Find a conformal equivalence between the following domains:
the strip $ S = \{ z \in \Bbb C \ | \ 0 < \Bbb Im(z) < 1 \} $ and the quadrant $ Q = \{z \in \Bbb C \ | \ \Bbb Re(z) > 0, \Bbb Im(z) > 0 \} $.
By considering a suitable bounded solution of Laplace's equation $ u_{xx} + u_{yy} = 0 $ on the strip S, find a non-constant, harmonic function on Q which is constant on each of the two boundaries of the quadrant.
I can manage the first part - use a variation on the conformal mapping $ z \mapsto e^z $, but then I am unable to determine the final bit.
This is one of my example sheet questions, which is completely non-examinable. I have already had the supervision on it, but was just hoping for a bit more detail (the supervisor wasn't able to explain it fully himself).
Set $$ f(z)=\exp(\pi z/2), $$ Then $f'(z)\ne 0$, $f$ is one-to-one on $$ S=\{z\in\mathbb C : 0<\mathrm{Im}\, z<1\}, $$ and $$ f[S]=Q=\{z\in\mathbb C: \mathrm{Re}\,z>0,\,\mathrm{Im}\,z>0\} $$
The harmonic function on $S$ is $v=\frac{\pi}{2}\mathrm{Im}\, z$ and $u=v\circ f^{-1}$ which is constant on the boundaries of $Q$.
In particular, $$u(x,y)=\mathrm{Im}\,\log z$$