Gamma Function

The gamma function $\Gamma (x)$ is an extension of the factorial function to real (and complex) numbers. Specifically, if $n\in \{1,2,3,\dots \}$, then $\Gamma (n)=(n-1)!$ For any positive real number $\alpha$, we define $\Gamma (\alpha$) as $\Gamma (\alpha )={\int }_0^{\infty }{x}^{\alpha -1}{e}^{-x}dx,\alpha >0$

Properties of the Gamma Function

For any positive real number $\alpha$:

1. $\Gamma (\alpha )={\int }_0^{\infty }{x}^{\alpha -1}{e}^{-x}dx$
2. ${\int }_0^{\infty }{x}^{\alpha -1}{e}^{-\lambda x}dx=\frac{\Gamma (\alpha )}{{\lambda }^{\alpha }},\text{for }lambda>0$
3. $\Gamma (\alpha +1)=\alpha \Gamma (\alpha )$
4. $\Gamma (n)=(n-1)!,\text{for }n=1,2,3,\dots$
5. $\Gamma (\frac{1}{2})=\sqrt{\pi }$

Related

• Gamma Distribution