Simple analytical approximations to the integrals of the Bessel functionsJν: application to the transmittance of a circular aperture

Abstract

Two accurate, yet simple, analytic approximations to the integral of the Bessel function J0 are presented. These first and second-order approximations are obtained by improving on the recently developed method known as two-point quasi-rational approximations. The accuracy of the first-order approximant is better than 0.05. The second-order approximant is practically indistinguishable from the true integral, even for very large values of the argument (overall accuracy is better than 0.002 05). Our approximants are, in addition, analytic and therefore replace with significant advantages both the well known power series and the asymptotic formulae of the integral. Approximants to the transmittance function of a plane wave through a circular aperture are derived, a problem which arises in diffraction theory and particle scattering. The second-order approximant to the transmittance is analytic too, and can be evaluated for small and large values of the argument, just with a hand-calculator. Its accuracy is better than 0.0011. As an extension, two first-order approximations to the integrals of the Bessel functions Jν, of fractional order ν, are derived.