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Question Number 34291 by math khazana by abdo last updated on 03/May/18

let B(x,y) = ∫_0 ^1 u^(x−1) (1−u)^(y−1)  du  and  Γ(x)= ∫_0 ^∞  t^(x−1)  e^(−t) dt  1) prove that Γ(x) = 2∫_0 ^∞  u^(2x−1)  e^(−u^2 ) du  2)give Γ(x)Γ(y) at form of double integrale  3)prove that B(x,y) =((Γ(x)Γ(y))/(Γ(x+y)))  4) calculate B(m,n) for m and n integr naturals

$${let}\:{B}\left({x},{y}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {u}^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{y}−\mathrm{1}} \:{du}\:\:{and} \\ $$$$\Gamma\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:{u}^{\mathrm{2}{x}−\mathrm{1}} \:{e}^{−{u}^{\mathrm{2}} } {du} \\ $$$$\left.\mathrm{2}\right){give}\:\Gamma\left({x}\right)\Gamma\left({y}\right)\:{at}\:{form}\:{of}\:{double}\:{integrale} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:{B}\left({x},{y}\right)\:=\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{B}\left({m},{n}\right)\:{for}\:{m}\:{and}\:{n}\:{integr}\:{naturals} \\ $$

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