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

1. Jeffrey, Alan; and Dai, Hui-Hui (2008). Handbook of Mathematical Formulas 4th Ed. Academic Press. ISBN 978-0-12-374288-9. pp. 234–235

External links and references

• Wolfram MathWorld on the Euler Integral