$ \int_{0}^{\infty} x^{(m+n)/2- 1} e^{-x(\frac{mz+n}{2})} dx $ (density of $F$ distribution)

Question

I'm deriving the density of the $F_{n, m}$ distribution and I ended up with the following integral

$$ \int_{0}^{\infty} x^{(m+n)/2- 1} e^{-x(\frac{mz+n}{2})} dx $$

This integral looks like it could be molded into a Gamma function but I don't see how. I can add the entire context to this question but I don't think it would help.

Answer 1

Consider the Mellin transform of $e^{-a x}$ $$ \int_0^\infty x^{s-1} e^{-\alpha x} \; dx = \alpha^{-s}\Gamma(s) $$ now set $s = (m+n)/2$ and $\alpha = (m z + n)/2$