 Axisymmetric gravitational potential of a displaced cylindrical interface

Consider a uniform-density cylinder with an axisymmetric radial Fourier mode $\eta =ϵe^{ikz}$. 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 ${\psi }_{\mathrm{in}}=A{I}_0(kr)e^{ikz}$ and ${\psi }_{\mathrm{out}}=B{K}_0(kr)e^{ikz}$. Continuity at $R$ and the derivative jump ${\psi }_{\mathrm{out}}^{\prime }-{\psi }_{\mathrm{in}}^{\prime }=4\pi G\rho ϵe^{ikz}$, together with $I_0(\kappa )K_0^{\prime }(\kappa )-I_0^{\prime }(\kappa )K_0(\kappa )=-1/\kappa$, 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.