= Fay solution
{c}
{title2=$q=2\delta\sum_{n\ge1}\sin(n\theta)/\sinh(n\delta Z)$}
The Fay solution is a periodic solution of the negative-flux <viscous Burgers equation> for $Z>0$. Its <Fourier series> has exponentially decreasing coefficients at each positive $\delta Z$. Expanding $1/\sinh(ns)=2\sum_{j\ge0}e^{-(2j+1)ns}$ and summing the sine series gives
$$
q=2\delta\sin\theta\sum_{j\ge0}\frac1{\cosh((2j+1)s)-\cos\theta},\qquad s=\delta Z.
$$
For $s\gg1$, the first harmonic gives $q\sim4\delta e^{-s}\sin\theta$. The <Fay shock-layer asymptotics> describe its opposite, small-$s$ regime.
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