Real Analysis: A Historical Approach / Edition 2

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Combining historical coverage with key introductory fundamentals, Real Analysis: A Historical Approach, Second Edition helps readers easily make the transition from concrete to abstract ideas when conducting analysis. Based on reviewer and user feedback, this edition features a new chapter on the Riemann integral including the subject of uniform continuity, as well as a discussion of epsilon-delta convergence and a section that details the modern preference for convergence of sequences over convergence of series. Both mathematics and secondary education majors will appreciate the focus on mathematicians who developed key concepts and the difficulties they faced.

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Editorial Reviews

From the Publisher
“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)
The textbook for a junior- or senior-level college introductory analysis course begins with a sampling of classic and famous problems first posed by the founders of mathematics, to illuminate the utility of infinite, power, and trigonometric series. Stahl (Mathematics, University of Kansas) then develops the basic tools of advanced calculus, and presents examples to reinforce concepts and demonstrate the validity of many of the historical methods and results. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780470878903
  • Publisher: Wiley
  • Publication date: 8/30/2011
  • Edition description: New Edition
  • Edition number: 2
  • Pages: 316
  • Product dimensions: 6.40 (w) x 9.30 (h) x 0.80 (d)

Meet the Author

SAUL STAHL, PhD, is Professor in the Department of Mathematics at The University of Kansas. He has published numerous journal articles in his areas of research interest, which include combinatorics, discrete mathematics, and topological graph theory. Dr. Stahl is the author of Introductory Modern Algebra: A Historical Approach and Introduction to Topology and Geometry, both published by Wiley. He was awarded the Carl B. Allendoerfer Award from the Mathematical Association of America for expository articles in both 1986 and 2006.

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Table of Contents

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



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