Relation Between Beta and Gamma functions

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Thread starter PLAGUE
Start date Jan 2, 2024
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PLAGUE

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Jan 2, 2024

TL;DR: Proof of Beta Gamma function relation

Handwritten derivation of the Beta-Gamma relation: Gamma(m) and Gamma(n) written as integrals, a double integral with a highlighted inner integral in e^(-x(1+y)) x^(m+n-1), and the result Beta(m,n) = Gamma(m)Gamma(n)/Gamma(m+n).

So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?

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pasmith

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Jan 2, 2024

The line asserts that [tex] \int_0^\infty e^{-x(1+ y)}x^{m+n-1}\,dx = \frac{\Gamma(m + n)}{(1 + y)^{m + n}}.[/tex] How would you apply the given formulae to arrive at that?

Relation Between Beta and Gamma functions

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