Zeitschrift für angewandte Mathematik und Physik

, Volume 65, Issue 6, pp 1077–1105 | Cite as

High-index asymptotics of spherical Bessel products averaged with modulated Gaussian power laws

  • Roman Tomaschitz


Bessel integrals of type \({\int_0^\infty {g(k)j_l^{(m)}(k)j_l^{(n)} (k)k^{2}{\rm d}k}}\) are investigated, where the kernel g(k) is a modulated Gaussian power-law distribution \({k^\mu{e}^{-ak^{2}-(b+{{\rm i}} \omega)k}}\), and the j l (m) are multiple derivatives of spherical Bessel functions. 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. Two methods allowing efficient numerical calculation of these integrals are presented, covering Bessel indices l in the currently accessible multipole range 0 ≤ l ≤ 104 and beyond. The first method is based on a representation of spherical Bessel functions by Lommel polynomials. Gaussian power-law averages can then be calculated in closed form as finite Hankel series of parabolic cylinder functions, which allow high-precision evaluation. The second method is asymptotic, covering the high-l regime, and is applicable to general distribution functions g(k) in the integrand; it is based on the uniform Nicholson approximation of the Bessel derivatives in conjunction with an integral representation of squared Airy functions. A numerical comparison of these two methods is performed, employing Gaussian power laws and Kummer distributions to average the Bessel products.

Mathematics Subject Classification (2000)

33C10 33F05 


Spherical Bessel functions Airy functions High-index asymptotics Nicholson approximation Lommel polynomials Hankel series Gaussian power-law densities Kummer distributions 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of PhysicsHiroshima UniversityHigashi-HiroshimaJapan

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