Tomaschitz, R. (2013). Bessel integrals in epsilon expansion: Squared spherical Bessel functions averaged with Gaussian power-law distributions, Applied Mathematics and Computation 225, 228-241, DOI: 10.1016/j.amc.2013.09.035
Abstract (ScienceDirect, SAO/NASA ADS)
Bessel integrals of type {int_0^infty {k^{μ+2}{e}^{-ak2-(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 power-law 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 high-precision evaluation. In the case of integer power-law 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 power-law 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 high-index Airy approximation over a wide range of integer Bessel indices l.
MSC 2000: 33C10, 33F05
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Title: |
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Bessel integrals in epsilon expansion: Squared spherical Bessel functions averaged with Gaussian power-law distributions |
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Authors: |
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Tomaschitz, Roman |
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AA(Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan) |
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Applied Mathematics and Computation, vol. 225, pp. 228-241 |
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12/2013 |
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Keywords: |
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Squared spherical Bessel functions, Regularization of Hankel series, Gaussian power-law densities, Kummer distributions, Airy approximation of Bessel integrals |
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description: Roman Tomaschitz (2013) Squared spherical Bessel functions averaged with Gaussian power-law distributions, Appl. Math. Comput. 225, 228.
Keywords: squared spherical Bessel functions, regularization of Hankel series, Hermite residuals, Gaussian power-law densities, Kummer distributions, Airy approximation of Bessel integrals, combinatorial identities for Hankel coefficients
Highlights
The high-index 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 high-precision 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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