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Applied asymptotic analysis. (English) Zbl 1101.41031
Graduate Studies in Mathematics 75. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4078-9/hbk). xv, 467 p. (2006).
This is a book of asymptotic analysis of both, integrals, and solutions of differential equations. The first part of the book consist of two chapters which are an introduction to the concepts of asymptotic analysis, definitions, relevant topics and several examples. The second part of the book consists of 4 chapters about asymptotic expansions of integrals with an exponential kernel: Watson’s lemma, Laplace’s method, steepest descents techniques and stationary phase method. These chapters are not only a formal presentation of the methods; the author goes deep inside the fundamentals of the methods giving a nice view of “why they work”. Beside of this, the author offers several interesting examples of special functions and physical problems. The third part of the book is divided into 5 chapters devoted to the asymptotic analysis of solutions of linear ordinary differential equations, boundary value problems and weakly nonlinear waves. The author makes a tour through the most important concepts of the matter: classification of regular and singular points, turning points, Stokes phenomenon, Borel summation, WKB method, singular perturbations and matching of asymptotic expansions among others. The last two chapters of the book are devoted to asymptotics of oscillatory phenomena and asymptotics of weakly nonlinear differential equations.

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 34E05 Asymptotic expansions of solutions to ordinary differential equations 34E15 Singular perturbations, general theory for ordinary differential equations