Half-strip conformal map by hyperbolic cosine
= Half-strip conformal map by hyperbolic cosine
{title2=$\zeta=\cosh(\pi z/b)$}
For $b>0$, the <conformal map> $\cosh(\pi z/b)$ takes $\{\operatorname{Re}z>0,\ 0<\operatorname{Im}z<b\}$ 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 $-1$.