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:

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

$${\displaystyle \Gamma (n)=(n-1)!}$$

See also[]

• Leonhard Euler
• List of topics named after Leonhard Euler

References[]

External links and references[]

• Wolfram MathWorld on the Euler Integral
• NIST Digital Library of Mathematical Functions dlmf.nist.gov/5.2.1 relation 5.2.1 and dlmf.nist.gov/5.12 relation 5.12.1

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