Computations of spheroidal harmonics with complex arguments: A review with an algorithm

This paper not only reviews the various methodologies for evaluating the angular and radial prolate and oblate spheroidal functions and their eigenvalues, but also presents an efficient algorithm which is developed with the software package Mathematica. Two algorithms are developed for computation of the eigenvalues λₘₙ and coefficients dᵣᵐⁿ. Important steps in programming are provided for estimating eigenvalues of the spheroidal harmonics with a complex argument c. Furthermore, the starting and ending points for searching for the eigenvalues by Newton's method are successfully obtained. As compared with the published data by Caldwell [J. Phys. A 21, 3685 (1988)] or Press et al. [Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992)] (for a real argument) and Oguchi [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 the intermediate coefficients d_ρ|rᵐⁿ, the prolate and oblate radial spheroidal functions of the second kind, and their first-order derivatives, as obtained by Flammer [Spheroidal Wave Functions (Stanford University Press, Stanford, CA, 1987)], are found to be inaccurate, although these tabulated data have been considered as exact referenced results for about half a century. The algorithm developed here for evaluating the spheroidal harmonics with the Mathematica program is also found to be simple, fast, and numerically efficient, and of a much better accuracy than the other results tabulated by Flammer and others, being able to produce results of 100 significant digits or more.