Forward-smooth high-order uniform Aharonov–Bohm asymptotics

The Aharonov–Bohm (AB) function, describing a plane wave scattered by a flux line, is expanded asymptotically in a Fresnel-integral based series whose terms are smooth in the forward direction and uniformly valid in angle and flux. Successive approximations are valid for large distance r from the flux (or short wavelength) but are accurate even within one wavelength of it. Coefficients of all the terms are exhibited explicitly for the forward direction, enabling the high-order asymptotics to be understood in detail. The series is factorally divergent, with optimal truncation error exponentially small in r . Systematic resummation gives further exponential improvement. Terms of the series satisfy a resurgence relation: the high orders are related to the low orders. Discontinuities in the backward direction get smaller order by order, with systematic cancellation by successive terms. The relation to an earlier scheme based on the Cornu spiral is discussed. (paper)