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

Weber integrals∫ 0 ∞k 2 + μe − a k 2j n 2( pk ) d k and Beltrami integrals∫ 0 ∞k 2 + μe − bkj n 2( pk ) d k are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functionsj n 2with 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.