Consider the equation

$$\nabla^{2} \phi=\delta(x) g(y)$$

on the two-dimensional strip $-\infty<x<\infty, 0 \leqslant y \leqslant a$, where $\delta(x)$ is the delta function and $g(y)$ is a smooth function satisfying $g(0)=g(a)=0 . \phi(x, y)$ satisfies the boundary conditions $\phi(x, 0)=\phi(x, a)=0$ and $\lim _{x \rightarrow \pm \infty} \phi(x, y)=0$. By using solutions of Laplace's equation for $x<0$ and $x>0$, matched suitably at $x=0$, find the solution of $(*)$ in terms of Fourier coefficients of $g(y)$.

Find the solution of $(*)$ in the limiting case $g(y)=\delta(y-c)$, where $0<c<a$, and hence determine the Green's function $\phi(x, y)$ in the strip, satisfying

$$\nabla^{2} \phi=\delta(x-b) \delta(y-c)$$

and the same boundary conditions as before.