So, my teacher showed me this proof and unfortunately, it is vacation now. I don't understand what happened in the marked line. Can anyone please explain?
2026-03-27 07:48:16.1774597696
How to prove the relation between Beta and Gamma functions?
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In double integrals $$ \int_c^d[\int_a^bf(x)g(y)dx]dy=\int_a^bf(x)dx\int_c^dg(y)dy $$
So $$\begin{align} \Gamma(m)\Gamma(n)&=\int_0^\infty e^{-xy}x^ny^{n-1}\int_0^\infty e^{-x}x^{m-1}dx\\&=\int_0^\infty y^{n-1}[\int_0^\infty e^{-x(1+y)}x^{m+n-1}dx]dy \end{align} $$