The resurgence properties of the large order asymptotics of the Anger--Weber function I

The aim of this paper is to derive new representations for the Anger--Weber function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these representations, we obtain a number of properties of the large order asymptotic expansions of the Anger--Weber function, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.