 Razor-thin disk Poisson kernel

The razor-thin disk Poisson kernel converts a horizontal Fourier transform of surface density of a disk into its gravitational potential. Write the razor-thin disk as $\rho (\mathbf{x},z)=\Sigma (\mathbf{x})\delta (z)$ and use the Poisson equation for Newtonian gravity ${\nabla }^2\Phi =4\pi G\rho$. For horizontal wavevector $\mathbf{k}$ of magnitude $k>0$, the transformed equation isA solution decaying away from the disk has the form $C(\mathbf{k})e^{-k|z|}$. Integrating across $z=0$ gives the derivative jump $-2kC=4\pi G\hat{\Sigma }$, hence the midplane relationThis multiplier turns the nonlocal horizontal gravity calculation into multiplication in Fourier space. It is useful in calculations of the self-gravity of an astrophysical disk, where longer wavelengths have a larger potential response at fixed density amplitude. The $k=0$ mode requires separate boundary information: an infinite uniform sheet has a gravitational potential proportional to $|z|$, which cannot decay at both vertical infinities.