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.