Half-strip conformal map by hyperbolic cosine ($\zeta =\cosh (\pi z/b)$)

For $b>0$, the conformal map $\cosh (\pi z/b)$ takes $\{\mathrm{Re}z>0,0<\mathrm{Im}z<b\}$ bijectively onto the upper half-plane. The exponential maps this half-strip onto the upper half-plane exterior to the unit circle; then $\cosh (\pi z/b)=(w+w^{-1})/2$ is a scaled Joukowski map. All critical points of the entire map are $z=inb$, mapping alternately to $1$ and $-1$.