Euler integral

For the Euler–Poisson integral, see Gaussian integral.

In mathematics, there are two types of Euler integral:

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

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

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

See also

• Leonhard Euler
• List of topics named after Leonhard Euler

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