## 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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### References:

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