamma (\alpha )=(\alpha -1)\Gamma (\alpha -1)$

$\forall \alpha \in \mathbb{N},\Gamma (\alpha )=(\alpha -1)!$

$\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$