Integral Representation of different Gamma-Functions

Question

I came across the relation $(Γ(x) Γ(y))/Γ(x + y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$

Can someone tell me how to prove this?

Thanks!

Answer 1

Using $\Gamma(c)=\int_0^\infty u^{c-1}e^{-u}du=2\int_0^\infty v^{2c-1}e^{-v^2}dv$, we substitute $a=r\sin\theta,\,b=r\cos\theta,\,t=\sin^2\theta$: $$\Gamma(x)\Gamma(y)=4\int_0^\infty a^{2x-1}e^{-a^2}da\int_0^\infty b^{2y-1}e^{-b^2 }db\\=4\int_0^{\pi/2}\sin^{2x-1}\theta\cos^{2y-1}\theta d\theta\int_0^\infty r^{2(x+y)-1}e^{-r^2}dr\\=\Gamma(x+y)\int_0^1 t^{x-1}(1-t)^{y-1}dt.$$The final integral over $t$ is called the Beta function, $\operatorname{B}(x,\,y)$.