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Invitation to Classical Analysis
Peter Duren, University of Michigan, Ann Arbor, MI
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Pure and Applied Undergraduate Texts
2012; 392 pp; hardcover
Volume: 17
ISBN-10: 0-8218-6932-9
ISBN-13: 978-0-8218-6932-1
List Price: US$74
Member Price: US$59.20
Order Code: AMSTEXT/17
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This book gives a rigorous treatment of selected topics in classical analysis, with many applications and examples. The exposition is at the undergraduate level, building on basic principles of advanced calculus without appeal to more sophisticated techniques of complex analysis and Lebesgue integration.

Among the topics covered are Fourier series and integrals, approximation theory, Stirling's formula, the gamma function, Bernoulli numbers and polynomials, the Riemann zeta function, Tauberian theorems, elliptic integrals, ramifications of the Cantor set, and a theoretical discussion of differential equations including power series solutions at regular singular points, Bessel functions, hypergeometric functions, and Sturm comparison theory. Preliminary chapters offer rapid reviews of basic principles and further background material such as infinite products and commonly applied inequalities.

This book is designed for individual study but can also serve as a text for second-semester courses in advanced calculus. Each chapter concludes with an abundance of exercises. Historical notes discuss the evolution of mathematical ideas and their relevance to physical applications. Special features are capsule scientific biographies of the major players and a gallery of portraits.

Although this book is designed for undergraduate students, others may find it an accessible source of information on classical topics that underlie modern developments in pure and applied mathematics.

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Readership

Undergraduates and research mathematicians interested in analysis, number theory, and special functions.

Reviews

"The author's goal is to revive and reinvigorate at the undergraduate level a host of topics that used to be or ought to be well known. ... Although I read the book in linear order, the chapters are mostly independent of each other and can be sampled at random. The book is thus an excellent source of supplementary material for upper-level undergraduate courses. Each chapter starts at an introductory level and rises to a fairly high peak. Two notable features of the book are substantial sets of good exercises at the ends of chapters and engaging historical vignettes scattered throughout (portraits of a dozen of the principal actors are included). ... It is a treat to watch a master landscape artist at work coloring the sky without using cerulean blue. ... The composition of the book is superior. ... This volume is a valuable guidebook for tourists in the old country of analysis. Pick up a copy and learn to speak like a native."

-- Harold P. Boas, The American Mathematical Monthly

"It seems just a beautiful book, and it is really needed. ... Had such a book existed when I was a student, I might have gone straight into analysis. ... The topics that you treat have not only fallen from the curriculum but have fallen from easy availability to students. An instructor has to present them on his own and refer students to scattered resources if they want to learn more. Thanks for taking the trouble to write the book."

-- Raz Stowe, Idaho State University, Pocatello, ID

"[T]his book offers a wealth of gems of classical real analysis in a very lucid and exciting style of presentation. No doubt, this wonderful collection of topics is the work of a highly passionate and experienced teacher, whose expository mastery becomes apparent everywhere in the book . . . Summing up, this excellent book is a highly useful and valuable complement to the existing standard texts in modern advanced calculus."

-- Werner Kleinert, Zentralblatt MATH

"This is a concise, very clearly-written undergraduate textbook in classical analysis, that includes a very broad selection of the most important theorems in the subject... The book starts with a brief development of the topology of the real line, focusing on limit theorems. After that, each chapter tends to be independent, although cross-references to other chapters are given when needed. Each chapter ends with a wealth of problems, of moderate difficulty: not drill or straightforward applications of the material in the chapter, but also noticeably easier than the results proven in the chapter. Very Good Feature: brief biographical and historical notes scattered throughout the text."

-- Allen Stenger, MAA Online

"This is a delightful book that every mathematician will want for leisure reading. Its highlights include a brief but thorough introduction to undergraduate analysis, followed by several chapters on special topics. My favorites include the three proofs of the Weierstrass approximation theorem, the story of the Tauberian theorems, and the discussion of the zeta function. The book is beautifully written, with historical anecdotes and interesting exercises. It has a place beside the books of Hardy, Landau, and Titchmarsh."

-- John Garnett

"This is the kind of book that should be in the library of every serious student of analysis. The material, while classical, is in its totality not readily available in this form, and so the author has done a great service in writing this text."

-- Elias M. Stein, Princeton University

"Peter Duren's Invitation to Classical Analysis is a beautiful book. It presents a rich selection of results in classical analysis clearly and elegantly. Every undergraduate student of mathematics would learn a lot from reading it."

-- Peter Lax

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