Uniform asymptotic smoothing of Stokes’s discontinuities

Abstract

Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form. In terms of the singulant F - the difference between the larger and smaller exponents, and real on the Stokes line - the change in the multiplier is the error function $\pi^{-\frac{1}{2}}\int^\sigma_{-\infty}dt \exp (-t^2) \text{where} \sigma = ImF/(2ReF)^{\frac{1}{2}}.$ The derivation requires control of exponentially small terms in the dominant series; this is achieved with Dingle's method of Borel summation of late terms, starting with the least term. In numerical illustrations the multiplier is extracted from Dawson's integral (erfi) and the Airy function of the second kind (Bi): the small exponential emerges in the predicted universal manner from the dominant one, which can be 10$^{10}$ times larger.