Find a conformal map from \(\ts{z\in \CC \st \abs{z} < 1, \Im(z) > 0} = \DD \intersect \HH\) to \(\DD\).
Solution
Note that \(z\mapsto z^2\) doesn’t actually work, because the image is \(\DD\sm \RR_{\geq 0}\) and has a slit deleted. Instead compose:
\(z\mapsto i{z-1\over z+1}\), which maps \(\DD\to \HH\) and restricts to map \(\DD \intersect \HH \to Q_1\).
\(z\mapsto z^2\), which maps \(Q_1\to \HH\)
\(z\mapsto {z-i\over z+i}\) which maps \(\HH\to \DD\).
Solution: Using Joukowski maps
In parts:
Use \(z\mapsto \inverseof{z}\) to send \(\HH \intersect \DD\) to \(Q_{34} \intersect \DD^c\).
Use \(z\mapsto -z\) to map this to \(\HH \intersect \DD^c\)
Use \(z\mapsto {1\over 2}(z+\inverseof{z})\) to map \(\HH \intersect \DD^c\) to \(\HH\)
Then use the Cayley map \(z\mapsto {z-i\over z+i}\) to map \(\HH\to \DD\).