# The gamma and beta function

## The gamma and beta function

$\displaystyle {\displaystyle \index{Gamma function} \EulerGamma@{z}:=\int_0^{\infty}t^{z-1}\expe^{-t}\,dt }$

Constraint(s): $\displaystyle {\displaystyle \realpart{z}>0}$

$\displaystyle {\displaystyle \EulerGamma@{z+1}=z\EulerGamma@{z}\quad\textrm{with}\quad\EulerGamma@{1}=1 }$
$\displaystyle {\displaystyle \EulerGamma@{z}\EulerGamma@{1-z}=\frac{\cpi}{\sin@{\cpi z}} }$

Constraint(s): $\displaystyle {\displaystyle z\notin\mathbb{Z}}$

$\displaystyle {\displaystyle \int_{-\infty}^{\infty}\expe^{-x^2}\,dx=2\int_0^{\infty}\expe^{-x^2}\,dx =\int_0^{\infty}t^{-1/2}\expe^{-t}\,dt=\EulerGamma@{1/2}=\sqrt{\cpi} }$
$\displaystyle {\displaystyle \int_{-\infty}^{\infty}\expe^{-\alpha^2x^2-2\beta x}\,dx =\sqrt{\frac{\cpi}{\alpha^2}}\,\expe^{\beta^2/\alpha^2} \quad\alpha,\beta\in\mathbb{R} }$

Constraint(s): $\displaystyle {\displaystyle \alpha\neq 0}$

$\displaystyle {\displaystyle \EulerGamma@{z}\EulerGamma@{z+1/2}=2^{1-2z}\sqrt{\cpi}\,\EulerGamma@{2z},\quad z\in\mathbb{C} }$

Constraint(s): $\displaystyle {\displaystyle 2z\neq 0,-1,-2,\ldots}$

$\displaystyle {\displaystyle \EulerGamma@{z}\sim\sqrt{2\cpi}\,z^{z-1/2}\expe^{-z} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{z}\rightarrow\infty}$

$\displaystyle {\displaystyle \EulerGamma@{x+\iunit y}\sim\sqrt{2\cpi}\,|y|^{x-1/2}\expe^{-|y|\cpi/2} }$

Constraint(s): $\displaystyle {\displaystyle |y|\rightarrow\infty}$

$\displaystyle {\displaystyle \frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\sim z^{a-b},\quad a,b\in\mathbb{C} }$

Constraint(s): $\displaystyle {\displaystyle |z|\rightarrow\infty}$

$\displaystyle {\displaystyle \index{Beta function} \EulerBeta@{x}{y}:=\int_0^1t^{x-1}(1-t)^{y-1}\,dt,\quad\realpart{x}>0 }$

Constraint(s): $\displaystyle {\displaystyle \realpart{y}>0}$

$\displaystyle {\displaystyle \EulerBeta@{x}{y}=\frac{\EulerGamma@{x}\EulerGamma@{y}}{\EulerGamma@{x+y}} \quad\realpart{x}>0 }$

Constraint(s): $\displaystyle {\displaystyle \realpart{y}>0}$

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-\infty}^{\infty}\frac{dt}{(r+\iunit t)^{\rho}(s-\iunit t)^{\sigma}} =\frac{(r+s)^{1-\rho-\sigma}\EulerGamma@{\rho+\sigma-1}}{\EulerGamma@{\rho}\EulerGamma@{\sigma}} }$