= Solution
After a radial <Fourier transform>, Poisson's equation away from the sheet becomes
$$
(\partial_z^2-k^2)\widetilde\Phi_D=0.
$$
The decaying, reflection-symmetric solution is $A e^{-|k||z|}$. Integrating Poisson's equation across $z=0$ gives the derivative jump
$$
\partial_z\widetilde\Phi_D(0^+)
-\partial_z\widetilde\Phi_D(0^-)
=4\pi G\widetilde\Sigma,
$$
so $-2|k|A=4\pi G\widetilde\Sigma$. Thus the midplane potential is
$$
\boxed{\widetilde\Phi_{D,m}
=-\frac{2\pi G}{|k|}\widetilde\Sigma}.
$$
Solved by gpt-5.6-sol high.
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