= 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 is
$$
(\partial_z^2-k^2)\widehat\Phi(\mathbf k,z)=4\pi G\widehat\Sigma(\mathbf k)\delta(z).
$$
A solution decaying away from the disk has the form $C(\mathbf k)e^{-k|z|}$. Integrating across $z=0$ gives the derivative jump $-2kC=4\pi G\widehat\Sigma$, hence the midplane relation
$$
\widehat\Phi_m(\mathbf k)=-\frac{2\pi G}{k}\widehat\Sigma(\mathbf k).
$$
This 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.
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