= Strip automorphism with a prescribed image of its midpoint
{title2=$g_\beta(z)=\frac h\pi\operatorname{Log}(A+B e^{\pi z/h})$}
On $S_h=\{z:0<\operatorname{Im}z<h\}$, let $\beta\in S_h$ and write $e^{\pi\beta/h}=A+iB$ with $B>0$. The <exponential function> $E(z)=e^{\pi z/h}$ is a <conformal map> from the strip bijectively onto the <upper half-plane>, sending $ih/2$ to $i$. The affine map $w\mapsto A+Bw$ is an automorphism of the half-plane sending $i$ to $E(\beta)$. Conjugate it by $E$, using the <principal complex logarithm> as the inverse. The displayed formula is a conformal automorphism of the strip and satisfies $g_\beta(ih/2)=\beta$. The branch range $0<\operatorname{Im}\operatorname{Log}w<\pi$ is what prevents an unwanted vertical period in this inverse.
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