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Resources for the Study of Real Analysis 4 AMS / MAA PRESS

VOL 4 VOL TEXTBOOKS Robert L Brabenec Robert Resources for the Study Study the for Resources of Analysis Real 243 pages • Spine: 1/2 in • 50lb paper • Softcover • 4-color Process (CMYK) • 4-color Process 243 pages • Spine: 1/2 in • 50lb paper • Softcover / MAA AMS

Resources for the Study of Real Analysis 4 AMS / MAA PRESS

VOL Robert L Brabenec —Paul Zorn, St. Olaf College —William Dunham, Muhlenberg College Trim Size 7 x 10 Size Trim TEXTBOOKS The book offers even more than its title suggests: a true trove of resources of trove true a suggests: title its than more even offers book The rough— who face the exciting—but often their teachers) for students (and to the routine calculations of elementary passage from the for here gems are There analysis. real of insights and arguments deeper students who need for the quickest average students who need review; ideas; and for everyone who need for teachers mathematical challenges; and culture. with a taste for mathematical highlights analysis— of mathematical smorgasbord a rich Brabenec has provided essays, and selected readings— including a host of problems, historical hungry. no one should go away from which by the author while gathered This book is a collection of materials a period of years. real analysis over for use as a It is intended teaching material for textbook, or to provide supplement to a traditional analysis seminars its historical development. or independent study in analysis and biographical information, a wide range The book includes historical and on a variety of topics, and many of problem types, selected readings materials are collected Since all these references for additional study. items most easily choose and students can into a single book, teachers may use the book as a supplement to Teachers suitable for their purpose. of the book on their own. their courses, while students may read much a real for supplement a as specifically written been has book other No analysis course. AMS / MAA

Resources for the Study of Real Analysis 10.1090/text/004

Resources for the Study of Real Analysis

by

Robert L. Brabenec Wheaton College

Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA CLASSROOM RESOURCE MATERIALS

Classroom Resource Materials is intended to provide supplementary classroom material for students-laboratory exercises, projects, historical information, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. Council on Publications Roger Nelsen, Chair Zaven A. Karlan, Editor William Bauldry Daniel E. Kullman Gerald Bryce Stephen B Maurer Sheldon P. Gordon Douglas Meade William J. Higgins Judith A. Palagallo Mic Jackson Wayne Roberts Paul Knopp

101 Careers in , edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus Mysteries and Thrillers, R. Grant Woods Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Conjecture and Proof, Miklos Laczkovich A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robert Edward Lewand Elementary Mathematical Models, Dan Kalman Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture, Margie Hale Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Geometry from Africa: Mathematical and Educational Explorations, Paulus Gerdes Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Amey Inverse Problems: Activities for Undergraduates, C. W. Groetsch Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math through the Ages: A Gentle History for Teachers and Others (Expanded Edition), William P. Berlinghoff and Fernando Q. Gouvea Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell Mathematical Modeling in the Environment, Charles Hadlock Mathematics for Business Decisions Part 1: Probability and Simulation (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Mathematics for Business Decisions Part 2: Calculus and Optimization (electronic textbook), Richard B. Thompson and Christopher G. Lamoureux Ordinary Differential Equations: A Brief Eclectic Tour, David A. Sanchez Oval Track and Other Permutation Puzzles, Robert B. Ash A Primer ofAbstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen A Radical Approach to Real Analysis, David M. Bressoud Resources for the Study of Real Analysis, Robert L. Brabenec She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probability and Simulation, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Tepching Statistics Using Baseball, Jim Albert Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, and Elyn Rykken

MAA Service Center P. 0. Box 91112 Washington, DC 20090-1112 1-800-331-lMAA FAX: 1-301-206-9789 www.maa.org This book is dedicated to the hundreds of Wheaton College students who have taken my course in real analysis during the past forty years. Their enthusiasm and contributions enriched these classes and helped to shape the material in this book. Contents

Preface xi

Review of Calculus 1 An Outline of a Traditional Course in Calculus 1 Calculus Review Problems 6

II Analysis Problems 19 Introduction 19 Section 1. Basic Problems Problem l. Properties of Even and Odd Functions 19 Problem 2. Use of the Cancellation Principle 22 Problem 3. Finding Maclaurin Representations 26 Problem 4. Evaluating Jd xk dx by Cavalieri Sums 29 Section 2. Supplementary Problems Problem 5. The Pervasive Nature of 31 Problem 6. Counterexamples 36 Problem 7. Some Examples of Unusual Functions 37 Problem 8. Identifying Interior, Exterior, Boundary, and Points 39 Problem 9. Exploring Uniform Continuity 40 Problem 10. Comparing the , the , and Raabe's Test 42 Problem 11. Various Proofs that L:'= 1 ;!,: = 1~t 47 viii Part . Contents

Problem 12. Finding a Value for I::= 1 n\ 51 Problem 13. Rearrangements of Conditionally 52 Problem 14. Finding for rr 61 Problem 15. A Function with of All Orders, but No Maclaurin Series 68 Problem 16. Evaluating Jd xk dx by Fermat's Method 69 Problem 17. Two Proofs that Jd (x lnx)ll dx = (-l)nn!/(n + l)

Ill Essays 125 Section 1. History and Biography Essay 1. The Time of Archimedes and Other Giants (pre-1660) 126 Essay 2. The Time of Newton and Leibniz (1660-1690) 129 Essay 3. The Time of Euler and the Bernoullis (1690-1790) 132 Essay 4. The Time of Cauchy, Fourier, and Lagrange (1790-1850) 135 Essay 5. The Time of Riemann and Weierstrass (1850-1900) 139 Essay 6. The Time of Lebesgue and Hardy (1900-1940) 142 Section 2. New Looks at Calculus Content Essay 7. Obtaining the Formulas 147 Essay 8. Tests for Convergence of Series 155 Section 3. General Topics for Analysis Essay 9. Proof Techniques in Analysis 163 Essay 10. Sets and Topology 170 Part . Contents Ix

