A Toroidal Harmonic Representation of the Yukawa-potential Kernel for a Circular Cylindrical Source

Abstract

A true cylindrical series expansion of the Yukawa-or screened Coulomb-potential kernel is developed for a finite circular cylindrical source through the application of a toroidal harmonic expansion. The Yukawa kernel is separated into a singular and nonsingular part. The singular part is expanded in terms of the associated toroidal harmonics and the nonsingular part is expanded in terms of an elementary binomial expansion.

The Magnetic Field from Cylindrical Arc Coils and Magnets: A Compendium with New Analytic Solutions for Radial Magnetization and Azimuthal Current

“…The "standard" Legendre elliptic integrals welldescribe the field from these regular cylindrical shapes because the series expansion converges on the exact solution rapidly and with a low number of terms. [37,70,71] These articles [44,57,69] are important as they demonstrate a general series that may similarly well-describe the geometry (source/field) and converge on the exact solution with a reasonably low number of terms. What the articles do not provide is a detailed error analysis or method for choosing an appropriate number of terms.…”

“…[ 56 ] Using a similar approach to [44], the axisymmetric analytic solution is given by ref. [57] as a compact series of hypergeometric functions.…”

“…The previously discussed articles by Selvaggi for diametric [ 44 ] and radial magnetization [ 57 ] use an integral‐series Green's function for the axisymmetric geometry. For axial magnetization, Selvaggi [ 69 ] demonstrates a similar analytic solution for both the axisymmetric and non‐axisymmetric cases using a series containing hypergeometric functions — importantly, claiming a “quick” convergence.…”