The Application of Non-Integral Legendre Functions to Potential Problems

There are a number of important potential problems having axial symmetry and involving a conical boundary whose solutions require the use of zero-order Legendre functions of real but non-integral degree. The numerical evaluation of such problems is relatively simple once the zeros of these functions have been determined and when certain integrals have been evaluated. This paper presents tables and curves giving this information, together with a number of approximate expressions for these quantities. A number of potential distributions involving conducting cones, spheres, and rings are worked out to illustrate the application of these functions.