= Solution
For nonzero real <wavenumber> $k$, seek a <Fourier mode> of the <Newtonian gravitational potential> in the form $\Phi'=f(z)e^{ikx}$. Away from the razor-thin sheet, the <Poisson equation for Newtonian gravity> becomes $f''-k^2f=0$. Requiring the perturbation to decay on both sides and remain continuous gives $f=Ae^{-|k||z|}$.
Integrating the <Poisson equation for Newtonian gravity> through $z=0$ fixes the derivative jump:
$$
f'(0^+)-f'(0^-)=4\pi G\Sigma_0,\qquad-2|k|A=4\pi G\Sigma_0.
$$
Thus \b[the perturbing gravitational potential] is
$$
\boxed{\Phi'(x,z)=-\frac{2\pi G\Sigma_0}{|k|}e^{ikx-|k||z|}.}
$$
At the midplane this reduces to the <razor-thin disk Poisson kernel>. The observable real disturbance is the real part. Writing the perturbation amplitude as $\Sigma_a$ instead of $\Sigma_0$ gives the same formula with $\Sigma_a$; linearity makes it independent of the background density. The $k=0$ disturbance is a uniformly changed sheet and has potential proportional to $|z|$, rather than a decaying nonzero-<wavenumber> solution.
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