Gamma function

The Gamma function or Euler's Gamma function Γ(z)\Gamma(z) is a complex-valued function defined over the entire complex plane. It is the analytic continuation of the factorial function and is defined as

Γ(z)=∫0∞wz−1e−wdw\Gamma(z)=\int_{0}^{\infty}w^{z-1}e^{-w}dw

Some notable values are

Γ(12)=π,Γ(1)=1,Γ(z+1)=zΓ(z),Γ(n)=(n−1)!\Gamma\left( \frac{1}{2} \right)=\sqrt{ \pi },\quad \Gamma(1)=1,\quad \Gamma(z+1)=z\Gamma(z),\quad \Gamma(n)=(n-1)!