IV Selected Readings 175 Introduction 175 Selection 1. John Bernoulli and the Marquis de l'Hopital 177 Selection 2. The Euler-Stirling Correspondence 179 Selection 3. Developing Rigor in Calculus 187 Selection 4. The Contribution of Lebesgue 198 Selection 5. The Bernoulli Numbers and Some Wonderful Discoveries of Euler 202

Annotated Bibliography 207

Addltlonal References 217

Index 221

Preface

This book is a collection of materials I have gathered while teaching a real analysis course every year for more than thirty-five years. I prepared it with the hope it will benefit and enrich the experience both of students who take a real analysis course, as well as those who teach it. This collection is intended to supplement a traditional real analysis textbook, where teachers and students may choose items of interest to them. It is my conviction that such supplementary materials have a much greater chance of being used in an analysis course if they are readily available in one place. Part I contains materials that provide the greatest benefit if read before the real analysis course begins. Because such a course assumes knowledge of topics from calculus of one real variable, review of this material in advance frees the student to concentrate on new content and theoretical emphases in analysis without having to revisit the calculus at the same time. The outline of a traditional calculus course allows a student to check necessary topics for review, whereas the calculus review problems offer the opportunity to refresh necessary skills which may have lain dormant for some time. Many hints are provided with the problems to encourage students to carry out this review. The problems in Part II are intended to supplement the ones usually found in an analysis text. There is a wide variety of problem types, and each one has exercises for the student to attempt. Most of them contain explanatory detail about the historical background of the topic or how it fits with other parts of analysis. I try to present a topic from a variety of perspectives in order to enhance learning and understanding. The entries in Part ill are called essays, and represent short discussions of a particular topic from calculus or analysis. Some of these are content-oriented, whereas others are intended to give an alternate perspective after the student has first studied the material in a traditional manner. The collection of six essays based on time periods presents a historical overview of the development of analysis by concentrating on the main individdals who were influential in this development. xi xii Part . Preface

Part IV contains a collection of five supplementary readings to illustrate the variety of materials that are available beyond a standard textbook. The annotated bibliography contains many references that can be read with profit by faculty and students. There is information for each one to help readers decide which might best fit their interests and needs. We are fortunate to be living at a time when there is a growing interest in these kinds of supplementary materials-the past twenty years especially have seen many new entries of a historical, biographical, or expository nature. Let me explain how I use these materials in my analysis course. I encourage students to work on the calculus review problems before the analysis class begins. Since our course is taught in the fall semester, I give them the problems before they leave for summer va­ cation. At the beginning of the course, while presenting the abstract material on properties of real numbers and sequences, I use Essays 7 and 8 to review the material on derivatives and series from calculus. This not only give~ students a more familiar alternative to the abstract material, but it also provides an introduction to the spirit of careful organization and attention to detail that is essential for work in analysis. Students read one of the six historical essays a week and we spend some time in class discussion on the material. I use Essays 9 and 10 in the middle of the course to provide an alternative perspective for the topics of proofs and topology, and assign supplementary problems from Part II at appro­ priate places in the course. The material in chart form from Problems 8, 9, and 20 works well in class discussions. I like to assign some of the enrichment problems in Section 3 to teams of students and have them learn the material, write up solutions to the exercises, and make oral presentions to the rest of the class. Students enjoy this opportunity to make presentations. Acknowledgements. I would first like to express appreciation to my students who have used different versions of these materials in classes and contributed their suggestions for improvements. I am also grateful to Bill Dunham from Muhlenberg College and Judy Pala­ gallo from the University of Akron for reading various portions of this manuscript and offering valuable advice for improvement. My mathematics colleague at Wheaton, Terry Perciante, helped me prepare the figures for the book, and my computer science colleague, John Hayward, guided me through and around the pitfalls of working with a computer. Much of the work was accomplished during a sabbatical semester in 2002 at Cambridge University. My thanks go to Wheaton College for providing this sabbatical opportunity, to the Aldeen Fund for financial support with travel expenses, and to the Department of Pure Mathematics and Mathematical Statistics at Cambridge for their support during my time there. I am also grateful to my wife Bonnie for her support and good advice during my work on this project. Many of you will undoubtedly be aware of additional problems, topics, or bibliographic items that can be added should there be a revision of this book. I would be glad to receive these at [email protected]. Annotated Bibliography

[1] M. E. Baron, The Origins of the lnfinitesmal Calculus, Pergamon Press, Oxford, 1969. In the Preface, the author states "the origins of the infinitesmal calculus lie, not only in the significant contributions of Newton and Leibniz, but also in the centuries-long struggle to investigate area, volume, and arc by purely ge­ ometric means." She decided to emphasize the geometric techniques and methods employed first by the Greek, Hindu, and Arabic cultures, and then by the European predecessors of Newton and Leibniz. Her early study was enhanced by Boyer's book (see [4]) including its extensive references. This book is thorough and well­ referenced. Its scholarly development makes it more suitable for faculty than stu­ dents. [2] Eric Temple Bell, Men ofMathematics, Simon and Schuster, New York, 1961. This book was written in 1935 at a time when expository books about mathemat­ ics and mathematicians were not readily available. It came as a breath of fresh air with its lively accounts of biographical details and mathematical accomplishments for many of the greatest mathematicians of history. Ever since, it has inspired many to learn more about mathematics. While some have criticized the author for factual inaccuracies he includes to enliven the exposition, it remains a valuable resource. Compare the nature of the writing in this book with the more formal exposition in the Biographical Dictionary of Mathematicians (see [16]). [3] Garrett Birkhoff, A Source Book in Classical Analysis, Harvard University Press, Cambridge, MA, 1973. This source book of mathematical writings in the nineteenth century picks up where Struik's source book (see [31]) ends. The writings that are most helpful for a course in real analysis are in Chapters 1, 3, and 5. They include material from Cauchy, Fourier, Gauss, Abel, Bolzano, Riemann and Weierstrass. 207 208 Annotated Bibliography

