# Conformal mapping to the unit disk

I am studying for a prelim and I dont know how to do this: Find a conformal mapping that maps the set $$\{z\in\mathbb{C}: |z|>1, Re(z)>0\}$$ to the unit disk.

I tried using $$z^2$$ and $$1/z$$ but I don't think that works.

You are correct $$z^2$$ followed by $$1/z$$ is a good try, but it is Wrong! It does not respect the boundary requirement.
Hint: the map $$\begin{eqnarray*} w= \frac{1+z}{1-z} \end{eqnarray*}$$ permutes the regions the diagram below with the permutation $$(1234)(5678)$$.