= Dirichlet Green function for an infinite strip
On the strip $\mathbb R\times(0,a)$, with $a>0$, choose the convention $\Delta G=\delta$ and zero <Dirichlet boundary conditions> on both horizontal sides. The <Dirichlet Green function> is
$$
G(x,y;b,c)=-\frac1\pi\sum_{n=1}^\infty
\frac{e^{-n\pi|x-b|/a}}n
\sin\frac{n\pi y}{a}\sin\frac{n\pi c}{a},\qquad 0<c<a.
$$
To derive the coefficient, use the <Fourier sine series> completeness relation $\delta(y-c)=(2/a)\sum_{n\ge1}\sin(n\pi y/a)\sin(n\pi c/a)$. Each horizontal mode solves $(\partial_x^2-\kappa_n^2)g_n=\delta(x-b)$ with $\kappa_n=n\pi/a$. Its decaying <Green function> is $-e^{-\kappa_n|x-b|}/(2\kappa_n)$, since the first <derivative> has jump one. Multiplying by $2/a$ gives the displayed coefficient.
The series converges absolutely when $x\ne b$, vanishes on $y=0,a$, and decays as $|x-b|\to\infty$. At the <point source> $(b,c)$ it has the expected logarithmic singularity; the source equation is interpreted in the <distribution> sense. The strip width sets the longest horizontal decay scale $a/\pi$.
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