let give B(x,y)= ∫_0 ^1 u^(x−1) (1−u)^(y−1) du and (beta function) and Γ(x) =∫_0 ^∞ u^(x−1) e^(−u) du (x>0)(gamma function of euler) 1) prove that Γ(x)= 2∫_0 ^∞ u^(2x−1) e^(−u^2 ) du . 2) prove that B(x,y) = ((Γ(x).Γ(y))/(Γ(x+y))) .


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Question Number 28442 by abdo imad last updated on 25/Jan/18

$l e t g i v e B (x, y) = \int_{0}^{1} u^{x - 1} {(1 - u)}^{y - 1} d u a n d (b e t a f u n c t i o n)$ $a n d Γ (x) = \int_{0}^{\infty} u^{x - 1} e^{- u} d u (x > 0) (g a m m a f u n c t i o n o f e u l e r)$ $1) p r o v e t h a t Γ (x) = 2 \int_{0}^{\infty} u^{2 x - 1} e^{- u^{2}} d u .$ $2) p r o v e t h a t B (x, y) = \frac{Γ (x) . Γ (y)}{Γ (x + y)} .$
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