Tomaschitz, R. (2014). Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high-index asymptotics, Mathematical Methods in the Applied Sciences 37,1249-1272, DOI: 10.1002/mma.2882
Abstract (WILEYonline, SAO/NASA ADS, Zbl 06303271)
Weber integrals {int_0^infty {k^{2+μ}{e}^{-ak2}j_n^{2} (pk)dk}} and Beltrami integrals {int_0^infty {k^{2+μ}{e}^{-bk}j_n^{2} (pk)dk}} are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functions j {/n 2} with 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, based on 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 to their high-index asymptotics.
MSC 2000: 33C10, 33F05
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Title: |
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Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high-index asymptotics |
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Authors: |
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Tomaschitz, Roman |
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Affiliation: |
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AA(Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan) |
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Publication: |
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Mathematical Methods in the Applied Sciences, vol. 37, issue 9, pp. 1249-1272 |
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Publication Date: |
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06/2014 |
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AUTHOR |
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Keywords: |
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Squared spherical Bessel functions, Weber integrals, Beltrami integrals, Finite Legendre series, Gaussian power-law densities, High-index asymptotics, Debye expansion, Airy approximation |
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Abstract Copyright: |
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(c) 2013 John Wiley & Sons, Ltd. |
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DOI: |
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Bibliographic Code: |
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description: Roman Tomaschitz (2014) Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high-index asymptotics, Math. Meth. Appl. Sci. 37,1249.
Keywords: squared spherical Bessel functions, Weber integrals, Beltrami integrals, finite Legendre series, Gaussian power-law densities, high-index asymptotics, Debye expansion, Airy approximation
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