Find the conformal map that takes the upper half-plane conformally onto the half-strip \[ \ts{w=x+iy \st -\pi/2 < x < \pi/2,\, y>0} .\]
Solution
It’s well known that \(z\mapsto \sin(z)\) sends \(-\pi/2<\Re(z) < \pi/2\) with \(\Im(z) > 0\) to \(\HH\):
So take \(z\mapsto \arcsin(z)\).