In eleven chapters, Krantz's book succeeds in providing a reference work for "the working engineer or scientist" encompassing the essence of real analysis. By "real analysis", the author means advanced calculus, the theory of real functions sans Lebesque theory, although the Riemann-Stieltjes integral is covered.
The book follows the outline of a typical advanced calculus/real analysis book, including the topology of the real line and basic information about sequences and series, limits and continuity, derivatives, integrals, sequences and series of functions, power series, exponential and trigonometric functions, logarithms, the gamma function (including Stirling's formula), and Fourier series. There is more advanced material on metric spaces, including the Baire Category Theorem and the Ascoli-Arzela Theorem. The final chapter gives a brief treatment of differential equations, emphasizing Picard's iteration technique and touching on the method of characteristics, power series methods, and Fourier analytic methods. There is a ten-and-a-half page glossary of all important terms defined and discussed in the body of the book, a list of notation, a "Guide to the Literature" a bibliography, and an index. (In the Bibliography, there is a confusing reference to a differential equations book by Krantz and George Simmons. Both the title and publication date are incorrect.)
True to the idea of a handbook, there are good, but brief, explanations, well-chosen examples, and only a few proofs. In addition to the book's principal audience, students preparing for exams at either the undergraduate or master's level will find this a valuable resource.
Henry Ricardo (email@example.com) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.
Preface * Basics * Sequences * Series * The Topology of the Real Line * Limits and the Continuity of Functions * The Derivative * The Integral * Sequences and Series of Functions * Some Special Functions * Advanced Topics * Differential Equations * Glossary of Terms from Real Variable Theory * List of Notation * A Guide to the Literature * Bibliography * Index