Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 69 / 2 / b / Solution

Adopt the planar Fourier transform conventionExtend the disk potential to three dimensions. The Poisson equation for Newtonian gravity becomesFor $k>0$, decay away from an isolated disk and continuity across it give $\tilde{\varphi }=B(\mathbf{k})e^{-k|z|}$. Integration through the plane yields $-2kB=4\pi G\tilde{\Sigma }$, hence the razor-thin disk Poisson kernelThe given transform of $K(\mathbf{x})=1/|\mathbf{x}|$ is $\tilde{K}=2\pi /k$. The convolution theorem therefore gives the forward potential relationFor the reverse relation, $\widetilde{{\Delta }_2\varphi }=-k^2\tilde{\varphi }$, soInvert the transform to obtain the thin-disk potential-density inversionAn additive constant in $\varphi$ is annihilated by ${\Delta }_2$. These formulas hold for suitable isolated finite disks, and in a distributional or regularized sense for less regular profiles. The uniform zero-wavenumber sheet needs a separate vertical solution; the infinite Mestel disk requires potential differences or an infrared regularization, since its absolute forward integral diverges.