How to show that a complex function have a branch in a domain I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}} $$ has a branch on $G = \mathbb C \backslash [-1,1] $.
I'm having a hard time in finding the way to approach this kind of questions, so what I'm actually looking is on a way to approach that kind of questions and not the solution for this specific question.
Thanks!

It could be useful, even if not proper/formal, to integrate the function over a small loop around the suspicious point (withtin the domain of analiticity of the function in order to take advantage of Cauchy theorems): this should result in an expression involving quantities path-related (eg. the radius of a small circle: try with $\sqrt{z}$ around the origin) or phase-dependent. 
Another improper but useful way is to study the behaviour of the function around $z_0e^{\pm i\epsilon}$ and $z_0e^{i(2\pi\pm\epsilon)}$, $\epsilon>0$: this should result in different values for $f(z)$ even if evaluated in two "different (for a phase factor) but identified" points.  
