A naive procedure for computing angular spheroidal functions

TL;DR

This paper introduces a straightforward algorithm for calculating eigenvalues and eigenfunctions of the angular spheroidal wave equation, utilizing a series expansion and Newton's method, applicable even for complex parameters.

Contribution

It develops a novel, simple procedure based on a scarcely used method for efficient computation of spheroidal functions and eigenvalues.

Findings

The method accurately computes eigenvalues and eigenfunctions.

It is effective for complex prolateness parameters.

Results compare favorably with existing procedures.

Abstract

An algorithm for computing eigenvalues and eigenfunctions of the angular spheroidal wave equation, based on a known but scarcely used method, is developed. By requiring the regularity of the wave function, represented by its series expansion, the eigenvalues appear as the zeros of a one variable function easily computable. The iterative extended Newton method is suggested as especially suitable for determining those zeros. The computation of the eigenfunctions is then immediate. The usefulness of the method, applicable also in the case of complex values of the "prolateness" parameter, is illustrated by comparing its results with those of procedures used by other authors.