 Fay solution

ID: fay-solution

Z > 0

. Its Fourier series has exponentially decreasing coefficients at each positive

δ Z

. Expanding

1/ sinh (n s) = 2 \sum_{j \geq 0} e^{- (2 j + 1) n s}

and summing the sine series gives

q = 2 δ sin θ \sum_{j \geq 0} \frac{1}{c o s h (( 2 j + 1 ) s ) - c o s θ}, s = δ Z .

For

s ≫ 1

, the first harmonic gives

q \sim 4 δ e^{- s} sin θ

. The Fay shock-layer asymptotics describe its opposite, small-

s

regime.

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