It is a question in spirit of [this one](https://mathoverflow.net/questions/105457/complex-evaluation-of-a-classical-real-integral).
Is there a way to prove Euler's formula
$$
\int\_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$
using contour integration (and maybe something else, say, integration by parts or change of variables is ok, but double integration and Fubini theorem is not)? For $a=b=1/2$ we get the value of $\Gamma(1/2)$ which is essentially the value the cited question was about.