Euler integral

In mathematics, there are two types of Euler integral:

1. Euler integral of the first kind: the Beta function
   $\mathrm{\Beta}(x,y)= \int_0^1t^{x-1}(1-t)^{y-1}\,dt =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$
2. Euler integral of the second kind: the Gamma function
   $\Gamma(z) = \int_0^\infty t^{z-1}\,e^{-t}\,dt$

For positive integers m and n

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

$\Gamma(n) = (n-1)! \,$

See also

• Leonhard Euler
• List of topics named after Leonhard Euler