Find a conformal map from the strip \(\theset{z\in \CC \suchthat 0 < \Im(z) < 1}\) to \(\HH\).
Solution
In steps:
- Dilate by \(z\mapsto \pi z\) to get \(0<\Im(z) < \pi\).
- Exponentiate by \(z\mapsto e^z\) to get \(\HH\).
Why \(e^z\) works: apply \(\Log\) to \(\HH\), use polar coordinates to write \(w=re^{i\theta}\) with \(0<\theta<\pi\) and note \[ \Log(w) = \ln\abs{w} +i\Arg(w) = \ln(r) + i\theta ,\] and noting that the image of \(\ln(\wait)\) is all of \(\RR\).