Hyperasymptotics and the Stokes' phenomenon

Daalhuis, A. B. Olde

doi:10.1017/s0308210500030936

Cited by 11 publications

(5 citation statements)

References 8 publications

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“…If we restrict z to the right-half plane, the re-expansion of the remainder R N (z, µ, ν) can be done using only elementary functions. For a general theory of such re-expansions, see the papers of Olde Daalhuis [12,13]. Theorem 1.4.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the large argument asymptotics of the Lommel function via Stieltjes transforms

Nemes¹

2015

Asymptotic Analysis

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The aim of this paper is to investigate in detail the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.2010 Mathematics Subject Classification. 41A60, 30E15, 34M40.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the large argument asymptotics of the Lommel function via Stieltjes transforms

Nemes¹

2015

Asymptotic Analysis

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“…Factorial-over-power divergence [6]; Stokes' line smoothing [5]; Hyperasymptotics [11,34]; hyperterminants [12,13]. Stokes' smoothing [26,35].…”

Section: Hyperasymptoticsmentioning

confidence: 99%

Resurgent aspects of applied exponential asymptotics

Samuel¹,

Trinh²

2022

Preprint

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In many physical problems, it is important to capture exponentially-small effects that lie beyond-all-orders of a typical asymptotic expansion; when collected, the full expansion is known as the trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically in the context of singular non-linear perturbative differential or integral equations. Separate to applied exponential asymptotics, there exists a closely related line of development known as Écalle's theory of resurgence, which describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. This connection is realised through the process of Borel resummation. However, in contrast to singularly perturbed problems, Borel resummation and Écalle's resurgence theory have mainly focused on non-parametric asymptotic expansions (i.e. differential equations without a parameter). The relationships between these latter areas and applied exponential asymptotics has not been thoroughly examined, partially due to differences in language and emphasis. In this work, we explore these connections by developing an alternative framework for the factorial-over-power ansatz in exponential asymptotics that is centred on the Borel plane. Our work clarifies a number of elements used in apploed exponential asymptotics, such as the heuristic use of Van Dyke's rule and the universality of factorial-over-power ansatzes. Along the way, we provide a number of useful tools for probing more pathological problems in exponential asymptotics known to arise in applications; this includes problems with coalescing singularities, nested boundary layers, and more general late-term behaviours.

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“…These scales are manifested typically by the exponential prefactors of (often) divergent infinite series. Techniques exist for managing the divergence and analytical continuation of these series (Berry & Howls 1991;Howls 1992;Olde Daalhuis 1993;Howls et al 2004), which lead to exponentially improved numerical approximations that are uniformly valid in wider ranges of the underlying parameters (Olde Daalhuis 1998;Olde Daalhuis & Olver 1998). This approach can provide better error bounds (Boyd 1994) and algebraic methods to resolve complex geometric structures (Howls 1997;Delabaere & Howls 2002).…”

Section: Introductionmentioning

confidence: 99%

Exponential asymptotics and boundary-value problems: keeping both sides happy at all orders

Howls

2010

Proc. R. Soc. A.

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We introduce templates for exponential-asymptotic expansions that, in contrast to matched asymptotic approaches, enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter e → 0 + and have a single boundary layer at one end of the interval. For linear equations, the template is a transseries that takes the form of a sliding ladder of exponential scales. For nonlinear equations, the transseries template is a two-dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed. An exponential-asymptotic approach also reveals how boundary-value problems force the surprising presence of transseries in the linear case and negative powers of e terms in the series beyond all orders in the nonlinear case. We also demonstrate how these transseries can be resummed to generate multiplescales-type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter. Finally, we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion.

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Hyperasymptotics and the Stokes' phenomenon

Cited by 11 publications

References 8 publications

On the large argument asymptotics of the Lommel function via Stieltjes transforms

On the large argument asymptotics of the Lommel function via Stieltjes transforms

Resurgent aspects of applied exponential asymptotics

Exponential asymptotics and boundary-value problems: keeping both sides happy at all orders

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