Schwarz-Christoffel map of a half-plane joined to a strip ($z=\frac{2}{\pi }[R+{\tan }^{-1}(1/R)],R=\sqrt{Z^2-1}$)

The upper half-plane joined to the downward strip $-1<x<1$ has reentrant corners at $\pm 1$. The Schwarz-Christoffel formula gives $z^{\prime }(Z)=2R/(\pi Z)$, with $R\sim Z$ at upper infinity. Since $(R+{\tan }^{-1}(1/R){)}^{\prime }=R/Z$, integration gives the displayed map. The inverse trigonometric function is continued analytically in the upper half-plane so its corner values are $\pm \pi /2$. The origin is the lower channel end and infinity the upper open end. The complex potentials $2Z/\pi$ and $2\log Z/\pi$ respectively normalize unit horizontal velocity in the upper region and unit upward velocity in the channel.