The rotationally symmetric solution &fgr; (r,z) satisfying ∇2&fgr;=0 in the region exterior to a cylinder of radiusAon which the boundary values are given by &fgr; (A,z<0) =&fgr; (A,z≳L) =0 and &fgr; (A,z) =F(z) for 0<z<Lis represented by a Fourier‐Bessel integral, which can be approximated by a Fourier‐Bessel series. It is shown how the quality of the approximation can be improved by ensuring that the period of the series is sufficiently large. For a finite period the error of the approximation can be understood either in terms of induced charge distributions on two parallel planes, or by considering an infinity of image charge distributions atr=A. The error can be reduced by eliminating the images of potential distributions, which is done numerically through the addition of a suitably chosen auxiliary Fourier‐Bessel series.