Off-plane razor-thin Poisson kernel
= Off-plane razor-thin Poisson kernel
{title2=$\widetilde\Phi(\mathbf k,z)=-(2\pi G/k)\widetilde\Sigma(\mathbf k)e^{-k|z|}$}
For $k>0$, horizontal <Fourier transform> gives $(\partial_z^2-k^2)\widetilde\Phi=4\pi G\widetilde\Sigma\delta(z)$. Decay and <continuity> give an exponential on each side; its derivative jump fixes the coefficient. The uniform mode needs a separate linear vertical solution and a potential reference.