A change of variables in Riemann's proof of the functional equation of $\zeta(s)$

Question

In Riemann's functional equation proof it says: From $$\Gamma(s)=2\int_0^\infty e^{-x^{2}}x^{2s-1}dx,$$ after variable substitution, we get $$\Gamma\left(\frac{s}2\right)=n^{s} \pi^{\frac{s}2}\int_0^\infty x^{\frac{s}2-1}e^{-n^{2}\pi x}dx.$$ How is it derived? Any ideas?

Answer 1

The change of variables implicitly given is $x^2 \mapsto n^2 \pi x$. With this change of variables, the claim is now straightforwardly checked.

It may be valuable to note how one can quickly recognize which change of variables is used. Here, one can look at the argument of the exponential. As it was $-x^2$ and it becomes $-n^2 \pi x$, one sees that necessarily $x^2 \mapsto n^2 \pi x$.