In mathematics, there are two types of Euler integral:

  1. The Euler integral of the first kind is the beta function <math display="block">\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)}</math>
  2. The Euler integral of the second kind is the gamma function <math display="block">\Gamma(z) = \int_0^\infty t^{z-1}\,\mathrm e^{-t}\,dt</math>

For positive integers and , the two integrals can be expressed in terms of factorials and binomial coefficients:

<math display="block">\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</math>

<math display="block">\Gamma(n) = (n-1)! </math>

See also

  • Leonhard Euler
  • List of topics named after Leonhard Euler

References

  • 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