1 Gamma Function
The Gamma function is the most common extension of the [[factorial-function]] to [[complex-numbers]]. It appears of
The gamma function of a complex number \(z\) with strictly positive real component \(\mathfrak{R}(z)>0\) is defined by the integral \[ \Gamma(z)=\int_{0}^{\infty}{t^{z-1}e^{-t}}~d{t}. \]
This is known as the Euler integral of the second kind and can be shown to converge absolutely. Note that Euler’s integral of the first kind is the [[beta-function]].
For any positive integer \(z\in \mathbb{Z}_{+}\) we have that \[ \Gamma(z)=(z-1)!. \]
Proof: Using integration by parts we can compute \[ \begin{align} \Gamma(z+1) & =\int_{0}^{\infty}{t^ze^{-t}}~d{t} \\ & = [-t^ze^{-t}]_{0}^\infty+\int_{0}^{\infty}{zt^{z-1}e^{-t}}~d{t} \\ & = \underbrace{\lim_{ t \to \infty } (-t^ze^{-t})}_{=0}-\underbrace{(0^ze^{-0})}_{=0}+\underbrace{z\int_{0}^{\infty}{t^{z-1}e^{-t}}~d{t}}_{=z\Gamma(z)} \\ & =z\Gamma(z). \end{align} \] Furthermore, we can compute \[ \Gamma(1)=\int_{0}^{\infty}{t^0e^{-t}}~d{t}=\int_{0}^{\infty}{e^{-t}}~d{t}=1, \] hence the result follows by induction. \(\square\)