[4] C. B. Boyer, The History of the Calculus and its Conceptual Development, Dover, New York, 1959. This book was written in 1949 to provide a historical treatment of the "funda­ mental ideas of the subject (i.e. calculus) from their incipiency in antiquity to the final formulation of these in the precise concepts familiar to every student of the el­ ements of modem ." This quote is taken from the Preface. As a 300 page paperback, it provides a readable and helpful introduction to this topic. A substantial bibliography suggests additional resources. The author also wrote a complete history of mathematics text in 1968 1 which is distinguished by the in­ clusion of an extensive number of individuals and results, but each section is quite brief (about one page in length on average). [5] David Bressoud, A Radical Approach to Real Analysis, The Mathematical Associ­ ation of America, 1994. The Preface explains this book "is designed to be a first encounter with real analysis, laying out its context and motivation in terms of the transition from to those that are less predictable, especially Fourier series, and marking some of the traps into which even great mathematicians have fallen. The book begins with Fourier's introduction of trigonometric series and the problems they created for the mathematicians of the early nineteenth century. It follows Cauchy's attempts to establish a firm foundation for calculus, and considers his failures as well as his successes. It culminates with Dirichlet's proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet's proof." This is a radical approach as contrasted with a traditional approach of organiz­ ing the analysis course in the logical order of concept development. The order of topics in such a course is the properties, sequences, continuity, differ­ entiability, integrability, and sequences and series of functions, without regard for the historical development of analysis. The main focus of this book is on nineteenth century material, and especially trigonometric series. Many graphical and numeri­ cal investigations are provided to encourage active exploration by the reader. [6] David M. Burton, History ofMathematics: An Introduction, Third edition, Wm. C. Brown Publishers, Dubuque, IA, 1995. A distinguishing characteristic of Burton's book is its emphasis on well-written prose. Some mathematical details are omitted in favor of providing interesting nar­ rative of the topics he chooses to present. There are separate chapters devoted to Fibonacci sequences, the cubic and quartic equations, probability theory, and non­ Euclidean geometry, while algebra and analysis receive slight attention. The ma­ terial he does present on Newton and Leibniz in Chapter 8 is very good, but the material on Cauchy, Riemann, and Weierstrass in Section 11.3 is rather meager.

1Carl B. Boyer and Uta C. Merzbach, A History of Mathematics, Second edition, John Wiley and Sons, New York, 1989. Annotated Bibliography 209

[7] Ronald Calinger, editor, Classics of Mathematics, Prentice Hall, Inc, 1995. This is a collection of excerpts from the original writings of several mathemati­ cians that have been translated into English. There are also brief biographical se­ lections and comments about the importance of the material. Some samples that are pertinent to analysis include the following: (a) Gottfried Leibniz and the Fundamental Theorem of Calculus on pages 383-386 and 393-394. (b) and the Treatise of on pages 475-478. (c) Niels Abel and the on pages 537-538 and 594-596. (d) Augustin-Louis Cauchy and the Cours d'analyse on pages 597-603. (e) Karl Weierstrass and the on pages 604-610. [8] John H. Conway and Richard K. Guy, The Book ofNumbers, Springer-Verlag, New York, 1996. This interesting book is a collection of short descriptions about unusual proper­ ties of familiar numbers. While most fit best in a course on discrete mathematics, number theory, algebra or geometry, there are several items of interest for analysis. These include the use of Gregory's and Stormer's numbers for the of :n:, the Euler-Mascheroni2 number (y), Apery's number (((3)), Bernoulli num­ bers resulting from Faulhaber's formula, and various methods to find the pattern in a sequence of integers. [9] William Dunham, "A Historical Gem from Vito Volterra" Mathematics Magazine, 63 (1990) 234-237. Many examples that were considered "unusual" or even "pathological" at the time of their introduction have often led to an improved understanding of basic analysis concepts as continuity, differentiability, and integration. Many of these examples such as Dirichlet's everywhere discontinuous function (see Problem 7) are characterized by the use of one expression for rational domain values and a different expression for irrational domain values. This article deals with one such function that is continuous only for irrational values, using a proof by Volterra in the 1880s to demonstrate the impossibility of finding a function that is continuous only for rational values. [10] William Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington, D.C., 1999. This paperback is Volume 22 in the MAA's Dolciani Mathematical Exposi­ tions series. It begins with a short biographical sketch of Euler and contains eight chapters on topics where Euler significantly advanced the level of knowledge. In Chapter 2 on logarithms, Euler describes the early usage of series expansions to represent functions as ln (1 + x) and ex without the usual approach of derivatives and Maclaurin series. Chapter 3 on infinite series traces Euler's attempts over a

