COMPUTATIONS OF SPHEROIDAL HARMONICS WITH COMPLEX ARGUMENTS - A REVIEW WITH AN ALGORITHM

This paper not only reviews the various methodologies for evaluating t he angular and radial prolate and oblate spheroidal functions and thei r eigenvalues, but also presents an efficient algorithm which is devel oped with the software package MATHEMATICA. Two algorithms are develop ed for computation of the eigenvalues lambda(mn) and coefficients d(r) (mn). Important steps in programming are provided for estimating eigen values of the spheroidal harmonics with a complex argument c. Furtherm ore, the starting and ending points for searching for the eigenvalues by Newton's method are successfully obtained. As compared with the pub lished data by Caldwell [J. Phys. A 21, 3685 (1988)] or Press cr al. [ Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambrid ge University Press, Cambridge, 1992)] (for a real argument) and Oguch i [Radio Sci. 5, 1207 (1970)] (for a complex argument), the spheroidal harmonics and their eigenvalues estimated using this algorithm are of a much higher accuracy. In particular, a lot of tabulated data for th e intermediate coefficients d(rho \ r)(mn), the prolate and oblate rad ial spheroidal functions of the second kind, and their first-order der ivatives, as obtained by Flammer [Spheroidal Wave Functions (Stanford University Press, Stanford, CA, 1987)], are found to be inaccurate, al though these tabulated data have been considered as exact referenced r esults for about half century. The algorithm developed here for evalua ting the spheroidal harmonics with the MATHEMATICA program is also fou nd to be simple, fast, and numerically efficient, and of a much better accuracy than the other results tabulated by Flammer and others, bein g able to produce results of 100 significant digits or more. [S1063-65 1X(98)05511-1].