Gamma Function

$\Gamma (z)=(z-1)!={\int }_0^{\infty }{t}^{z-1}{e}^{-t}\mathrm{d}t$

• This is the analytic continuation of the factorial function

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

• Proven by integration by parts
• Like a factorial

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

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

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

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