Tomaschitz, R. (2014). Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and highindex asymptotics, Mathematical Methods in the Applied Sciences 37,12491272, DOI: 10.1002/mma.2882
Abstract (WILEYonline, SAO/NASA ADS, Zbl 06303271)
Weber integrals {int_0^infty {k^{2+μ}{e}^{ak^{2}}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 powerlaw densities. Finite series representations of the integrals are derived for integer powerlaw index μ, which admit highprecision 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 highn 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 highindex asymptotics.
MSC 2000: 33C10, 33F05
Title: 

Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and highindex asymptotics 
Authors: 

Tomaschitz, Roman 
Affiliation: 

AA(Department of Physics, Hiroshima University, 131 Kagamiyama, HigashiHiroshima 7398526, Japan) 
Publication: 

Mathematical Methods in the Applied Sciences, vol. 37, issue 9, pp. 12491272 
Publication Date: 

06/2014 
Origin: 

AUTHOR 
Keywords: 

Squared spherical Bessel functions, Weber integrals, Beltrami integrals, Finite Legendre series, Gaussian powerlaw densities, Highindex asymptotics, Debye expansion, Airy approximation 
Abstract Copyright: 

(c) 2013 John Wiley & Sons, Ltd. 
DOI: 


Bibliographic Code: 

description: Roman Tomaschitz (2014) Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and highindex asymptotics, Math. Meth. Appl. Sci. 37,1249.
Keywords: squared spherical Bessel functions, Weber integrals, Beltrami integrals, finite Legendre series, Gaussian powerlaw densities, highindex asymptotics, Debye expansion, Airy approximation
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