= Potential of a charged cylindrical shell with angular Fourier data
{title2=$\Phi_n=\frac{Rs_n}{2n}(r_</r_>)^n$}
For an infinite circular cylinder of radius $R$, assume the <electrostatic potential> is independent of the axial coordinate. An angular <Fourier mode> $s_n\cos n\theta$, or $s_n\sin n\theta$, in the prescribed jump $\partial_r\Phi|_{R^+}-\partial_r\Phi|_{R^-}=-\sigma(\theta)$ produces the same angular mode in the potential, multiplied by
$$
\frac{Rs_n}{2n}\begin{cases}(r/R)^n,&r<R,\\(R/r)^n,&r>R.\end{cases}
$$
Here $r_<=\min(r,R)$ and $r_>=\max(r,R)$ in the abbreviated header formula. <Separation of variables> selects $r^n$ inside by regularity and $r^{-n}$ outside by decay. Continuity equates their boundary values, and the derivative jump is minus $2n/R$ times that value. A nonzero angular mean of the <surface charge density> introduces a logarithmic term, so zero mean is necessary for a potential decaying at infinity in this two-dimensional reduction.
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