let-give-B-x-y-0-1-u-x-1-1-u-y-1-du-and-beta-function-and-x-0-u-x-1-e-u-du-x-gt-0-gamma-function-of-euler-1-prove-that-x-2-0-u-2x-1-e-u-2-du

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)} .