Map \(\CC\sm[-1, 1]\) to \(\DD\).
Solution
In steps:
Send \(-1\to 0\) and \(1\to \infty\) with \(z\mapsto {z+1\over z-1}\). Checking that \(f(0) = -1\), this yields \(\CC\sm\RR_{\leq 0}\).
Unwrap with \(z\mapsto \sqrt{z}\) to obtain the right half-plane \(-\pi/2<\Arg(z) < \pi/2\).
Apply the rotated Cayley map \(z\mapsto {z-1\over z+1}\) to map this to \(\DD\).
Note that the composition is \[ {\sqrt{z+1\over z-1} -1 \over \sqrt{z+1\over z-1} +1} &= { \qty{ \sqrt{z+1} - \sqrt{z-1}}^2 \over (z+1)-(z-1) } \\ &= z - \sqrt{z^2-1} ,\] which has inverse \(z\mapsto {1\over 2}\qty{z+{1\over z}}\).