TITLE: Green function of the half-plane
QUESTION [5 upvotes]: There is an equation for Green function of a half-plane:
$$ \Delta G = \delta (x - x_{0})\delta(y - y_{0}), \quad G(x, 0) = 0.$$
Of course, the Green function which satisfies the boundary condition is represented as
$$
G(x, y, x_{0}, y_{0}) = G_{0}(x, y, x_{0}, y_{0}) - G_{0}(x, y, x_{0}, -y_{0}),
$$ 
where $G_{0}(x, y, x_{0}, y_{0})$ is the Green function for a plane, so I need to find a Green function for a plane.
After using Fourier transform (and setting $x_{0}, y_{0}$ to zero) I got
$$-(k_{x}^{2} + k_{y}^2)\tilde {G_{0}} = \frac{1}{2 \pi} \Rightarrow \tilde G_{0} = -\frac{1}{2 \pi}\frac{1}{k_{x}^{2} + k_{y}^{2}}.$$
But after that I got divergent integral. Did I make a mistake?

REPLY [4 votes]: You didn't make a mistake, but it does mean that the Green function doesn't have a nice Fourier transform (else the integral would converge).
You can also try, for instance, to look for spherically symmetric Green functions $G=G(r)$, which will give you an easily-solved ODE for $G(r)$ (but the point $r=0$ will need some special attention).
Another thing is that if you get an integral somewhere of the form
$$ I(r)=\int_{-\infty}^{\infty} \frac{e^{-|k|r}}{|k|}\,dk, $$
you can (not rigorously at all) differentiate it w.r.t. $r$, integrate the result, then integrate back. It depends on your tolerance towards such cheap tricks. Even if you get your answer using a cheap trick, you can in the end just verify that it does, in fact, solve the equation.