Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high‐index asymptotics

Abstract

Weber integrals and Beltrami integrals are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functions 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 on the basis of 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 with their high‐index asymptotics. Copyright © 2013 John Wiley & Sons, Ltd.