  The aim of this paper is to derive new representations for the Hankel and
Bessel functions, 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 large order asymptotic expansions of the Hankel and Bessel
functions due to Debye, including explicit and realistic error bounds,
asymptotics for the late coefficients, exponentially improved asymptotic
expansions, and the smooth transition of the Stokes discontinuities.
