Real Analysis E-Book

Saul Stahl

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 383

Veröffentlichungsjahr: 2012

Beschreibung

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.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface to the Second Edition

A Focused and Historical Approach

Intended Audience for this Book

Pedagogy

Acknowledgments

Chapter 1: Archimedes and the Parabola

1.1 The Area of the Parabolic Segment

1.2* The Geometry of the Parabola

Chapter 2: Fermat, Differentiation, and Integration

2.1 Fermat’s Calculus

Chapter 3: Newton’s Calculus (Part 1)

3.1 The Fractional Binomial Theorem

3.2 Areas and Infinite Series

3.3 Newton’s Proofs

Chapter 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

Chapter 5: Euler

5.1 Trigonometric Series

Chapter 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

Chapter 7: Sequences and Their Limits

7.1 The Definitions

7.2 Limit Theorems

Chapter 8: The Cauchy Property

8.1 Limits of Monotone Sequences

8.2 The Cauchy Property

Chapter 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

Chapter 10: Series of Functions

10.1 Power Series

10.2 Trigonometric Series

Chapter 11: Continuity

11.1 An Informal Introduction

11.2 The Limit of a Function

11.3 Continuity

11.4 Properties of Continuous Functions

Chapter 12: Differentiability

12.1 An Informal Introduction to Differentiation

12.2 The Derivative

12.3 The Consequences of Differentiability

12.4 Integrability

Chapter 13: Uniform Convergence

13.1 Uniform and Nonuniform Convergence

13.2 Consequences of Uniform Convergence

Chapter 14: The Vindication

14.1 Trigonometric Series

14.2 Power Series

Chapter 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 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

Real Analysis

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Library of Congress Cataloging-in-Publication Data:

Stahl, Saul.Real analysis : a historical approach / Saul Stahl. — 2nd ed.p. cm.Includes index.ISBN 978-0-470-87890-3 (hardback)1. Mathematical analysis. 2. Functions of real variables. I. Title.QA300.S882 2011515′.8—dc22

2011010976

oBook ISBN: 978118096864ePDF ISBN: 978118096840ePub ISBN: 978118096857

This book is dedicated to the memory of my parents and my brother, Finkla, Moses, and Dan Stahl

Preface to the Second Edition

A Focused and Historical Approach

The need for rigor in analysis is often presented as an end in itself. Historically, however, the excitement and impatience that characterized the first century and a half of calculus and its applications induced mathematicians to place this issue on the proverbial back burner while they explored new territories. Ironically, it was developments in physics, especially the study of sound and heat, that brought the pathological behavior of trigonometric series into the foreground and forced the mathematical world to pay closer attention to foundational issues.

It is the purpose of this text to provide a picture of analysis that reflects this evolution. Rigor is therefore introduced as an explanation of the convergence of series in general and of the puzzling behavior of trigonometric series in particular.

The first third of this book describes the utility of infinite, power, and trigonometric series in both pure and applied mathematics through several snapshots from the works of Archimedes, Fermat, Newton, and Euler offering glimpses of the Greeks’ method of exhaustion, pre-Newtonian calculus, Newton’s concerns, and Euler’s miraculously effective, though often logically unsound, mathematical wizardry. The infinite geometric progression is the scarlet thread that unifies Chapters 1 to 5 wherein the nondifferentiability of Euler’s trigonometric series provides an example that clarifies the need for this careful examination of the foundations of calculus.

Chapters 6 to 10 consist of a fairly conventional discussion of various aspects of the completeness of the real number system. These culminate in Cauchy’s criterion for the convergence of infinite series which is in turn applied to both power and trigonometric series. Sequential continuity and differentiability are discussed in Chapters 11 and 12 as is the maximum principle for continuous functions and the mean value theorem for differentiable ones. Chapter 13 covers a discussion of uniform convergence proving the basic theorems on the continuity, integrability, and differentiability of uniformly convergent series and applying them to both power and trigonometric series. While the exposition here does not follow the historical method, a considerable amount of discussion and quoted material is included to shed some light on the concerns of the mathematicians who developed the key concepts and on the difficulties they faced.

Chapter 14, The Vindication, uses the tools developed in Chapters 6 to 13 to prove the validity of most of the methods and results of Newton and Euler that were described in the motivational chapters.

Intended Audience for this Book

This text and its approach have been used in a junior/senior-level college introductory analysis course with a class size between 20 and 50 students. Roughly 40% of these students were mathematics majors, of whom about one in four planned to go on to graduate work in mathematics; another 40% were prospective high school teachers. The balance of the students came from a variety of disciplines, including business, economics, and biology.

Pedagogy

Experience indicates that about three quarters of the material in this text can be covered by the above described audience in a one-semester college-level course. In the context of this fairly standard time constraint, a trade-off between the historical background material and the more theoretical last two chapters needs to be made. One strategy favors covering uniform convergence [through Cauchy’s theorem about the continuity of uniformly convergent series of continuous functions (13.2.1-3)] instead of the more traditional Maximum Principle and Mean Value Theorem.

Each section is followed by its own set of exercises that vary from the routine to the challenging. Hints or solutions are given for most of the odd-numbered exercises. Each chapter concludes with a summary. The excerpts in the appendices can be used as independent reading or as an in-class line-by-line reading with commentary by the professor.

This book contains sections which can be considered optional material for a college course; these sections are indicated by an asterisk. The least optional of the optional sections is 4.2, which describes the rudiments of Newton’s polygon method. While the details look daunting, the material can be covered in two 50-minute-long lectures. The purpose of Section 6.4 is to provide an elementary and self-contained proof of the existence of irrational numbers. Section 9.4 is included as an explanation of some concerns of higher mathematics. All of these optional sections reinforce the importance of infinite series and infinite processes in mathematics.

The Riemann integral proved to be a thorny problem. Within the limitations of one semester it is impossible to do justice to historical motivation, the traditional contents of the course, uniform convergence, and the Riemann integral. Therefore I have decided to state and make use of the integrability of continuous functions without proof. This is applied for the first time in the proof of Theorem 13.2.4, a place to which a typical class is unlikely to get in one semester.

For the sake of completeness, a new Chapter 15 has been included in the second edition of this book. Its purpose is to establish the rigorous foundations of the Riemann integral. Riemann’s definition of this integral, appeared in an (unpublished) manuscript on trigonometric series. However, the only example he gave is too technical for the purposes of this text. For that reason, the development in this new chapter is the traditional one in terms of partitions and the least upper and greatest lower bounds.

SAUL STAHL

Lawrence, Kansas

Acknowledgments

Above all I wish to acknowledge the help given to me by Fred Galvin, who carefully read and corrected major portions of the first edition. Fred van Vleck and Seungly Oh proofread parts of the manuscript. I am indebted to them. I am also indebted to my colleague Bill Paschke and the late Pawel Szeptycki for their patience when they were subjected to my ramblings and half-baked ideas. My thanks are also due to Susanne Steitz-Filler, Christine Punzo, Jacqueline Palmieri and all the folks at John Wiley & Sons, as well as Larisa Martin, for their help in preparing this edition. Thanks also to Judy Roitman for making some of her class notes available to me.

Your comments are welcome at [e-mail]

S. S.