2Lorenzo Mascheroni calculated the value for Euler's constant to 32 decimal places in 1790, but there were errors after the twentieth decimal place. 210 Annotated Bibliography

period of years to find the exact value for

Chapter 4 presents some connections between infinite series and the primes of num­ ber theory, and contains various proofs that the series of reciprocals of the primes di­ verges to infinity. Chapter 5 introduces the imaginary numbers and has three proofs of Euler's identity that eix = cosx + i sinx. [11] William Dunham, Journey Through Genius, Penguin Books, New York, 1991. This little book is popular with students because of its readability and interesting historical and biographical details. Four of the chapters are appropriate for an anal­ ysis course. Chapter 4 portrays the Greek period in the time of Archimedes and his idea of approximating areas and volumes by some very distinctive methods. Chap­ ter 7 presents some of the gifted individuals in the 17th century who preceded , and includes his work with the general and its use to approximate rr. Chapter 8 contains the work of Leibniz and the Bemoullis to pro­ vide an approach to calculus that differed from Newton's, and includes an unusual proof of the of the harmonic series. Chapter 9 presents the prolific life and accomplishments of and his discovery of the formula

The author is writing a similar book presenting some important results from analy­ sis which is to be published by Princeton University Press. This new book will be a valuable addition to this list of references. [12] C. H. Edwards, Jr., The Historical Development of the Calculus, Springer-Verlag, New York, 1979. This book presents the historical development of calculus throughout the entire period of mathematical history. More than half of the book, seven chapters, deals with the contribution of individuals prior to the time of Newton and Leibniz. There is a chapter each on Newton and Leibniz, and three chapters on the subsequent development of this subject. The author does a nice job in presenting each result in terms of the approach that was originally used. Emphasis is on the methods used to solve calculus problems throughout history, and many exercises are included. [13] Howard Eves, An Introduction to the History of Mathematics, 6th ed., Saunders College Publishing, 1990. This history text was written in 1964 with the goal of presenting the history of mathematics to prospective secondary teachers of mathematics. It is distinguished by extensive problem sets at the end of every chapter. Among other additions, the sixth edition includes ten cultural connections written by Jamie Eves. The material on calculus is mainly in Chapters 11 and 12, although there is a very nice chronol­ ogy of rr in Chapter 4. Annotated Bibliography 211

[14] John Fauvel and Jeremy Gray, editors, The History of Mathematics: A Reader, Macmillan Press, London, 1987. This collection of readings was assembled to support a course on the history of mathematics that was offered at the Open University in England. Its value for students is enhanced because the selections have been translated into English when necessary, and because of many unusual items such as obituary notices and excerpts from letters. For example, the reader can find a description of the first meeting between John Bernoulli and the Marquis de }'Hospital which can enliven a calculus presentation ofl'Hospital's rule (see Selection 1 in Part IV). Most of the entries that pertain to calculus and analysis are found in Chapters 12-14 and 18.

[15] A. Gardiner, Infinite Processes: Background to Analysis, Springer-Verlag, New York, 1982. The purpose of this book is to carefully examine the infinite processes that arise in elementary mathematics as a prologue to analysis. The longest unit is Part II which rigorously develops the concept of number. Part ill develops the concept of geometry, and the status of number in this setting. The author points out that ge­ ometry needed to be set aside for the careful presentation of calculus around 1870, based on an arithmetic treatment of real numbers. Part IV discusses the question: What is a function?

[16] Charles Coulston Gillispie, Editor-in-chief, Biographical Dictionary ofMathemati­ cians, Charles Scribner's Sons, New York, 1995. This is a four volume set that provides detailed biographical sketches of all im­ portant mathematicians, along with some commentary about the content and sig­ nificance of their work. A copy should be in every mathematics department library. The material on mathematicians is also contained in the 17 volume set titled the Scientific Dictionary ofBiography.

[17] Judith Grabiner, "Is Mathematical Truth Time Dependent?" The American Mathe­ matical Monthly, 81 (1974) 354-365. Judith Grabiner has written articles and books and given lectures about some of the foundational issues in calculus. The AMS-MAA Joint Lecture Series has a video of a talk she delivered at the January 1991 annual meetings in San Francisco with the title "Was Newton's Calculus Just a Dead End: Maclaurin and the Scottish Connection." In this article, she looks at the changing level of rigor in calculus from the time of Newton and Leibniz at the end of the 17th century until the time of Cauchy and Weierstrass in the 19th century, and asks for the reasons of this revolution in thought. After giving some examples of the reliance by the 18th century workers in calculus on any method that gives new results, she lists several reasons why stan­ dards changed so dramatically in the 19th century to produce a much higher level of rigor. Individuals such as Cauchy, Riemann, and Weierstrass took 18th century approximation techniques and transformed them into the clear definitions and rig­ orous proofs that characterized 19th century in calculus. This article is included as Selection 3 in Part IV. 212 Annotated Bibliography

