Euler integral

In mathematics, there are two types of Euler integral:^[1]

1. Euler integral of the first kind: the Beta function

$\mathrm{\Beta}(x,y)= \int_0^1t^{x-1}(1-t)^{y-1}\,dt =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$

2. Euler integral of the second kind: the Gamma function

$\Gamma(z) = \int_0^\infty t^{z-1}\,e^{-t}\,dt$

For positive integers m and n

$\mathrm{\Beta}(n,m)= {(n-1)!(m-1)! \over (n+m-1)!}={n+m \over nm{n+m \choose n}}$

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

See also

• Euler integral (thermodynamics)
• 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