On the electric field at the surface of a uniformly-charged cylindrical shell

In some recent papers, I have shown that the electric field at the surface of a uniformly-charged spherical shell (or a conducting sphere, since the charge distribution is the same) evaluates to half the field discontinuity across its surface. For a cylindrical shell, however, only a simple application of Gauss’s law for infinitely long shells is found in textbooks, which yields a field that is null inside the shell and, outside it, decays with the inverse of the distance to the central axis. Nothing is said for points located exactly at the shell. For those points, the amount of charge surrounded by a Gaussian surface coinciding with the shell is undefined, which makes the application of Gauss’s law inconclusive. In this note, by treating a cylindrical shell as a collection of identical charged rings, I derive that electric field in terms of elliptic integrals and then I show that, for very long shells, it reduces to half the field discontinuity across the shell.