# Green's functions (Fourier Series)

Homework Statement:
Prove that the Green's function satisfying (1) subject to the boundary conditions (2) can be expressed in the form of (3).
Relevant Equations:

\partial^2_x G + \partial^2_y G = \delta(x-x0) \delta(y-y0)

G(0,y) = G(x,0) = G(x,b), G(x,y)< \infty

\begin{gather}
y<y_0: G = -\frac{2}{\pi} \int_0^{\infty}
\frac{sin (k x_0) sin (kx) sinh(ky) sinh (k(b-y_0))}{k sinh (kb)} \\

y>y_0: G = -\frac{2}{\pi} \int_0^{\infty}
\frac{sin (k x_0) sin (kx) sinh(ky_0) sinh (k(b-y))}{k sinh (kb)}
\end{gather}
In order to obtain equation (3), I think I have to do the Fourier transform in the x direction:

\tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx

So I have:

-k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2 \tilde{G}(k,y,x_0,y_0}{\partial y^2} = e^{-i k x_0} \delta (y-y_0)

And then solve this one-dimensional equation for ##\tilde{G}##. I'm not sure doing this is correct, because the variable ##y## is delimited and ##x## is not.

In case this is right, do I have to demonstrate first any property? And finally, would it be correct to solve the 1D equation with the Wronskian method?