Find a conformal map from the sector \(\ts{\Arg(z) \in (0, \alpha)} \to \DD\).
Solution
The picture:
In steps:
Map the sector to \(\HH\) using \(z\mapsto z^{\pi/\alpha}\), choosing a branch cut for \(\Log\) along \(\RR_{\leq 0}\).
Map \(\HH\to \DD\) using the standard \(z\mapsto {z-i\over z+i}\).