Consider the function $$f(z) = \frac{1}{z} \ln \left( \frac{1-z}{1+z} \right).$$ This is clearly multivalued. There has to be two branch points at $\pm1$. Are there any other singularities, such as poles arising from the factor of $1/z$ or $\infty$? Does it depend on the choice of branch cuts?
For a simpler function such as $\ln(z)$, I know that there are branch points at $z=0$ and $z=\infty$. A cut can be chosen to use multiple copies of the complex plane in order to define a single valued analytic function. Can someone please help me do such a construction for $f$?