Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1708.07929

Accurate calculation of oblate spheroidal wave functions

Arnie L. Van Buren
View original source ↗Read source document ↗Export metadata ↓

Abstract

Alternative expressions for calculating the oblate spheroidal radial functions of both kinds R1ml and R2ml are shown to provide accurate values over very large parameter ranges using 64 bit arithmetic, even where the traditional expressions fail. First is the expansion of the product of a radial function and the angular function of the first kind in a series of products of the corresponding spherical functions, with the angular coordinate being a free parameter. Setting the angular coordinate equal to zero leads to accurate values for R2ml when the radial coordinate xi is larger than 0.01 and l is somewhat larger than m. Allowing it to vary with increasing l leads to highly accurate values for R1ml over all parameter ranges. Next is the calculation of R2ml as an integral of the product of S1ml and a spherical Neumann function kernel. This is useful for smaller values of xi. Also used is the near equality of pairs of low order eigenvalues when the size parameter c is large that leads to accurate values for R2ml using neighboring accurate values for R1ml. A modified method is described that provides accurate values for the necessary expansion coefficients when c is large and l is near m and traditional methods fail. A resulting Fortran computer program Oblfcn almost always provides radial function values with at least 8 accurate decimal digits using 64 bit arithmetic for m up to at least 1000 with c up to at least 2000 when xi is greater than 0.000001 and c up to at least 5000 when xi is greater than 0.01. Use of 128 bit arithmetic extends the accuracy to 15 or more digits and extends xi to all values other than zero. Oblfcn is freely available.

Explore related subjects

math.CA

Keep this discovery

BibTeXRIS

Arnie L. Van Buren. 2017-08-26. Accurate calculation of oblate spheroidal wave functions. https://arxiv.org/abs/1708.07929

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the log-concavity of the composite Bessel function $x^{\alpha}J_{\nu }\left( \beta x^{\gamma}\right) $

For a twice differentiable function $f:\left( a,b\right) \rightarrow \mathbb{R}$ define $v\left( f\right) =f^{\prime}f^{\prime}-f^{\prime\prime }f.$ It is well known that the positivity of $v\left( f\right) $ implies that the function $\left\vert f\right\vert $ is strictly log-concave on each subinterval which does not contain zeros of $f.$ In this paper we provide criteria for the positivity of $v\left( F\right) $ for the composite Bessel function $F\left( x\right) =J_{\alpha,\beta,\gamma,\nu}\left( x\right) :=x^{\alpha}J_{\nu}\left( \beta x^{\gamma}\right) $ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu.$

math.CA↗
arXiv2026

Riesz capacity ratios with negative exponents

We investigate sharp inequalities for ratios of Riesz capacities with negative exponents by combining computational experiments with rigorous analysis. For finite subsets of the line, we prove positivity of equilibrium masses when $-1<p<0$, enabling numerical tests of conjectured extremal ratios. In the plane, comparisons of the disk with regular polygon vertex sets reveal a cascade of transitions among the tested competitors and suggest a precise conjecture for the equilibrium measure of odd polygons, for which we give a partial proof. Numerical intersections of equality curves show that the regions where these sets outperform the disk are not simply nested. Similar numerical intersections occur in three dimensions between the regular-simplex equality curve and those of explicit five-point and six-point configurations. Motivated by the dimensional dependence of these comparisons, we prove that for each fixed $p<-2<q<0$, the regular simplex has a larger capacity ratio than the ball in all sufficiently large dimensions. Accompanying Python and Mathematica code supports reproduction and further testing of the conjectures.

math.CA↗
arXiv2026

Shorter proof of dimension-free $L^p$ estimates for maximal Riesz transforms

We provide a shorter and more direct proof of $L^p$ estimates for maximal Riesz transforms (of an arbitrary order) in terms of the corresponding Riesz transforms, with a constant independent of the dimension of the Euclidean space $\mathbb R^d$. This result was originally proved by Mateu, Orobitg, P\'erez and Verdera with a constant depending on the dimension, and improved to a dimension-free inequality by Kucharski, Wr\'obel and Zienkiewicz.

math.CA↗