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On the asymptotics of Bessel functions in the Fresnel regime. (English) Zbl 1321.33007

Summary: We introduce a version of the asymptotic expansions for Bessel functions \(J_\nu(z)\), \(Y_\nu(z)\) that are 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 formulae 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.

MSC:

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33F05 Numerical approximation and evaluation of special functions
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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