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

Abstract

The aim of this paper is to derive new representations for the Hankel functions, the Bessel functions and their derivatives, exploiting the reformulation 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 these representations, we obtain a number of properties of the asymptotic expansions of the Hankel and Bessel functions and their derivatives of nearly equal order and argument, including explicit and numerically computable error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

41 pages, 2 figures. Accepted for publication in Mathematische Annalen. arXiv admin note: substantial text overlap with arXiv:1309.2209. text overlap with arXiv:1309.2209, arXiv:1408.0674

References in corpus (2)

• An explicit formula for the coefficients in Laplace's method
• The resurgence properties of the large order asymptotics of the Hankel and Bessel functions