Strip automorphism with a prescribed image of its midpoint ($g_{\beta }(z)=\frac{h}{\pi }\mathrm{Log}(A+B{e}^{\pi z/h})$)

On $S_h=\{z:0<\mathrm{Im}z<h\}$, 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 $0<\mathrm{Im}\mathrm{Log}w<\pi$ is what prevents an unwanted vertical period in this inverse.