# Euler integral

(Redirected from Euler integral (disambiguation))

In mathematics, there are two types of Euler integral:[1]

1. Euler integral of the first kind: the beta function
${\displaystyle \mathrm {\mathrm {B} } (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}}$
2. Euler integral of the second kind: the gamma function
${\displaystyle \int _{0}^{\infty }t^{z-1}\,\mathrm {e} ^{-t}\,dt}$

For positive integers m and n

${\displaystyle \mathrm {B} (n,m)={(n-1)!(m-1)! \over (n+m-1)!}={n+m \over nm{n+m \choose n}}}$
${\displaystyle \Gamma (n)=(n-1)!}$