Euler integral

In mathematics, there are two types of Euler integral:

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

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

\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}}}

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

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