I have the following in my notes, but I can't remember how it works. Please help!
$\nabla^2\psi=0, \quad\psi\to 0\quad\text{as}\quad x^2+y^2\to\infty, \quad\psi (x,y,0)$ is continuous
Then by using Green's function, we get the solution to be
\psi(x',y',z')={z'\over 2\pi}\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty [(x-x')^2+(y-y')^2+z'^2]^{-3\over 2}\psi(x,y,0)\,\,\,dxdy\;.
(This part I am sure about.)
The primed x',y',z' are the variables introduced when using the Green's function G(\vec{x};\vec{x'}).
Why does this satisfy the boundary conditions? I am thinking that this solution is equivalent to
\psi(x,y,z)={z\over 2\pi}\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty [(x-x')^2+(y-y')^2+z^2]^{-3\over 2}\psi(x',y',0)\,\,\,dx'dy'\;.
But doesn't this imply that $\psi(x,y,0)\equiv 0? $ -- Not supposed to be true.
As an aside, are harmonic functions always spherically symmetrical?
Also, is it possible to actually evaluate that integral?