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

Abstract

Abstract. In this paper, we derive a new representation for the Anger-Weber function, employing the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373-396). As a consequence of this representation, we deduce a number of properties of the large order asymptotic expansion of the Anger-Weber function, including explicit and realistic error bounds, asymptotic approximations for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. IntroductionIn the first part of this series of papers [7], we proved new resurgence-type representations for the remainder term of the asymptotic expansion of the Anger-Weber function A −ν (λν) with complex ν and λ ≥ 1. These resurgence formulas have different forms according to whether λ > 1 or λ = 1. The main goal of this paper is to derive a similar representation for the Anger-Weber function A ν (λν) with complex ν and λ > 0. Our derivation is based on the reformulation of the method of steepest descents by Howls [4]. Using this representation, we obtain a number of properties of the large order asymptotic expansion 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.Our first theorem describes the resurgence properties of the asymptotic expansion of A ν (λν) for λ > 0. The notations follow the ones given in [11, p. 298]. Throughout this paper, empty sums are taken to be zero. Theorem 1.1. Let λ > 0 be a fixed positive real number, and let N be a non-negative integer. Then we have it (λit) dt.

Uniform asymptotic expansions for Lommel, Anger‐Weber, and Struve functions

“…If 𝜈 > 𝜇 + 1, the leading terms in its series expansion at 𝑧 = 0 are the same as those for 𝑠 𝜇,𝜈 (𝑧), given by ( 6) and (7), with 𝑘 running from 0 to 𝐾, where 𝜇 + 1 + 2𝐾 < 𝜈. If 𝜈 − 𝜇 ≠ 1, 3, 5, …, subsequent terms come from (6) and the series for 𝐽 𝜈 (𝑧) (Ref.…”

“…10.73] for a similar approach for 𝑆 𝜇,𝜈 (𝑧). For example, if ℜ(𝜈) > ℜ(𝜇) + 3, we have from ( 6), (7), and (24) as 𝑧 → 0…”

Uniform asymptotic expansions for Lommel, Anger-Weber and Struve functions

“…Most of these come from integral methods. For recent work we mention [7] and [8] where Nemes considered asymptotic expansions due to Watson [15,Sect. 10.15] for the Anger-Weber functions A ±ν (z) for large complex ν.…”

“…In [6] explicit formulas were given for (s)th coefficient in terms of the (2s)th derivative of a certain function. In [7] and [8] the coefficients were expressed as a double sum of generalised Bernoulli polynomials, a recursion formula equivalent to (4.27) was derived, and bounds for the error terms were given. In all of these the expansions (in terms of our notation) z is positive and ν is complex.…”