Define \[ G \definedas \theset{z\in \CC\suchthat \Re(z) > 0, \, \abs{z-1} > 1} .\]
Find all of the injective conformal maps \(G\to \DD\). These may be expressed as compositions of maps, but explain why this list is complete.
Solution
Use that every element of \(\Aut(\DD)\) is of the form \(f(z) = \lambda\psi_a(z)\), and the region \(G\) is conformally equivalent to \(\DD\). Thus every element of \(\Aut(G)\) can be conjugated to an element of \(\Aut(\DD)\) by a conformal map \(F: G\to \DD\). One such map is gotten by rotating \(z\mapsto iz\) to get \(\HH \intersect \abs{z}>1\), then applying a Joukowski map \(z\mapsto z+\inverseof{z}\) to get \(\HH\). So \[ f\in \Aut(G) \implies f = \inverseof{F} \circ \psi_{a} \circ F \text{ for some } \psi_a \in \Aut(\DD) .\]