Algebraic analysis of singular perturbation theory(Translations of Mathematical Monographs 227)By Takahiro Kawai and Yoshitsugu Takei (Translated by Goro Kato):130 pp., US$29.00 ISBN 0-8218-3547-5(American Mathematical Society, Providence, RI, 2005)
Algebraic analysis of singular perturbation theory(Translations of Mathematical Monographs 227)By Takahiro Kawai and Yoshitsugu Takei (Translated by Goro Kato):130 pp., US$29.00 ISBN 0-8218-3547-5(American Mathematical Society, Providence, RI, 2005)
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奇异微扰理论的代数分析（数学专着 227 版）作者：Takahiro Kawai 和 Yoshitsugu Takei（加藤吾郎翻译）：130 页，29.00 美元 ISBN 0-8218-3547-5（美国数学会，普罗维登斯，罗德岛州，2005 年）
DOI：
10.1112/blms/bdn065
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发表时间：
2008
影响因子：
0.9
通讯作者：
V. Novokshenov
中科院分区：
文献类型：
--
作者：
V. Novokshenov
做渐近分析是艺术而不是科学。作为一个例子，我可以参考博伊德的非常有启发性的论文[2]。历史上，渐近分析最令人兴奋的主题出现在奇异摄动理论中，包括斯托克斯现象的故事（见[1，5]）和匹配渐近展开法。虽然这些理论现在已经得到了严格的证明，但正如Littlewood [4]所说的那样，这些理论留下了“一种悖论和大胆的味道”。许多涉及奇异摄动的渐近展开式来自数学物理的线性偏微分方程，主要来自量子力学和衍射理论。对于这些问题，物理学家重新发明了一种称为WKB的渐近积分方法（以Wentzel，Kramers和Brillouin命名）;然而，它在很久以前就被数学家称为Liouville-Green方法。由这种方法产生的形式渐近展开式本质上是不一致的，它们在整个积分域中都是无效的。一致渐近性的方法导致了匹配渐近展开法，它已成为解决重要物理问题的有力工具，例如分层介质中的波传播。另一方面，给出了形式WKB渐近性的数学证明。
Doing asymptotics is much more art than science. As an illustration, I can refer to the very instructive paper by Boyd [2]. Historically, the most exciting topics of asymptotic analysis have appeared in the theory of singular perturbations, including the story of the Stokes phenomenon (see [1, 5]) and the method of matched asymptotic expansions. Though strictly proved now, these theories left ‘an aroma of paradox and audacity’, as Littlewood [4] put it, talking of the divergent asymptotic series arising here.A number of asymptotic expansions involving singular perturbations came from linear PDEs of mathematical physics, mainly from quantum mechanics and diffraction theory. For those problems, physicists had re-invented a method of asymptotic integration called WKB (after Wentzel, Kramers and Brillouin); however, it had been known to mathematicians long before as the Liouville–Green method. The formal asymptotic expansions produced by this method are essentially non-uniform; they are not valid in the entire domain of integration. The way to make uniform asymptotics has led to the method of matched asymptotic expansions, which has become a powerful tool for solving important physical problems, such as wave propagation in stratified media. On the other hand, it has also given mathematical proofs of formal WKB asymptotics.