Find a conformal map from the upper half-disc to the upper half-plane.
Solution
The map \(z\mapsto {1+z\over 1-z}\) sends \(\DD\) onto the right half-plane \(\theset{\Re w>0}\). Since \(\Im{1+z\over 1-z}={2\Im z\over\abs{1-z}^2}\), it sends \(\DD \intersect \HH\) onto the first quadrant \(Q_1=\theset{\Re w>0,\ \Im w>0}\).
The map \(z\mapsto z^2\) sends \(Q_1\) onto \(\HH\).