Asymptotic analysis and perturbation theory (Q2847379)

This textbook differs from many known textbooks on the asymptotic analysis by very detailed explanation of subtleties of the basic notions. Every new definition is supplied with illustrative examples, plots and tables if nesessary. The readership of the book will be very diverse. It will be useful to beginning students as well as to experienced researchers.NEWLINENEWLINEThis textbook will be most useful for instructors in classrooms and for individual self-training in the various asymptotic analysis tools. The author introduces asymptotic notation on the beginning pages and then applies this tool to familiar problems for beginning students, including limits, inverse functions, and integrals.NEWLINENEWLINEWith varying levels of problems in each section, this self-contained text makes the difficult subject of asymptotics easy to comprehend. Every chapter contains a list of exercises for independent training. Suitable for beginners, the book assumes no prior knowledge of differential equations. The book emphasizes problem solving and only some proofs are given.NEWLINENEWLINEWe decribe shortly the contents. The book contains nine chapters. Chapter~1~(Introduction to asymptotics) introduces basic definitions (definition of \(\sim\) and \(\ll\), big \(O\) and little \(o\) notation, asymptotic series, dominant balance and so on). Chapter~2~(Asymptotics to integrals) includes integrating of Tailor series, Laplace's method, properties of \(\Gamma(x)\), Watson's lemma, a review of complex variables (analytic functions, contour integrals, Gevrey asymptotics, asymptotics of oscillatory functions), the method of stationary phase, the method of steepest descents. Chapter~3~(Speeding up convergence) includes the Shanks transformation, the Richardson extrapolation, the Euler summation, the Borel summation, continued fractions, Padé approximations. Chapter~4~(Differential equations) is devoted to the basic properties of linear and nonlinear differential equations. Chapter~5~(Asymptotic series solutions for differential equations) presents some ways of determining the asymptotic series for several types of differential equations mainly by the method of dominant balance, in particular, the Airy equation, the parabolic cylinder equation, the Bessel equation and some nonlinear equations. Chapter~6~(Difference equations) describes recurrent difference equations, regular and singular points of linear homogeneous equations, Stirling series for the gamma function, Taylor series solution, Frobenius-like series. Chapter~7~(Perturbation theory) describes regular and singular perturbation problems for differential equations, boundary value problems, the Van Dyke method, multiple boundary layers. Chapter~8~(WKBJ theory) gives an outline of the WKBJ approximations through some simple examples. Chapter~9~(Multiple-scale analysis) shortly describes the Poincaré-Lindstedt method, the multiple-scale procedure, the two-variable expansion method.NEWLINENEWLINEThe book ends with an appendix-guide to the special functions, answers to odd-numbered problems, a bibliography and an index.NEWLINENEWLINEThe textbook is highly recommended for first acquaintance with the subject. The only imperfection is the bibliography which does not give direct references to well-known classical monographs on the asymptotic analysis and perturbation theory.