Tomaschitz, R. (2013). Bessel integrals in epsilon expansion: Squared spherical Bessel functions averaged with Gaussian powerlaw distributions, Applied Mathematics and Computation 225, 228241, DOI: 10.1016/j.amc.2013.09.035
Abstract (ScienceDirect, SAO/NASA ADS)
Bessel integrals of type {int_0^infty {k^{μ+2}{e}^{ak^{2}(b+{i} ω)k}j_l^{2} (pk)dk}} are studied, where the squared spherical Bessel function j {_{/l }^{2}} is averaged with a modulated Gaussian powerlaw density. These integrals define the multipole moments of Gaussian random fields on the unit sphere, arising in multipole fits of temperature and polarization power spectra of the cosmic microwave background. The averages can be calculated in closed form as finite Hankel series, which allow highprecision evaluation. In the case of integer powerlaw exponents μ, singularities emerge in the series coefficients, which requires ɛ expansion. The pole extraction and regularization of singular Hankel series is performed, for integer Gaussian powerlaw densities as well as for the special case of Kummer averages (a = 0 in the exponential of the integrand). The singular ɛ residuals are used to derive combinatorial identities (sum rules) for the rational Hankel coefficients, which serve as consistency checks in precision calculations of the integrals. Numerical examples are given, and the Hankel evaluation of Gaussian and Kummer averages is compared with their highindex Airy approximation over a wide range of integer Bessel indices l.
MSC 2000: 33C10, 33F05
Title: 

Bessel integrals in epsilon expansion: Squared spherical Bessel functions averaged with Gaussian powerlaw distributions 
Authors: 

Tomaschitz, Roman 
Affiliation: 

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

Applied Mathematics and Computation, vol. 225, pp. 228241 
Publication Date: 

12/2013 
Origin: 

AUTHOR 
Keywords: 

Squared spherical Bessel functions, Regularization of Hankel series, Gaussian powerlaw densities, Kummer distributions, Airy approximation of Bessel integrals 
Abstract Copyright: 

ELSEVIER 
DOI: 


Bibliographic Code: 

description: Roman Tomaschitz (2013) Squared spherical Bessel functions averaged with Gaussian powerlaw distributions, Appl. Math. Comput. 225, 228.
Keywords: squared spherical Bessel functions, regularization of Hankel series, Hermite residuals, Gaussian powerlaw densities, Kummer distributions, Airy approximation of Bessel integrals, combinatorial identities for Hankel coefficients
Highlights
The highindex evaluation of integrals containing squared spherical Bessel functions is studied.
The integrals arise as spectral averages in multipole expansions of spherical Gaussian random fields.
A highprecision integration technique based on finite Hankel series in epsilon regularization is developed.
An Airy approximation of the integrals is derived using uniform Nicholson asymptotics.
The finite series evaluation is compared with the Airy approximation over an extended range of Bessel indices.
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