Electric potentials and field lines for uniformly-charged tube and cylinder expressed by Appell's hypergeometric function and integration of $Z(u|m) \mathrm{sc}(u|m)$

Abstract

The closed-form expressions of electric potentials and field lines for a uniformly-charged tube and cylinder are presented using elliptic integrals and Appell's hypergeometric functions, where field lines are depicted by introducing the concept of the field line potential in axisymmetric systems, whose contour lines represent electric field lines outside the charged region, thought of as an analog of the conjugate harmonic function in the presence of non-uniform metric. The field line potential for the tube shows a multi-valued behavior and enables us to define a topological charge. The integral of $Z(u|m)\operatorname{sc}(u|m)$, where $ Z $ and $ \operatorname{sc} $ are the Jacobi zeta and elliptic functions, is also expressed by Appell's hypergeometric function as a by-product, which was missing in classical tables of formulas. In the Addendum appended after the main article, several relevant references are provided and the decomposition of the solution by ``degrees of transcendence'' is proposed. The attached supplementary calculations provide detailed derivations of several formulas including the integral in the title and discuss the resemblance between the field line potential for tube and the electric potential for disk.