($\mathbb{D}^c \intersect \mathbb{H}$ to $\mathbb{H}$)
Find a conformal map from \(\DD^c \intersect \HH\) to \(\HH\) using cross-ratios.
Solution
The unit circle and the real line both pass through \(\pm1\), so a fractional linear transformation sending \(-1\to0\) and \(1\to\infty\) sends both to lines through \(0\). Take the cross-ratio sending \((i, -1, 1)\to (1,0,\infty)\):
This is the cross-ratio \[ f(z) = {z+1\over z-1} {i-1\over i+1} .\]
The image of \(\DD^c \intersect \HH\) is the 1st quadrant \(\HH \intersect \Re(z) > 0\). Send this to \(\HH\) with \(z\mapsto z^2\).