Analytical solution of the field integrals of a cylindrical grid element*

Many axisymmetric physical problems related to the calculation of gravitational or electro-magnetic fields can be favorably described by a Poisson potential problem - or the associated field problem - on a cylindrical grid. Numerical integration of the field integrals on such a grid can become expensive in terms of computing time or requires to compromise the accuracy, respectively. Here the analytical solution of the field integrals over a cylindrical volume element are presented and a very efficient implementation is discussed. The results are useful, e.g., for modeling space-charge effects in beams in cylindrical coordinates or gravitational effects in galaxies. The numerical implementation is based on modified relations originally derived by Carlson. A new limiting value and a new transformation relation of the complex elliptic integral of third kind are presented. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).