PlanetMath: alternate integral representation of beta function (2)

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alternate integral representation of beta function (2)

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

$\displaystyle \int_{0}^{1} x^{p-1} (1 - x)^{q-1} \,dx$	$\displaystyle = \int_{0}^{\infty} \frac{1}{(1+s)^{p+1}} \left(\frac{s}{1+s}\right)^{q-1}\,ds$
	$\displaystyle = \int_{0}^{\infty} \frac{s^{q-1}}{(1 + s)^{p+q}} \,ds$

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

$\displaystyle \int_{0}^{\infty} \frac{s^{p-1}}{(1 + s)^{p+q}} \,ds$

$\displaystyle = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$

"alternate integral representation of beta function (2)" is owned by karstenb.

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This is version 5 of alternate integral representation of beta function (2), born on 2008-12-03, modified 2008-12-03.
Object id is 11299, canonical name is AlternateIntegralRepresentationOfBetaFunction2.
Accessed 378 times total.

Classification:

33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions)

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