= Fay shock-layer asymptotics
{c}
{title2=$q\sim(\pi/Z)\tanh[\pi\theta/(2\delta Z)]$}
For $|\theta|\ll1$ and $s=\delta Z\ll1$, the <Fay solution> becomes $q\sim4\delta\theta\sum_{j\ge0}[((2j+1)s)^2+\theta^2]^{-1}$. The partial-fraction identity for the <hyperbolic tangent> sums this to the displayed smooth shock layer. It resolves the odd transition from $-\pi/Z$ to $+\pi/Z$ on angular thickness $O(\delta Z)$; at $\theta=0$ both sides vanish and the comparison is read through their derivatives.
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