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Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high‐index asymptotics
Author(s) -
Tomaschitz Roman
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2882
Subject(s) - bessel function , mathematics , mathematical analysis , series (stratigraphy) , multipole expansion , trigonometric integral , power series , bessel polynomials , slater integrals , legendre polynomials , spherical harmonics , series expansion , asymptotic expansion , bessel process , order of integration (calculus) , struve function , legendre function , gaussian , orthogonal polynomials , gegenbauer polynomials , quantum mechanics , physics , classical orthogonal polynomials , paleontology , trigonometry , biology
Weber integrals∫ 0 ∞k 2 + μe − a k 2j n 2( pk ) d k and Beltrami integrals∫ 0 ∞k 2 + μe − bkj n 2( pk ) d k are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functionsj n 2with Gaussian or exponentially cut power‐law densities. Finite series representations of the integrals are derived for integer power‐law index μ , which admit high‐precision evaluation at low and moderate Bessel index n . At high n , numerically tractable uniform asymptotic approximations are obtained on the basis of the Debye expansion of modified spherical Bessel functions in the case of Weber integrals. The high‐ n approximation of Beltrami integrals can be reduced to Legendre asymptotics. The Airy approximation of Weber and Beltrami integrals is derived as well, and numerical tests are performed over a wide range of Bessel indices by comparing the exact finite series expansions of the integrals with their high‐index asymptotics. Copyright © 2013 John Wiley & Sons, Ltd.