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In mathematics, there are two types of Euler integral:

1. The Euler integral of the first kind is the beta function $${\displaystyle \mathrm {\mathrm {B} } (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt={\frac {\Gamma (z_{1})\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}}$$
2. The Euler integral of the second kind is the gamma function $${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}\,\mathrm {e} ^{-t}\,dt}$$

For positive integers m and n, the two integrals can be expressed in terms of factorials and binomial coefficients: $${\displaystyle \mathrm {B} (n,m)={\frac {(n-1)!(m-1)!}{(n+m-1)!}}={\frac {n+m}{nm{\binom {n+m}{n}}}}=\left({\frac {1}{n}}+{\frac {1}{m}}\right){\frac {1}{\binom {n+m}{n}}}}$$ $${\displaystyle \Gamma (n)=(n-1)!}$$

See also

• Leonhard Euler
• List of topics named after Leonhard Euler

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.

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

• Gamma and related functions
• Set index articles on mathematics