= Axisymmetric gravitational potential of a displaced cylindrical interface
{title2=$\psi_{\rm in}=-4\pi G\rho R K_0(kR)I_0(kr)\eta$}
Consider a uniform-density cylinder with an axisymmetric radial <Fourier mode> $\eta=\epsilon e^{ikz}$. Its displaced boundary gives a first-order <surface mass density> $\rho\eta$, although the interior density perturbation vanishes. The gravitational <Poisson equation> reduces to <Laplace equation> away from the interface. Regularity internally and decay externally give $\psi_{\rm in}=AI_0(kr)e^{ikz}$ and $\psi_{\rm out}=BK_0(kr)e^{ikz}$. Continuity at $R$ and the derivative jump $\psi'_{\rm out}-\psi'_{\rm in}=4\pi G\rho\epsilon e^{ikz}$, together with $I_0(\kappa)K'_0(\kappa)-I'_0(\kappa)K_0(\kappa)=-1/\kappa$, give the title amplitude. The prime differentiates the <modified Bessel function> argument. This axial mode is different from an azimuthal mode on a <gravitational potential of a displaced cylindrical interface>.
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