Euler integral

In mathematics, there are two types of Euler integral:


 * 1) The Euler integral of the first kind is the beta function $$\mathrm{\Beta}(z_1,z_2) = \int_0^1t^{z_1-1}(1-t)^{z_2-1}\,dt = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1+z_2)}$$
 * 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: $$\Beta(n,m) = \frac{(n-1)!(m-1)!}{(n+m-1)! } = \frac{n+m}{nm \binom{n+m}{n}} = \left( \frac{1}{n} + \frac{1}{m} \right) \frac{1}{\binom{n+m}{n}}$$ $$\Gamma(n) = (n-1)! $$

External links and references

 * Wolfram MathWorld on the Euler Integral
 * NIST Digital Library of Mathematical Functions dlmf.nist.gov/5.2.1 relation 5.2.1 and dlmf.nist.gov/5.12 relation 5.12.1