alternate integral representation of beta function (2)

Substitute $x:=\frac{1}{1+s}$, $dx=\frac{-1}{{(1+s)}^2}ds$:

${\displaystyle {\int }_0^1}{x}^{p-1}{(1-x)}^{q-1}𝑑x$ $={\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{1}{{(1+s)}^{p+1}}}{\left({\displaystyle \frac{s}{1+s}}\right)}^{q-1}𝑑s$

$={\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{s^{q-1}}{{(1+s)}^{p+q}}}𝑑s$

Since $B(p,q)=B(q,p)$ this gives:

${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{s^{p-1}}{{(1+s)}^{p+q}}}𝑑s$ $={\displaystyle \frac{\mathrm{\Gamma }(p)\mathrm{\Gamma }(q)}{\mathrm{\Gamma }(p+q)}}$

Title	alternate integral representation of beta function (2)
Canonical name	AlternateIntegralRepresentationOfBetaFunction2
Date of creation	2013-03-22 18:34:31
Last modified on	2013-03-22 18:34:31
Owner	karstenb (16623)
Last modified by	karstenb (16623)
Numerical id	8
Author	karstenb (16623)
Entry type	Result
Classification	msc 33B15