Real Analysis: A Historical Approach

285 Pages·1999·3.61 MB·English

by  Saul Stahl

REAL ANALYSIS
A Historical Approach
Saul Stahl
Real Analysis:
A Historical Approach

Real Analysis:
A Historical Approach
Saul Stahl
A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
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Copyright 0 1999 by John Wiley & Sons, Inc. All rights reserved.
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Librwy of Congress Cataloging in Publication Dam:
Stahl, Saul.
Real analysis : a historical approach / Saul Stahl.
P. cm.
" AWiley-Into science publication."
Includes bibliographical references and index.
ISBN 0-471-31852-3 (cloth : ak paper)
1. Mathematical analysis. 2. Functions of real variables.
1. Title.
QA300.S882 1999
515-dc21 99-21917
CIP
Printed in the United States of America
10987654321
Preface
A Focused and Historical Approach
The need for rigor in analysis is often presented as an end in itself. Histor-
ically, 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 territo-
ries. 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 offer-
ing glimpses of the Greeks' method of exhaustion, preNewtonian 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 the crucial counterexample that clarifies
the need for this careful examination of the foundations of calculus.
v
vi PREFACE
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 expo-
sition 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 conver-
gence (through Cauchy's theorem about the continuity of uniformly conver-
gent 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
PREFACE vii
of the optional sections is 4.2 which describes the rudiments of Newton's Poly-
gon 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 math-
ematics. 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 limita-
tions 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.
SAUL STAHL
Lawrence, Kansas

Acknowledgments
Fred van Vleck proofread portions of the manuscript. I am indebted to him
and to my colleagues Bill Paschke and Pawel Szeptycki for their patience when
they were subjected to my ramblings and half baked ideas. My thanks are
also due to Jessica Downey, Andrew Prince, and Sharon Liu of John Wiley &
Sons, Inc. and to Larisa Martin and Sandra Reed for their help in converting
my notes into a book.
Your comments are welcome at Stahl©math.ukans.edu
S. S.
ix