Euler integral

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

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

{\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. Euler integral of the second kind: the Gamma function

\Gamma (z)=\int _{0}^{\infty }t^{{z-1}}\,e^{{-t}}\,dt

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

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

References

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

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