Mureşan, Marian, A concrete approach to classical analysis, Springer, New York, 2009. xviii+433 pp. $69.95. ISBN 978-0-387-78932-3

A concrete approach to classical analysis

This is an excellent book that gives much more than its modest title suggests. It has a solid base of classical results indicated by the titles of the first eight chapters: Sets and numbers; Vector spaces and metric spaces; Sequences and series; Limits and continuity; Differential calculus onR; Integral calculus on R; Differential calculus on Rn; Double integrals, triple integrals, and line integrals. What makes this book valuable and unique is the use and development of the more recent ideas in experimental mathematics.

Even in the above-mentioned chapters this leads to the discussion of such topics as a discrete form of L’Hˆopital’s rule, Stirling numbers, unimodal sequences, nowhere differentiable functions, full study of the mean-value theorem and its consequences, and polylogarithms. One of the many aspects of experimental mathematics is the study of the important constants of mathematics, and this is taken up throughout the book with a full discussion of the classical properties of e, π, log 2,p2. There is one chapter entitled Constants in which the work of Borwein and others on very quickly convergent estimates for π, the so-called BBP formulae, are discussed and various Ramanujan results given.

A final chapter is devoted to Asymptotic and combinatorial estimates. All in all this is a very exciting list of topics, a book that covers the usual material but from an interesting perspective. In addition the exercises are often nontrivial, and no answers are given but sources are in the extensive bibliography. There are two caveats: the English is a little strange, although never so much so as to hinder understanding; the theorems, propositions, lemmas, and corollaries are numbered differently and this does not make looking for references easy.

Reviewed by P. S. Bullen

©Copyright American Mathematical Society 2009