2021/12/15 by Kiersten Meigs, Zachary Slepian, Meigs, Kiersten +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Cosmology and Nongalactic Astrophysics (astro-ph.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical functions and polynomials #astro-ph.CO #math.CA
paper · pdf · doi:10.48550/arxiv.2112.07809
20 pages, 1 figure, submitted
arxiv created 2021/12/15 · openalex publication_date 2021/12/15 · arxiv updated 2021/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We here present a method of performing integrals of products of spherical Bessel functions (SBFs) weighted by a power-law. Our method, which begins with double-SBF integrals, exploits a differential operator D defined via Bessel's differential equation. Application of this operator raises the power-law in steps of two. We also here display a suitable base integral expression to which this operator can be applied for both even and odd cases. We test our method by showing that it reproduces previously-known solutions. Importantly, it also goes beyond them, offering solutions in terms of singular distributions, Heaviside functions, and Gauss's hypergeometric, 2\rm F1 for all double-SBF integrals with positive semi-definite integer power-law weight. We then show how our method for double-SBF integrals enables evaluating arbitrary triple-SBF overlap integrals, going beyond the cases currently in the literature. This in turn enables reduction of arbitrary quadruple, quintuple, and sextuple-SBF integrals and beyond into tractable forms.