Summary: The purpose of this communication is to present an algorithm for evaluating zero-order Hankel transforms using ideas first put forward by

*L. N. G. Filon* [|On a quadrature formula for trigonometric integrals, Proc. R. Soc. Edinb. 49, 38-47 (1928/29;

JFM 55.0946.02)] 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

$h\left(p\right)$ and the latter is

${J}_{0}\left(rp\right)p)$. Here only

$h\left(p\right)$ is approximated by a quadratic over the basic subinterval instead of the entire integrand

$h\left(p\right){J}_{0}\left(rp\right)p$ being approximated. Since only

$h\left(p\right)$ 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

$exp\left(irp\right)$ is periodic and translationally invariant, whereas

${J}_{0}\left(rp\right)$ is an almost periodic decaying function.