  In this paper we derive new representations for the incomplete gamma
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 asymptotic expansions
of the incomplete gamma function with large arguments, including explicit and
realistic error bounds, asymptotics for the late coefficients, exponentially
improved asymptotic expansions, and the smooth transition of the Stokes
discontinuities.
