Green’s function and image system for the Laplace operator in the prolate spheroidal geometry

In the present paper, electrostatic image theory is studied for Green’s function for theLaplace operator in the case where the fundamental domain is either the exterior or theinterior of a prolate spheroid. In either case, an image system is developed to consist ofa point image inside the complement of the fundamental domain and an additional symmetriccontinuous surfaceimage over a confocal prolate spheroid outside the fundamental domain, although theprocess of calculating such an image system is easier for the exterior than for theinterior Green’sfunction. The total charge of the surface image is zero and itscentroid is at the origin of the prolate spheroid. In addition, if the source is on thefocal axis outside the prolate spheroid, then the image system of the exteriorGreen’s functionconsists of a point image on the focal axis and a line image on the line segment betweenthe two focal points