Euler integral

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الموسوعة الحرة

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

  1. The Euler integral of the first kind is the beta functionB(z1,z2)=∫01tz1−1(1−t)z2−1dt=Γ(z1)Γ(z2)Γ(z1+z2){\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 (ض_{1}+ض_{2})}}}
  2. The Euler integral of the second kind is the gamma function[2]Γ(z)=∫0∞tz−1e−tdt{\displaystyle \Gamma (z)=\int _{0}^{\infty}t^{z-1}\,\mathrm {e} ^{-t}\,dt}

For positive integersm 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+mnm(n+mn)=(1n+1m)1(n+mn){\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. pp. 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.