evaluation of beta function using Laplace transform


The beta integral can be evaluated elegantly using the convolution theorem (http://planetmath.org/LaplaceTransform) for Laplace transformsDlmfMathworldPlanetmath.

Start with the following Laplace transform:

s-α=[tα-1Γ(α)]=0e-sttα-1Γ(α)𝑑t

Since s-qs-p=s-q-p, the convolution theorem imples that

tq-1Γ(q)*tp-1Γ(p)=tq+p-1Γ(q+p)

Writing out the definition of convolutionMathworldPlanetmath, this becomes

0t(t-s)q-1Γ(q)sp-1Γ(p)𝑑s=tq+p-1Γ(p+q)

Setting t=1 and simplifying, we conclude that

01xp-1(1-x)q-1𝑑x=Γ(p)Γ(q)Γ(p+q)

QED

Title evaluation of beta functionDlmfDlmfMathworldPlanetmath using Laplace transform
Canonical name EvaluationOfBetaFunctionUsingLaplaceTransform
Date of creation 2013-03-22 14:37:36
Last modified on 2013-03-22 14:37:36
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Derivation
Classification msc 33B15