An evaluation of Legendre functions and their zeros for application to conical boundary value problems

The solution of boundary value problems involving the simplest of conical geometries requires the use of Legendre functions of real argument and non-integer degree. Furthermore, to satisfy the boundary conditions, the values of the non-integer degree which cause the Legendre functions to vanish for a given apex angle must be known. In contrast to the wealth of information available on Legendre functions of integer degree, the literature on the functions when the degree is non-integer is quite limited. Tables of zeros and plots of the functions versus the degree are virtually non-existent. This paper is devoted to the development of a method by which the zeros and the function values may be obtained when the degree is non-integer. Tables and plots of zeros and function values for 75 degree values for each of 36 argument values are included. These tables and plots provide the data which has been used to numerically evaluate a problem with a given conical configuration.Electrical and Computer Engineering, Department o