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Real Analysis: A Historical Approach, 2nd Edition

Real Analysis: A Historical Approach, 2nd Edition

Saul Stahl

ISBN: 978-1-118-09686-4 October 2011 316 Pages

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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.

Preface to the Second Edition


1. Archimedes and the Parabola

1.1 The Area of the Parabolic Segment

1.2 The Geometry of the Parabola

2. Fermat, Differentiation, and Integration

2.1 Fermat’s Calculus

3. Newton’s Calculus (Part 1)

3.1 The Fractional Binomial Theorem

3.2 Areas and Infinite Series

3.3 Newton’s Proofs

4. Newton’s Calculus (Part 2)

4.1 The Solution of Differential Equations

4.2 The Solution of Algebraic Equations

Chapter Appendix. Mathematica implementations of Newton’s algorithm

5. Euler

5.1 Trigonometric Series

6. The Real Numbers

6.1 An Informal Introduction

6.2 Ordered Fields

6.3 Completeness and Irrational Numbers

6.4 The Euclidean Process

6.5 Functions

7. Sequences and Their Limits

7.1 The Definitions

7.2 Limit Theorems

8. The Cauchy Property

8.1 Limits of Monotone Sequences

8.2 The Cauchy Property

9. The Convergence of Infinite Series

9.1 Stock Series

9.2 Series of Positive Terms

9.3 Series of Arbitrary Terms

9.4 The Most Celebrated Problem

10. Series of Functions

10.1 Power Series

10.2 Trigonometric Series

11. Continuity

11.1 An Informal Introduction

11.2 The Limit of a Function

11.3 Continuity

11.4 Properties of Continuous Functions

12. Differentiability

12.1 An Informal Introduction to Differentiation

12.2 The Derivative

12.3 The Consequences of Differentiability

12.4   Integrability

13. Uniform Convergence

13.1 Uniform and Non-Uniform Convergence

13.2 Consequences of Uniform Convergence

14. The Vindication

14.1 Trigonometric Series

14.2 Power Series

15. The Riemann Integral

15.1 Continuity Revisited

15.2 Lower and Upper Sums

15.3 Integrability

Appendix A. Excerpts from "Quadrature of the Parabola" by Archimedes

Appendix B. On a Method for 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



“Stahl’s book, though relatively modest in its historical ambit, is a workmanlike and very readable introduction to real analysis with a distinctive flavour provided by a plethora of accessible exercises, many of which are historically motivated.”  (The Mathematical Gazette, 1 March 2014)