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!

Note that $f$ is the composition of two functions: a Möbius transformation and the square root. So, the domain of the Möbius transformation is $\mathbb{C}\backslash\{1\}$ and its image is all the plane (Why?). On the other hand, the square root is only defined in a branch of the logarithm. 
Where does the Möbius transformation maps the interval $[-1,1]$? Since it maps $-1$ to $0$, $0$ to $-1$ and $1$ to infinity, it maps the interval to the nonpositive axis! (Why?) So the principal branch of the logarithm is defined in the image of $\mathbb{C}\backslash[-1,1]$ under the Möbius transformation, which is $\mathbb{C}\backslash(-\infty,0]$.
