In mathematics, there are two types of Euler integral:

1. The Euler integral of the first kind is the beta function B ( z 1 , z 2 ) = ∫ 0 1 t z 1 − 1 ( 1 − t ) z 2 − 1 d t = Γ ( z 1 ) Γ ( z 2 ) Γ ( z 1 + z 2 ) {\displaystyle \mathrm {\mathrm {B} } (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt={\frac {\Gamma (z_{1})\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}}
2. The Euler integral of the second kind is the gamma function Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t {\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: B ( n , m ) = ( n − 1 ) ! ( m − 1 ) ! ( n + m − 1 ) ! = n + m n m ( n + m n ) = ( 1 n + 1 m ) 1 ( n + m n ) {\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}}}} Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!}