Real Analysis

A Historical Approach

Saul Stahl

• Publisher: Wiley
• Year: 2012
• Print ISBN: 9780470878903
• eBook ISBN: 9781118096857
• Edition: 2
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

A provocative look at the tools and history of real analysis

This new edition of Real Analysis: A Historical Approach continues to serve as an interesting read for students of analysis. Combining historical coverage with a superb introductory treatment, this book helps readers easily make the transition from concrete to abstract ideas.

The book begins with an exciting sampling of classic and famous problems first posed by some of the greatest mathematicians of all time. Archimedes, Fermat, Newton, and Euler are each summoned in turn, illuminating the utility of infinite, power, and trigonometric series in both pure and applied mathematics. Next, Dr. Stahl develops the basic tools of advanced calculus, which introduce the various aspects of the completeness of the real number system as well as sequential continuity and differentiability and lead to the Intermediate and Mean Value Theorems. The Second Edition features:

• A chapter on the Riemann integral, including the subject of uniform continuity

• Explicit coverage of the epsilon-delta convergence

• A discussion of the modern preference for the viewpoint of sequences over that of series

Throughout the book, numerous applications and examples reinforce concepts and demonstrate the validity of historical methods and results, while appended excerpts from original historical works shed light on the concerns of influential mathematicians in addition to the difficulties encountered in their work. Each chapter concludes with exercises ranging in level of complexity, and partial solutions are provided at the end of the book.

Real Analysis: A Historical Approach, Second Edition is an ideal book for courses on real analysis and mathematical analysis at the undergraduate level. The book is also a valuable resource for secondary mathematics teachers and mathematicians.

Table of contents

• Cover
• Half Title page
• Title page
• Copyright page
• Dedication
• Preface to the Second Edition
• Acknowledgments
• Chapter 1: Archimedes and the Parabola
• Chapter 2: Fermat, Differentiation, and Integration
• Chapter 3: Newton’s Calculus (Part 1)
• Chapter 4: Newton’s Calculus (Part 2)
• Chapter 5: Euler
• Chapter 6: The Real Numbers
• Chapter 7: Sequences and Their Limits
• Chapter 8: The Cauchy Property
• Chapter 9: The Convergence of Infinite Series
• Chapter 10: Series of Functions
• Chapter 11: Continuity
• Chapter 12: Differentiability
• Chapter 13: Uniform Convergence
• Chapter 14: The Vindication
• Chapter 15: The Riemann Integral
• Appendix A: Excerpts from “Quadrature of the Parabola” by Archimedes
• Appendix B: On a Method for the Evaluation of Maxima and Minima by Pierre de Fermat
• Appendix C: From a Letter to Henry Oldenburg on the Binomial Series (June 13, 1676) by Isaac Newton
• Appendix D: From a Letter to Henry Oldenburg on the Binomial Series (October 24, 1676) by Isaac Newton
• Appendix E: Excerpts from “Of Analysis by Equations of an Infinite Number of Terms” by Isaac Newton
• Appendix F: Excerpts from “Subsiduum Calculi Sinuum” by Leonhard Euler
• Solutions to Selected Exercises
• Bibliography
• Index