Euler integral

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

1. The Euler integral of the first kind is 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. The Euler integral of the second kind is the gamma function

\({\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}\,\mathrm {e} ^{-t}\,dt} \)

For positive integers m and n, the two integrals can be expressed in terms of factorials and binomial coefficients:

\( {\displaystyle \mathrm {B} (n,m)={\frac {(n-1)!(m-1)!}{(n+m-1)!}}={\frac {n+m}{nm{\binom {n+m}{n}}}}=\left({\frac {1}{n}}+{\frac {1}{m}}\right){\frac {1}{\binom {n+m}{n}}}} \)
\( {\displaystyle \Gamma (n)=(n-1)!} \)