[18) Judith Grabiner, The Origins of Cauchy's Rigorous Calculus, The MIT Press, Cam­ bridge, MA, 1981. Augustine-Louis Cauchy was largely responsible for the initiation of rigor into calculus. The author develops the position that Cauchy's crucial insight was that "by means of the limit concept, the calculus could be reduced to the algebra of inequalities." She also shows in this book that while all parts of calculus received a new formulation at the hands of Cauchy, many predecessors contributed to this and many successors carried out his work. The contributions of Euler and Lagrange during the 18th century are highlighted. Gauss and Bolzano are presented as two individuals who had similar ideas as Cauchy but who were distracted by other pur­ suits. Riemann and Weierstrass are later contemporaries of Cauchy who pushed his ideas further. The author devotes the second half of the book to a presentation of Cauchy's results on the calculus, showing how it "made obsolete many of the earlier treatments of limits, convergence, continuity, derivatives, and ." A helpful appendix contains English translations of some of Cauchy's major contributions to the foundations of the calculus. [ 19] Ivor Grattan-Guinness, The Development ofthe Foundations ofMathematical Anal­ ysis from Euler to Riemann, The MIT Press, Cambridge, MA, 1970. This book opens with a discussion of the vibrating string problem as it was understood in the 18th century by d' Alembert and Euler. Work on this question continued during the next century, leading to the method of trigonometric series as developed by Daniel Bernoulli, Lagrange, and then Joseph Fourier, who used this device for his solution of the heat diffusion equation. Although there is some discussion of more general ideas from the development of analysis in the 19th cen­ tury by Cauchy, Dirichlet, Riemann, and Weierstrass, the dominant theme is that of Fourier series and convergence questions related to it. Many examples are presented in careful detail to illustrate how various questions were handled in their historical context. The book ends with Riemann's integrability criterion, and the author notes that this led to a new era of analysis in which the ideas of Cantor's set theory and Lesbegue's measure theory dominate. There is also an appendix which presents the various for series of constants that were developed during the first half of the 19th century. [20] Ivor Grattan-Guinness, editor, From the Calculus to Set Theory 1630-1910, Prince­ ton University Press, 2000. This book was first published by Gerald Duckworth and Co. in London in 1980, but was out of print before that decade ended. Fortunately, a paperback version is now available. The book contains six chapters, each written by a different author on a topic of their speciality. In the first chapter, Kristi Moller Pederson presents the beginning of the ideas of calculus during the period 1630 to 1660, while in Chapter 2 Henk Bos continues this development from Newton and Leibniz until the middle of the 18th century. I. Grattan-Guinness writes Chapter 3 to trace the emergence of analysis ideas from physical problems in the 18th century through the search for foundations during most of the 19th century. Thomas Hawkins discusses in Chapter 4 the development of the concept that culminated with Lebesgue Annotated Bibliography 213

measure and integration in 1902. Joe Dauben presents the basic results of Cantor's set theory in Chapter 5, and Robert Bunn writes the concluding chapter on the foundations of mathematics, especially in logic, that had developed by the time Russell and Whitehead wrote Principia mathematica around 1910. The first four chapters of the book provide an excellent historical framework for the development of analysis, along with several specific examples from the mathematics of this time period. [21] E. Hairer and G. Wanner,Analysis by Its History, Springer-Verlag, New York, 1995. The authors choose to treat the topics of analysis in their historical order. Chap­ ter 1 deals with the origins of the elementary functions using the mathematical reasoning of Descartes, Newton, and Euler. Chapter 2 introduces the differential and integral calculus in the language of Leibniz, the Bernoullis, and Euler. Chap­ ter 3 presents the theoretical foundations of calculus from the nineteenth century, correcting errors and replacing intuitive reasoning of the previous two centuries. The final chapter treats calculus of several variables. An abundance of interesting quotations enliven the book. A wide range of major theorems are included, well beyond the scope of a usual one semester course in real analysis. [22] Thomas Hawkins, Lebesgue's Theory of Integration: Its Origins and Development, Chelsea Press, New York, 1975. The first two chapters present Riemann's development of the concepts of func­ tion and integral in the 1850s as a significant improvement over the work of his predecessors such as Cauchy. The next two chapters discuss Cantor's theory of sets and the subsequent development of ways to measure these sets during the period from 1870 to 1900. The pioneering results of Lebesgue to create a modern theory of integration in the opening decades of the twentieth century are given in Chapter 5. The author carefully places the new theory in the setting of its historical context and development as a reasonable consequence of the work of many individuals in the fifty years preceding the year 1902. [23] Omar Hijab, Introduction to Calculus and Classical Analysis, Springer-Verlag, New York, 1997. This book provides a non-traditional choice for a real analysis text. The author chooses to include many significant applications and devotes a quarter of the book to this topic. Some of the more unusual inclusions for a book at this level are the Euler-Maclaurin formula, the , continued fractions, infinite products, and Laplace and Fourier transforms. On the pedagogical side, he chooses to avoid E -8 arguments, replacing them by sequence results based, in turn, on the sup and inf concepts. The integral is then defined in terms of area under the graph, and the interchange of limits and integral theorems are based on the monotone convergence theorem. In the Preface, the author states that he "chose several of the jewels of classical eighteenth and nineteenth century analysis and inserted them at the end of the book, inserted the axioms for reals at the beginning, and filled in the middle with (and only with) the material necessary for clarity and logical completeness." Each teacher will need to decide whether this book is suitable as a text for his or her students, or if it works better as a reference source for ideas. 214 Annotated Bibliography

[24] Dan Kalman, "Six Ways to Sum a Series," The College Mathematics Journal, Vol. 24, No. 5, 1993, pp. 402-421. This interesting article delivers just what its title promises. The series referred to is Euler's series, namely

00 1

n=ILn2

which converges to T(2 /6. A rich collection of mathematical results and methods are presented, including unusual trigonometric identities, de Moivre's formula, the tranformation of a double integral using Jacobians, residue theory from complex analysis, Fourier series, and vector spaces. Most of these results are adapted from other sources. Additional proofs that

continue to regularly occur in the literature. It seems a shame there are so many different proofs for the value of

but not one for the exact value of

[25] Victor J. Katz, A History ofMathematics, Addison-Wesley Educational Publishers, Inc., New York, 1998. Katz's book was the first history of mathematics text to give serious attention (i.e., three chapters) to the contribution from cultures other than those of the early Greek and the modem European periods that are traditionally emphasized. His work is also distinguished by his attention to mathematical details that support the con­ cepts presented in the historical narrative. Three chapters are devoted to calculus and analysis for a total of 176 pages. [26] Morris Kline, Mathematical Thought from Ancient to Modem Tzmes, Oxford Uni­ versity Press, New York, 1972. This lengthy book has an entry on most major topics in the development of mathematics. Five of the fifty chapters are devoted to calculus and real analysis, beginning with Chapter 17 titled ''The Creation of the Calculus" and ending with Chapter 40 titled "The Instillation of Rigor in Analysis." Other chapters deal with parts of analysis such as four chapters on differential equations, two chapters on the , and single chapters on functions of a complex variable, the­ ory of functions of a real variable, functional analysis, and divergent series. Kline is able to express the main ideas in a readable manner. Annotated Bibliography 215

