Euler's Inegrals. Euler's gamma and beta functions are the following parameterdependent integrals for \(x, y>0\) :

\[\Gamma(x):=\int_{0}^{\infty} e^{-t} t^{x-1} d t \quad \text { and } \quad B(x, y):=\int_{0}^{1} t^{x-1}(1-t)^{y-1} d t\]

(i) Show that

\[\Gamma(x) \Gamma(y)=4 \int_{(0, \infty)^{2}} e^{-u^{2}-v^{2}} u^{2 x-1} v^{2 y-1} d(u, v) .\]

(ii) Use part (i) to prove that

\[B(x, y)=\frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}\]