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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 theFresnel regime), as opposed to the standard expansions that are applicable onlyin the Fraunhofer regime (i.e. when $|z| > \nu^2$). As expected, in theFraunhofer regime our asymptotics reduce to the classical ones. The approach isbased on the observation that Bessel's equation admits a non-oscillatory phasefunction, and uses classical formulas to obtain an asymptotic expansion forthis function; this in turn leads to both an analytical tool and a numericalscheme for the efficient evaluation of $J_\nu(z)$, $Y_\nu(z)$, as well asvarious related quantities. The effectiveness of the technique is demonstratedvia several numerical examples. We also observe that the procedure admitsfar-reaching generalizations to wide classes of second order differentialequations, to be reported at a later date.

On the asymptotics of Bessel functions in the Fresnel regime