Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy Original Research Article

The purpose of this communication is to present an algorithm for evaluating zero-order Hankel tranforms using ideas first put foward by Filon in the context of finite range Fourier integrals. In Filon quadrature philosophy, the integrand is separated into the product of an (assumed) slowly varying component and a rapidly oscillating one (in our problem, the former is h(p) and the latter is J0(rp)p; see equation (1.2). Here only h(p) is approximated by a quadratic over the basic subinterval instead of the entire integrand h(p)J0(rp)p being approximated. Since only h(p) has to be approximated, only a relatively small number of subintervals is required. In addition, the error incurred is relatively independent of the magnitude of r. There is a profound difference between the finite range Fourier integral and the zero-order Hankel transform in that exp(irp) is periodic and translationally invariant, whereas J0(rp) is an almost periodic decaying function.