= Vorticity-sheet eigenfunction of inviscid Couette flow
{title2=$\widehat w_\xi=G_{|\alpha|}(z,\xi)$}
= Vorticity-sheet eigenfunctions of inviscid Couette flow
{synonym}
Let $G_a$ be the <Dirichlet Green function> of $D^2-a^2$ on $[-1,1]$, with $a=|\alpha|$:
$$
G_a(z,\xi)=-\frac{\sinh[a(z_<+1)]\sinh[a(1-z_>)]}{a\sinh(2a)}.
$$
It is continuous and has derivative jump one, so $(D^2-a^2)G_a=\delta(z-\xi)$. The <Dirac delta multiplication identity> gives $(z-\xi)\delta(z-\xi)=0$, proving it is a <generalized eigenfunction> of the <inviscid Couette continuous spectrum>. Integrating $G_a(z,\xi)q_0(\xi)e^{-i\alpha\xi t}$ reconstructs the evolving vertical <velocity> from its initial vorticity.
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