Hyperbolic-cosine half-strip map
= Hyperbolic-cosine half-strip map
The map $z\mapsto\cosh(\pi z)$ is a conformal bijection from
$$
\{z:\Re z>0,\ 0<\Im z<1\}
$$
onto the upper half-plane. Its three boundary intervals map respectively to $(1,\infty)$, $(-1,1)$ and $(-\infty,-1)$.