Paper detail

On the asymptotics of Bessel functions in the Fresnel regime

We introduce a version of the asymptotic expansions for Bessel functions $J_ν(z)$, $Y_ν(z)$ that is valid whenever $|z| > ν$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the Fraunhofer regime (i.e. when $|z| > ν^2$). 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 $J_ν(z)$, $Y_ν(z)$, 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.