On the asymptotics of Bessel functions in the Fresnel regime

TL;DR

This paper develops new asymptotic expansions for Bessel functions valid in the Fresnel regime, enabling efficient evaluation and analysis beyond the classical Fraunhofer regime, with demonstrated numerical effectiveness.

Contribution

It introduces a novel asymptotic expansion for Bessel functions applicable when |z| > ν, deep in the Fresnel regime, extending classical results and providing practical numerical tools.

Findings

New asymptotic expansions valid for |z| > ν

Reduction to classical expansions in the Fraunhofer regime

Numerical demonstrations of the method's effectiveness

Abstract

We introduce a version of the asymptotic expansions for Bessel functions $J_\nu(z)$, $Y_\nu(z)$ that is valid whenever $|z| > \nu$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the Fraunhofer regime (i.e. when $|z| > \nu^2$). As expected, in the Fraunhofer regime our asymptotics reduce to the classical ones. The approach is based on the observation that Bessel's equation admits a non-oscillatory phase function, and uses classical formulas to obtain an asymptotic expansion for this function; this in turn leads to both an analytical tool and a numerical scheme for the efficient evaluation of $J_\nu(z)$, $Y_\nu(z)$, as well as various related quantities. The effectiveness of the technique is demonstrated via several numerical examples. We also observe that the procedure admits far-reaching generalizations to wide classes of second order differential equations, to be reported at a later date.