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Introduction to asymptotics. A treatment using nonstandard analysis. (English) Zbl 0915.34001
Singapore: World Scientific. x, 165 p. \$ 26.00; £18.00 (1997).
Besides the usual basic topics in asymptotics as the asymptotic behaviour of Laplace-type integrals (method of steepest descent), Fourier integrals (method of stationary phase), power series (in particular entire functions) and solutions to ordinary differential equations (WKB approximation, connection formulas), the book deals with more difficult questions as optimal remainders (error bounds), uniform asymptotics (saddle points near poles, endpoints or near each other) and hyperasymptotics (Stokes’ phenomenon). The first half of the text is written by means of the terminology of nonstandard analysis, for which a brief introduction is given in the appendix. The second half is written in the framework of standard analysis. Remarkable are illustrative examples, both analytical and numerical, and several exercises for the reader. Let us mention that the asymptotic symbol $\sim$ is defined on p. 5, but used e.g. on p. 21 in another sense without explanation, though for the last sense there is also used the symbol $\approx$, e.g. on p. 39, also without explanation.
##### MSC:
 34-01 Textbooks (ordinary differential equations) 26E35 Nonstandard real analysis 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)