let-B-x-y-0-1-t-x-1-1-t-y-1-dt-withx-gt-0and-y-gt-0-prove-that-B-x-y-x-y-x-y-

Tinku Tara June 4, 2023 Integration 0 Comments

Question Number 40892 by abdo.msup.com last updated on 28/Jul/18

let B(x,y) =∫_0 ^1 t^(x−1) (1−t)^(y−1) dt withx>0and y>0 prove that B(x,y)= ((Γ(x).Γ(y))/(Γ(x+y)))

$${let}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt} \\ $$$${withx}>\mathrm{0}{and}\:{y}>\mathrm{0}\:{prove}\:{that} \\ $$$${B}\left({x},{y}\right)=\:\frac{\Gamma\left({x}\right).\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$

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let-B-x-y-0-1-t-x-1-1-t-y-1-dt-withx-gt-0and-y-gt-0-prove-that-B-x-y-x-y-x-y-

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