Theory and computation of spheroidal wavefunctions

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