[27] Reinhard Laubenbacher and David Pengelley, Mathematical Expeditions, Springer­ Verlag, New York, 1999. This book is authored by two members of the mathematics faculty at New Mex­ ico State University, where there has been a serious attempt to use original sources at many levels of mathematics instruction. This book gives a historical introduction to five main areas of mathematics, followed by some representative selections of original work by important figures in the development of each topic. In Chapter 3 on analysis, the selections are chosen from the writings of Archimedes, Cavalieri, Leibniz, Cauchy, and Abraham Robinson. [28] George F. Simmons, Calculus Gems, McGraw Hill, New York, 1992. This small paperback was originally an appendix to Simmons' calculus text in the 1980s. Part 1 contains brief biographies of several individuals who contributed in some way to the development of the ideas of calculus. These are uneven in that some of the most important individuals, such as the Bernoullis, Fourier, and Cauchy have very brief selections, while some minor individuals have extended entries. Part 2 contains several "gems," which are outlines of proofs for some interesting results not typically found in a calculus text. Some of the more interesting ones include the catenary curve, the Bernoulli numbers, the general Maclaurin series for tan x, and several proofs that

See Selection 5 in Part IV for one of these gems. [29] Saul Stahl, Real Analysis: A Historical Approach, John Wiley & Sons, New York, 1999. The first five chapters contain examples of the work of Archimedes, Fermat, Newton and Euler on topics from infinite series, power series, and trigonometric series. The middle of the book develops the basic properties of real,numbers, se­ quences, continuity, differentiability, and infinite series. Material on the is omitted to make room for material on the historical movitation for the concepts of convergence. Many interesting problems are presented, including a sig­ nificant amount of material on Fourier series. [30] John Stillwell. Mathematics and Its History, Springer-Verlag, New York, 1989. Chapter 8 briefly describes several of the discoveries in the 17th century by individuals such as Cavalieri, Fermat, and Wallis leading up to the calculus as formulated by Newton and Leibniz. Chapter 9 on infinite series contains mate­ rial about interpolation formulas, generating functions, and the zeta function. Each of the twenty chapters concludes with short biographical sketches of mathemati­ cians who were influential in the development of the mathematics discussed in that chapter. [31] Dirk J. Struik, A Source Book in Mathematics 1200-1800, Harvard University Press, Cambridge, MA, 1969. 216 Annotated Bibliography

This book contains five chapters with selections of mathematical writings which have been translated into English. Most selections are brief, averaging about six pages, with a short introduction for each one. Chapters 4 and 5 contain 38 selections dealing with topics from analysis, written by individuals such as Kepler, Cavalieri, Fermat, Pascal, Leibniz, Newton, Euler, and the Bemoullis. This author has also written A Concise History ofMathematics, published in a Dover paperback in 1987. [32) Robert M. Young, Excursions in Calculus: An Interplay of the Continuous and Discrete, The Mathematical Association of America, 1992. The claim is made that "the purpose of this book is to explore-within the con­ text of elementary calculus-the rich and elegant interplay that exists between the two main currents of mathematics, the continuous and the discrete." As might be expected, there are many results from number theory and the search for patterns, dealing with their implications for and applications to many topics in the calcu­ lus. There is a collection of more than 400 exercises, with a reference provided for the solution of each one. More than 400 references are also included for additional reading and research. [33) Hans Niels Jahnke, editor, A History of Analysis, American Mathematical Society, 2003. I include this reference at the end of the Annotated Bibliography since I only became aware of it shortly before my book went to press. It is a joint effort of several individuals who each wrote a chapter on part of the history of analysis from antiquity until the end of the nineteenth century. There was much interaction between the writers to ensure a unified text, most of which was translated from the original German into English. It is an excellent, detailed, scholarly treatment which will be of great benefit to the instructor. Additional References

Several references to helpful books and articles that are not listed in the Annotated Bibli­ ography occur throughout the text. These are listed below and the number in parentheses following each entry refers to the page in the text where that reference may be found.

Donald J. Albers, G. L. Alexanderson, Constance Reid, International Mathematical Con­ gresses: An Illustrated History, Springer-Verlag, New York, 1987. (page 142) E. N. da C. Andrade, A Brief History of the Royal Society, Royal Society, London, 1960. (page 129) Robert G. Bartle, The Elements of Integration, John Wiley & Sons, Inc., New York, 1966. (page 122) Bruno Belhoste, Augustin-Louis Cauchy: A Biography, Springer-Verlag, New York, 1991. .(page 135) Bruce C. Berndt, Ramanujan 's Notebooks, Part I, Springer-Verlag, New York, 1985. (page 145) Carl B. Boyer and Uta C. Merzbach, A History ofMathematics, Second edition, John Wiley & Sons, New York, 1989. (pages 139 and 144) Robert L. Brabenec, Introduction to Real Analysis, PWS-Kent Publishing Company, Boston, 1990. (page 126) R. Creighton Buck, Advanced Calculus, McGraw-Hill Book Company, New York, 1965. (page 107) W. K. Buhler, Gauss: A Biographical Study, Springer-Verlag, New York, 1981. (page 138) Gale E. Christianson, In the Presence of the Creator: Isaac Newton and His Times, The Free Press, New York, 1984. (page 129) Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, Cambridge, Massachusetts, 1979. (page 142) 217 218 Additional References

