Real Analysis: A Historical Approach, 2nd Edition (Pure and Applied Mathematics)

316 Pages·2011·9.28 MB·English

by  Saul Stahl

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Real Analysis
PURE AND APPLIED MATHEMATICS
A Wiley Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN III, DAVID A. COX, PETER HILTON,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
Real Analysis:
A Historical Approach
Second Edition
Saul Stahl
The University of Kansas
Department of Mathematics
Lawrence, KS
)WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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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 2011
515'.8—dc22 2011010976
Printed in the United States of America.
oBook ISBN: 978118096864
ePDF ISBN: 978118096840
ePub ISBN: 978118096857
10 9 8 7 6 54 3 2 1
This book is dedicated to
the memory of my parents and my brother,
Finkla, Moses, and Dan Stahl
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Contents
Preface to the Second Edition xi
Acknowledgments xv
1 Archimedes and the Parabola 1
1.1 The Area of the Parabolic Segment 1
1.2* The Geometry of the Parabola 8
2 Fermat, Differentiation, and Integration 13
2.1 Fermat's Calculus 13
3 Newton's Calculus (Part 1) 19
3.1 The Fractional Binomial Theorem 19
3.2 Areas and Infinite Series 23
3.3 Newton's Proofs 30
4 Newton's Calculus (Part 2) 35
4-1 The Solution of Differential Equations 35
4-2* The Solution of Algebraic Equations 40
Chapter Appendix: Mathematica Implementations of
Newton's Algorithm 48
VII
νιϊι CONTENTS
5 Euler 51
5.1 Trigonometrie Series 51
6 The Real Numbers 61
6.1 An Informal Introduction 61
6.2 Ordered Fields 64
6.3 Completeness and Irrational Numbers 71
6.4* The Euclidean Process 76
6.5 Functions 80
7 Sequences and Their Limits 85
7.1 The Definitions 85
7.2 Limit Theorems 92
8 The Cauchy Property 103
8.1 Limits of Monotone Sequences 103
8.2 The Cauchy Property 111
9 The Convergence of Infinite Series 115
9.1 Stock Series 115
9.2 Series of Positive Terms 120
9.3 Series of Arbitrary Terms 124
9.4* The Most Celebrated Problem 132
10 Series of Functions 139
10.1 Power Series 139
10.2 Trigonometric Series 145
11 Continuity 149
11.1 An Informal Introduction 149
11.2 The Limit of a Function 150
11.3 Continuity 155
11.4 Properties of Continuous Functions 163
12 Differentiability 169
12.1 An Informal Introduction to Differentiation 169
12.2 The Derivative 171
12.3 The Consequences of Differentiability 177