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单词 EvaluationOfBetaFunctionUsingLaplaceTransform
释义

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

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更新时间:2025/5/4 20:25:20