Unusual solutions of the Laplace equation

Abstract

Standard textbook treatments of electrostatic boundary-value problems in 2-D polar coordinates and 3-D spherical polar coordinates do not consider angular and radial coordinates on an equal basis. Series solutions are traditionally expressed in terms of oscillating angular functions and monotonic radial functions, but it is also possible to choose oscillating radial functions and monotonic angular functions. This unusual choice of functions is useful for the class of boundary-value problems where the potential is specified along the radial coordinate. Examples of such boundary-value problems are worked out.