John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Prob­ lem in Mathematics, Joseph Henry Press, Washington D.C., 2003. (page 147) Marcus Du Sautoy, The Music of the Primes, HarperCollins Publisher, New York, 2003. (page 147) Underwood Dudley, Elementary Number Theory, Second edition, W. H. Freeman and Company, San Francisco, 1978. (page 169) William B. Ewald, ed., From Kant to Hilbert: A Source Book in the Foundations of Math­ ematics, Clarendon Press, Oxford, 1996. (page 140) Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis, Holden-Day, Inc., San Francisco, 1964. (page 36) I. Grattan-Guinness, ed., Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge Inc., New York, 1994. (page 170) Jeremy J. Gray, The Hilbert Challenge, Oxford University Press, Oxford, 2000. (page 144) Thomas L. Hankins, Jean d'Alembert: Science and the Enlightenment, Clarendon Press, Oxford, 1970. (page 134) G. H. Hardy, A Mathematician's Apology, Cambridge University Press, Cambridge, 1969. (page 145) Julian Havil, Gamma: Exploring Euler's Constant, Princeton University Press, Princeton, New Jersey, 2003. (pages 33 and 146) John Herival, Joseph Fourier: The Man and the Physicist, Clarendon Press, Oxford, 1975. (page 136) Joseph E. Hofmann, Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity, Cambridge University Press, Cambridge, 1974. (page 130) loan James, Remarkable Mathematicians, Cambridge University Press, Cambridge, 2002. (page 144) Robert Kanigel, The Man Who Knew Infinity, Charles Scribner's Sons, New York, 1991. (page 145) Don H. Kennedy, Little Sparrow: A Portrait of Sophia Kovalevsky, Ohio University Press, Athens, Ohio, 1983. (page 141) James Kirkwood, An Introduction to Analysis, PWS-Kent Publishing Company, Boston, 1989. (page 43) Ann Hibner Koblitz, A Convergence of Lives Sofia Kovalevskaia: Scientist, Writer, Revo­ lutionary, Rutgers University Press, New Brunswick, New Jersey, 1993. (page 141) Eli Maor, e: The Story of a Number, Princeton University Press, Princeton, New Jersey, 1994. (page 82) Michael Monastyrsky, Riemann, Topology, and Physics, translated by James King and Vic­ toria King and edited by R. 0. Wells, Jr., Birkhauser, Boston, 1999. (page 140) James R. Newman, editor, The World of Mathematics, Simon and Schuster, New York, 1956. (page 143) Constance Reid, Hilbert, Springer-Verlag, New York, 1970. (page 142) Ranjan Roy, 'The Discovery of the Series Formula for rr by Leibniz, Gregory, and Nilakan­ tha," Mathematics Magazine, Vol. 63, No. 5, December 1990, pp. 291-306. (page 127) Additional References 219

George Sarton, "Lagrange's Personality," Proceedings of the American Philosophical So­ ciety, Vol. 88, No. 6, 1944. (page 134) James A. Sellers, ''Beyond Mere Convergence," PRIMUS, Volume XII, Number 2, June 2002, pp. 157-164. (page 94) N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, Published electronically as http://www.research.att.com/"'njas/sequences. (page 95) Ian Tweddle, James Stirling, Scottish Academic Press, Edinburgh, 1987. (pages 179-187) Alfred van der Poorten, "A Proof that Euler Missed," The Mathematical Intelligencer, Vol. 1, No. 4, 1978, pp. 195-203. (page 82) David V. Widder, Advanced Calculus, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1961. (page 119) Raymond L. Wilder, Introduction to the Foundations ofMathematics, John Wiley & Sons, New York, 1965. (page 166) Benjamin H. Yandell, The Honors Class: Hilbert's Problems and Their Solvers, A K Pe­ ters, Natick, MA, 2002. (page 142)

Index

Abel, Niels Henrik (1802-1829), 136, 138, Apollonius (262-190 B.C.), 126, 146 146,194,195,207,209 approximation, 52, 61, 64, 97, 98, 112, 161, Abelian functions, 141 162,181,182,184,193,194,211 absolute continuity, 200 Archimedean axiom, 168, 172 , 45-47, 53, 59, 163 Archimedes (287-212 B.C.), 48, 96-99, Academiae Algebrae, 110 126-128, 133,146,215 Academie des Sciences, 136, 138, 139, 198, arcsin, 29, 47 201 arctan, 28, 48, 62, 103 Acta Eruditorum, 131, 132, 177 area,8, 105,123,127,129,150,207 Actae Lipslienses, 179 Argand, J. R. (1768-1822), 138 ~o, 123 Aristotelian, 129 algebra, 144,171,189,190,208 arithmetic mean sequence, 116-119 of inequalities, 194 arithmetization of analysis, 141, 170 algebraic curve, 177 Ars Conjectandi, 110, 132 algebraic numbers, 82, 146, 171 astronomy,99, 138,143,187 alternating harmonic series, 54, 57 axiomatic, 143,144,164,187,191 , 41, 52, 58, 82, 162, 163 , 45, 54, 58, 59, 73, 162 Baire, Rene (1874-1932), 198,200 generalized form, 53, 55, 58 Banach, Stefan (1892-1945), 144 American Mathematical Society, 142 Barrow, Isaac (1630-1677), 129, 131, 178 anagram, 131 Basel, 132, 133, 177, 187 Analyse des lnfmiment Petits, 178 University of, 132, 133 analysis, 36, 75, 126, 136, 138, 141, 144, Berkeley, Bishop George (1685-1753), 135, 146,154,163,169,170,186,190,191, 192 208,212,216 Berlin, 134 The Analyst, 192 Berlin Academy, 82, 131, 133, 134, 137, analytic geometry, 127 139, 141, 193 ,21,32, 72, 78,90,91,95, Berlin, University of, 132, 139-142 114, 115, 140 Bernoulli, 137,146,178,188,210,213,215, Apery, Roger (1916-1994), 82,209 216

