= Schwarz-Christoffel map of a half-plane joined to a strip
{c}
{title2=$z=\frac2\pi[R+\tan^{-1}(1/R)],\quad R=\sqrt{Z^2-1}$}
The <upper half-plane> joined to the downward strip $-1<x<1$ has reentrant corners at $\pm1$. The <Schwarz-Christoffel formula> gives $z'(Z)=2R/(\pi Z)$, with $R\sim Z$ at upper infinity. Since $(R+\tan^{-1}(1/R))'=R/Z$, 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.
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