How to show that $\{z\in D:exp(\frac{z+1}{z-1})=w\}$ is countable

Question

I want to show that, for every w in the unit disk except zero, the set $\{z\in D:exp(\frac{z+1}{z-1})=w\}$ is countable.

I had the idea of solving the equation by writting $z=re^{it}$ and by writting w in its exponential form. I have then these two equations:

$exp(\frac{r^2-1}{1+r^2-2rcos(t)})=|w|$

$\frac{-2rsin(t)}{1+r^2-2rcos(t)}=a$

where $|w|$ is the norm of w and $a$ is its angle.

I tried to solve them but I'm blocked. Can someone help me solve them or help me find another way to show that the set is countable?

Answer 1

Hint: you have a mobius transformation inside the exponent. Can you see where does it map the unit disk?