Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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TL;DR: In this article, the authors implique l'accord avec les conditions générales d'utilisation (http://www.compositio.org/legal.php).
Abstract: © Foundation Compositio Mathematica, 1962-1964, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
379 citations
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353 citations
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01 Jan 2006TL;DR: Asymptotic analysis of exponential integrals has been studied extensively in the literature, see as discussed by the authors for a survey of asymptotics of exponential and weakly nonlinear waves.
Abstract: Fundamentals: Themes of asymptotic analysis The nature of asymptotic approximations Asymptotic analysis of exponential integrals: Fundamental techniques for integrals Laplace's method for asymptotic expansions of integrals The method of steepest descents for asymptotic expansions of integrals The method of stationary phase for asymptotic analysis of oscillatory integrals Asymptotic analysis of differential equations: Asymptotic behavior of solutions of linear second-order differential equations in the complex plane Introduction to asymptotics of solutions of ordinary differential equations with respect to parameters Asymptotics of linear boundary-value problems Asymptotics of oscillatory phenomena Weakly nonlinear waves Appendix: Fundamental inequalities Bibliography Index of names Subject index.
343 citations
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TL;DR: In this paper, the authors explore quadrature methods for highly oscillatory integrals, which approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency.
Abstract: In this paper, we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome is two families of methods, one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of [Filon (Filon 1928 Proc. R. Soc. Edinb. 49 , 38–47)][1]. Both kinds of methods approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency. We determine asymptotic properties of these methods, proving, perhaps counterintuitively, that their performance drastically improves as frequency grows. The paper is accompanied by numerical results that demonstrate the potential of this set of ideas.
[1]: #ref-4
335 citations