= Thin-disk potential-density inversion
{title2=$\widetilde\Sigma=-k\widetilde\Phi/(2\pi G)$}
For an isolated <razor-thin disk>, define the two-dimensional <Fourier transform> by $\widetilde g(\mathbf k)=\int e^{-i\mathbf k\cdot\mathbf x}g(\mathbf x)d^2x$. At $k>0$, the <razor-thin disk Poisson kernel> gives the displayed inversion. Since the transform of $1/|\mathbf x|$ is $2\pi/k$, inversion also gives $\Sigma=(4\pi^2G)^{-1}(1/|\mathbf x|)*\Delta_2\Phi$. The forward relation is $\Phi=-G(1/|\mathbf x|)*\Sigma$. An additive potential constant is irrelevant; a uniform infinite sheet and scale-free divergent potentials require separate zero-mode or potential-difference treatment rather than an arbitrary decaying boundary condition.
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