Euler integral
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In mathematics, there are two types of Euler integral:
- Euler integral of the first kind: the Beta function
<math> \Beta(x,y)= \int_0^1t^{x-1}(1-t)^{y-1}\,dt =\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}<math> - Euler integral of the second kind: the Gamma function
<math>
\Gamma(z) = \int_0^\infty t^{z-1}\,e^{-t}\,dt<math>
For positive integers m and n
- <math>\Beta(n,m)= {(n-1)!(m-1)! \over (n+m-1)!}={n+m \over nm{n+m \choose n}}<math>
- <math>\Gamma(n) = (n-1)! \,<math>