Fay shock-layer asymptotics ($q\sim (\pi /Z)\tanh [\pi \theta /(2\delta Z)]$)

For $|\theta |\ll 1$ and $s=\delta Z\ll 1$, the Fay solution becomes $q\sim 4\delta \theta {\sum }_{j\ge 0}[((2j+1)s{)}^2+{\theta }^2{]}^{-1}$. 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.