Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives

Abstract

Abstract.Alternative expressions for calculating the prolate spheroidal radial functions of the second kind (c. £) and their first derivatives with respect to £ are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so. The first alternative expression is obtained from the expansion of the product of (c, £) and the prolate spheroidal angular function of the first kind (c, rj) in a series of products of the corresponding spherical functions. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all (o) parameter ranges. The second alternative expression for Rml (c, £) involves an integral of the product of (c, rf) and a spherical Neumann function kernel. It provides accurate values when £ is near unity and I -mis not too large, even when c becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.

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“…As such they can provide accurate results when c ξ 2 − 1 is very large if one uses the partial sum of the series up to and including the term where the relative contribution is a minimum. We were able to demonstrate this for Mc (2) n using techniques described in [6] for computing accurate values of the prolate radial spheroidal functions of the second kind.…”

“…In [6] we showed how a similar expression for the prolate spheroidal radial functions leads to the traditional expressions for computing these functions. We also showed how it can be used to compute accurate values of the second kind R (2) mn (ξ, c) over larger ranges than previously attainable by choosing progressively smaller values of v, starting with π/2 and ending with 0, in order to maintain accuracy as the order n increases.…”

“…We compute the radial functions of the third kind from (12) using techniques described in [6] for R (2) mn (ξ, c). First we factor out of the series the expansion coefficient and the Bessel function K for r = n. We then sum the series forward from the r = n term, starting with unity for Mc (3) n and n for Ms (3) n .…”

“…In [6] we explored similar integral expressions for the radial spheroidal functions of the second kind R (2) mn (ξ, c) in terms of the angular spheroidal functions S mn (η, c). The kernel function for n -m even was the product of the spherical Neumann function y m (kR) and sin m (ϑ), where R is the spherical radius and ϑ is the spherical polar angle.…”

“…In [6] we described new procedures for calculating accurate values of the radial prolate spheroidal functions of the second kind over parameter ranges much wider than previously possible. In the present paper we successfully apply the procedures developed for prolate spheroidal geometry to the accurate calculation of radial Mathieu functions.…”