Is it possible to solve the following equation for F(x,y) analytically? I was trying to solve it using conformal mapping (e.g. sin(z) transformation) however, I couldn't find an answer.
$$ \frac{d^2 F}{dx^2} + \frac{d^2 F}{dy^2} = \delta(y) ( H(x) - H(-x) ) $$
where $\delta(y)$ is the delta function and $H(x)$ is the Heaviside step function.
If it is helpful, we can assume that the PDE domain is finite: $|x| < x_0$ and $|y| < y_0$.