Why do I need to use a branch cut from $0$ to $\infty$ when evaluating $\int_0^{\infty} \frac{x^{s-1}}{1+x}$?
I know there is a pole at $x=-1$
I can't seem to understand the purpose of the branch cut? Is it because if i write $f(x) = \frac{x^{s-1}}{1+x}$, $f(|x|) \ne f(|x|e^{i2\pi})$