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.

Readership

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

Reviews

"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