Euler integral

In mathematics, there are two types of Euler integral:

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

\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

See also: Euler integral, Beta function, Factorial, Gamma function, Integral, Leonhard Euler, Mathematics, Positive integer