On the $z=0$ plane I have the boundary conditions $V=\delta(x)\delta(y)$ I want to solve for $z>0$. Helmholtz equation is $\nabla ^2 V +k^2 V=0$
I though that spherical harmonics would be useful. The function should be independent of $\phi$ and I should have $V=0$ on the $z=0$ plane so I must multiply by $\cos(\theta)$
In general (for odd $n$) $$(j_n(kr)-iy_n(kr))P_n(\cos (\theta))=h_n^{(2)}P_n(\cos (\theta))$$
where $j$, $y$, $h$ are spherical Bessel/Hankel functions seem to satisfy these conditions, but there should be a definitive answer. How do I find it?
The solution can be expressed as a fourier transform integral, but I am interested in its spherical harmonic expansion (or closed form)
$$\frac{1}{(2\pi)^2}\int \int \exp\bigg( - z\sqrt{-k^2+ k_x ^2+k_y^2 } -ik_x x-ik_yy \bigg)dk_xdk_y$$