Details concerning fast and accurate electrostatic field calculation techniques based on the 3-dimensional surface charge simulation method by means of Fourier-type charges are presented. The techniques range over (a) the direct potential and field coefficient formulas, (b) a computation of Legendre function of the second kind with half integer degree—toroidal function Qm−½(x), (c) a high-speed integral and (d) an introduction of the associated toroidal function Sm−½(x). As an example for application, a tilted hemispherically tipped cylindrical rod electrode with a spherical dielectric is analysed and the results are demonstrated.
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Sato et al. (1986) studied this question.