Off-plane razor-thin Poisson kernel ($\tilde{\Phi }(\mathbf{k},z)=-(2\pi G/k)\tilde{\Sigma }(\mathbf{k})e^{-k|z|}$)

For $k>0$, horizontal Fourier transform gives $({\partial }_z^2-k^2)\tilde{\Phi }=4\pi G\tilde{\Sigma }\delta (z)$. 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.