Editing Euler integral

{{for|the Euler–Poisson integral|Gaussian integral}}
In [[mathematics]], there are two types of '''Euler integral''':<ref>{{cite book | last1 = Jeffrey | first1 = Alan | last2 = Dai | first2 = Hui-Hui | url = https://www.worldcat.org/oclc/180880679 | title = Handbook of mathematical formulas and integrals | date = 2008 | publisher = Elsevier Academic Press | isbn = 978-0-12-374288-9 | edition = 4th | location = Amsterdam | oclc = 180880679 | pages = 234–235}}</ref>

# 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>
# The ''Euler integral of the second kind'' is the [[gamma function]]<ref>{{Cite book |last=Jahnke |first=Hans Niels |title=A history of analysis |date=2003 |publisher=American mathematical society |isbn=978-0-8218-2623-2 |series=History of mathematics |location=Providence (R.I.) |page=116-117}}</ref> <math display="block">\Gamma(z) = \int_0^\infty  t^{z-1}\,\mathrm e^{-t}\,dt</math>

For [[positive integer]]s {{mvar|m}} and {{mvar|n}}, the two integrals can be expressed in terms of [[factorial]]s and [[binomial coefficient]]s:
<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==
{{Reflist}}

==External links and references==

* NIST Digital Library of Mathematical Functions [https://dlmf.nist.gov/5.2 dlmf.nist.gov/5.2.1] relation 5.2.1 and [https://dlmf.nist.gov/5.12 dlmf.nist.gov/5.12] relation 5.12.1

[[Category:Gamma and related functions]]


{{sia|mathematics}}