Advanced mathematical methods with Maple (Q2761407)

As a teacher of a course in mathematical analysis who tries to integrate the use of Maple I find this an extremely valuable book. In fact I do not know any comparable work. There is one caveat, however: as indicated in the title, the level of mathematical sophistication is rather high. Therefore most of the material will have to be filtered and supplemented (or preceded) by other material which is more generally available.NEWLINENEWLINENEWLINEThe mathematical content of the book is impressive. Roughly speaking, it treats advanced classical mathematical analysis with an unusually strong numerical flavor and an unusual stress on nonlinear differential equations and dynamical systems.NEWLINENEWLINENEWLINEIn 130 pages the first three chapters provide a user-friendly introduction to computer-assisted algebra with Maple. The first chapter (Introduction to Maple) contains a very lucid explanation of the use of free (unbound) versus bound (assigned) variables and of the rules for substitution as well as the three different types of quotes which Maple distinguishes. The author introduces plotting and graphs as well as simple programming tools. Differentiation, integration and the numerical evaluation of integrals are also discussed.NEWLINENEWLINENEWLINEChapter 2 (Simplification) discusses the tools needed to bring Maple output in a user-desired form. This involves the educated use of commands like `expand', `assume', `factor', `simplify', `combine', `convert', `collect', `series'. Using such commands is a skill that requires some training but is often essential for a successful use of Maple.NEWLINENEWLINENEWLINEChapter 3 (Functions and procedures) describes Maple as a high-level programming language, composition of functions and the total derivative of an expression. SAVE and READ are also discussed, together with output in PostScript form.NEWLINENEWLINENEWLINEIn the remaining 730 pages Maple is constantly used but rarely mentioned. The full stress is on the mathematical content and sometimes on physical applications.NEWLINENEWLINENEWLINEChapter 4 (Sequences, series and limits) discusses the classical results but also accelerated convergence, Aitken's \(\Delta^2\)-process and Richardson's extrapolation which are usually relegated to numerical analysis courses.NEWLINENEWLINENEWLINEChapter 5 is a fairly brief treatment of asymptotic expansions. In Chapter 6 continued fractions and Padé approximants are introduced. This naturally leads to the hypergeometric function.NEWLINENEWLINENEWLINEChapter 7 introduces the Green's function, a device allowing one (as the author stresses) to write the solution of particular linear differential equations in terms of integrals. There is no previous introduction to the theory of linear differential equations, so the student is supposed to have a background knowledge.NEWLINENEWLINENEWLINEIn Chapter 8 (Fourier series and systems of orthogonal functions) a fairly standard introduction to these topics is given. There is, however, also a section on the asymptotic behaviour of Fourier coefficients and another one on the Gibbs phenomenon.NEWLINENEWLINENEWLINEIn Chapter 9 (Perturbation theory) the use of Maple is less obvious. Both regular and singular perturbations are discussed. Newton's method for the solution of systems of nonlinear equations is given.NEWLINENEWLINENEWLINEChapter 10 is a long treatment of Sturm-Liouville equations. Chapter 11 (Special functions) introduces all classical orthogonal polynomials, Bessel functions, the Airy function and Mathieu functions.NEWLINENEWLINENEWLINEChapter 12 (Linear systems and Floquet theory) deals with systems of linear, but possibly nonautonomous differential equations. It recollects without proofs the basic results for such systems and then concentrates on periodically forced systems. There is a full page description on swinging the giant censer in the cathedral of Santiago de Compostela; this introduces parametric pumping. Hamiltonian systems are also introduced in this chapter.NEWLINENEWLINENEWLINEChapter 13 (Integrals and their approximation) describes both symbolic and numerical methods, using Maple extensively in both cases. Chapter 14 (Stationary phase approximations) can be seen as an elaboration of this, concentrating on integrals with highly oscillatory integrands. Chapter 15 (Uniform approximations for differential equations) applies similar ideas to certain differential equations. The remaining part of the book (Chapters 16 through 19) form a good introduction to the study of dynamical systems in 210 pages. We can recommend it to the reader interested in this subject.NEWLINENEWLINENEWLINESummarizing, this is an ideal companion text for advanced undergraduate students of mathematics and physics. There is a large number of exercises (the author claims over 1000 and he must be right) with solutions provided on the internet.NEWLINENEWLINENEWLINEThe many examples provided in the book are another of its strong points. A minor drawback is that they are practically all from physics, excluding recent applications in biology, chemistry etcetera. A book to be recommended!