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The aim of this paper is to derive new representations for the Hankelfunctions, the Bessel functions and their derivatives, exploiting thereformulation of the method of steepest descents by M. V. Berry and C. J. Howls(Berry and Howls, Proc. R. Soc. Lond. A 434 (1991) 657--675). Using theserepresentations, we obtain a number of properties of the asymptotic expansionsof the Hankel and Bessel functions and their derivatives of nearly equal orderand argument, including explicit and numerically computable error bounds,asymptotics for the late coefficients, exponentially improved asymptoticexpansions, and the smooth transition of the Stokes discontinuities.

The resurgence properties of the Hankel and Bessel functions of nearly equal order and argument