Theory and Computation of the Spheroidal Wave Functions

TL;DR

This paper introduces a Mathematica package for high-precision computation of spheroidal wave functions, handling complex parameters and special cases, advancing the accuracy and scope of existing software.

Contribution

It provides a new software tool that computes spheroidal wave functions with arbitrary precision for complex parameters, surpassing previous limitations.

Findings

Computes spheroidal wave functions to arbitrary precision.

Includes special cases and asymptotic expansions.

Provides high-precision numerical examples.

Abstract

In this paper we report on a package, written in the Mathematica computer algebra system, which has been developed to compute the spheroidal wave functions of Meixner [J. Meixner and R.W. Schaefke, Mathieusche Funktionen und Sphaeroidfunktionen, 1954] and is available online (www.physics.uwa.edu.au/~falloon/spheroidal/spheroidal.html). This package represents a substantial contribution to the existing software, since it computes the spheroidal wave functions to arbitrary precision for general complex parameters mu, nu, gamma and argument z; existing software can only handle integer mu, nu and does not give arbitrary precision. The package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter gamma. The spheroidal wave functions of Flammer [C. Flammer, Spheroidal Wave Functions, 1957] are included as a special case of Meixner's more general functions. This paper presents a concise review of the general theory of spheroidal wave functions and a description of the formulas and algorithms used in their computation, and gives high-precision numerical examples.