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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.

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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 $(1234)(5678)$.

enter image description here

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