Euler integral

Template:For In mathematics, there are two types of Euler integral:<ref>Template:Cite book</ref>

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<ref>Template:Cite book</ref> <math display="block">\Gamma(z) = \int_0^\infty t^{z-1}\,\mathrm e^{-t}\,dt</math>

For positive integers Template:Mvar and Template:Mvar, 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

External links and 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

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