Euler integral

In mathematics, there are two types of Euler integral:[1]

  1. The Euler integral of the first kind is the beta function
    B ( z 1 , z 2 ) = ∫ 0 1 t z 1 − 1 ( 1 − t ) z 2 − 1 d t = Γ ( z 1 ) Γ ( z 2 ) Γ ( z 1 + z 2 ) {\displaystyle \mathrm {\mathrm {B} } (z_{1},z_{2})=\int _{0}^{1}t^{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[2]
    Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t {\displaystyle \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:

B ( n , m ) = ( n − 1 ) ! ( m − 1 ) ! ( n + m − 1 ) ! = n + m n m ( n + m n ) = ( 1 n + 1 m ) 1 ( n + m n ) {\displaystyle \mathrm {B} (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}}}}
Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!}

See also

References

  1. ^ Jeffrey, Alan; Dai, Hui-Hui (2008). Handbook of mathematical formulas and integrals (4th ed.). Amsterdam: Elsevier Academic Press. p. 234–235. ISBN 978-0-12-374288-9. OCLC 180880679.
  2. ^ Jahnke, Hans Niels (2003). A history of analysis. History of mathematics. Providence (R.I.): American mathematical society. p. 116-117. ISBN 978-0-8218-2623-2.

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


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