Chapter 1 discusses the mathematical aspects of the asymptotic theory. It starts with basic definitions, using for this purpose the so-called coordinate asymptotic expansions. Then the chapter turns to asymptotic analysis of integrals and describe, among others, the method of steepest descent. However, the main attention is with parametric asymptotic expansions. These are used in a wide variety of fluid–dynamic problems, for which purpose a number of asymptotic techniques has been developed. We discuss in Chapter 1 the method of matched asymptotic expansions, the method of multiple scales, the method of strained coordinates, and the WKB method.
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