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Euler integral

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

1. The Euler integral of the first kind is 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. 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

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

External links and references

Wolfram MathWorld on the Euler Integral

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