Gamma Function

#Math

$\displaystyle \Gamma(z)=(z-1)! =\int_{0}^{\infty} t^{z-1}e^{-t} , \mathrm{d}t$

• This is the analytic continuation of the factorial function

$\displaystyle \Gamma(z+1)=z\Gamma(z)$

• Proven by integration by parts
• Like a factorial

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

• Can be shown by raw computation or by techniques in complex analysis

$\displaystyle \Gamma\left( \frac{2n+1}{2} \right)=\sqrt{ \pi }\prod_{i = 1}^{n} \left( \frac{2i-1}{2} \right)$

• Combines $\displaystyle \Gamma(z+1)=z\Gamma(z)$ with $\displaystyle \Gamma\left( \frac{1}{2} \right)=\sqrt{ \pi }$