221 222 Index

Bernoulli, Daniel (1700-1782), 132, 133, calculus outline, 1-5 212 calculus review problems, 9-17 Bernoulli, James (1654-1705), 48, 50, llO, derivatives, 10-12 131, 132, 134, 135, 144 infinite series, 14-17 Bernoulli, John (1667-1748), 48, 70, 72, integration, 12-14 131-135, 144,175,177,178,211 limits and continuity, 9-10 Bernoulli, Nicholas (1687-1759), 132, 187 Cambridge University, 129, 131, 144, 145 Bernoulli numbers, 27, 31, 101, 102, cancellation principle, 15, 22-24, 156, 167 109-111, 132, 134,202-206,209,215 Cantor, Georg (1845-1918), ll2, 123, 125, Berzelius, 190 137,139,142,143,146,171 beta function, 103, 107, 108 Cantor set, 123 binomial cardinal number, 123, 142 coefficient, 203 Carnot, L. M. N. (1753-1823), 138 expansion, 11, 35, 128 Cartesian plane, 128 series, 128, 209 Cartesian product, 170 theorem, 31, 147, 148, 188, 210 catenary, 132,177,215 black plague, 129 Catherine (wife of Peter the Great), 133 Bolyai, Janos (1802-1860), 127, 138 Catherine the Great (1729-1796), 134 Bolzano, Bernhard (1781-1848), 139, 190, Cauchy, Augustin Louis (1789-1857), 126, 193,195,207,212 135-138, 141,143,146,170,176,190, Bolzano-Weierstrass theorem, 139, 145, 191, 193-196,207-209,211-213,215 165,166, 172-174 Cauchy-Lipschitz method, 194 forsequences,42, 167,172 Cauchy-Riemann equations, 145 for sets, 167, 169, 172 , 155, 163 Book of Squares, 19 Cavalieri, Bonaventura (1598-1647), 26, Boole, George (1815-1864), 144 29-31,69, 70, 72,100,101,128,215, Boolean algebra, 170 216 Borel, Emile, (1871-1956), 198, 199 Cavalieri sums, 29-31, 69, llO boundary points, 39, 40, 166, 173 Cayley, Arthur (1821-1895), 144 bounded celestial mechanics, 138 derivatives,42, 199 Cesaro, Ernesto (1859-1906), ll9 function,36,38,41, 199 , ll, 147-149, 152, 190 sequence,33,35, 155,157,158,162,163, characteristic triangle, 128 165,166 Charles Il (1630-1685), 129 set, 41 Ck-summable, ll8, ll9, 121 variation, 199,201 closed interval, 122, 171, 172 bounds for a series closed set, 41, 171, 173, 174 lower, 15, 53-60 cluster point of a sequence, 119 upper, 15, 53-60 clusterpointofaset, 166,167,169, 172-174 Bourbaki, 144 combinations, 128 brachistochrone, 48 Commentarii, 183 Brahe, Tycho (1546-1601), 128 compact set, 41, 121, 164, 173, 174 Brouwer, L. E. J. (1881-1966), 143 comparison test, 103, 157 Brunswick, House of, 130 direct, 157, 159, 162 Bullitt, William, 145, 146 improper integrals, 103 limit form, 16, 84, 156, 159 c, 123 ratio form, 46, 159, 160 calculating machine, 130 complete ordered field, 171 calculus, 126, 129-132, 134-137, 141, 143, completeness axiom, see least upper bound 147,148,151,154,155,175,178,187, axiom 189,192, 193,202,207-209,211,212, complex analysis, 140, 141, 147, 167, 191, 214,215 214 Index 223 complex numbers, 138 Darwinian revolution, 187 Comptes Rendes, 136 d'Alembert, Jean Le Rond (1717-1783), computer, 99, 127, 132 134,135,137,192,195,212 computer algebra system, 101 De analysi, 131 concave, 12,152 decreasing function, 33, 35, 119, 160, 161 , 52, 53 decreasing sequence, 53, 55, 80 conics, 126 Dedekind, Richard (1831-1916), 139, 141, conjectures,41, 73,86,92-95, 128 146, 166, 193 connected set, 174 DeMorgan, Augustus (1806-1871), 144 construction problems, 96 definite integral, 32, 123, 194 continuity, 37, 68, 143, 172, 173, 190 de Moivre's formula, 214 unifonn,40-42, 74,143 Denjoy, Arnaud (1884-1974), 200 , 36-39, 75, 112, 136, dense set, 199 144,154,160,164,165,168,171,194, density property, 37, 172 200 De quadratura curvarum, 131 nowhere differentiable, 39, 141, 143 derivative,37,68, 128,130,132,183,195 continuum problem, 142 bounded,41,42, 199 contradiction, 164,165 formulas, 147-155 contrapositive, 42, 156, 164, 165, 167 derived set, 40, 172, 173 convergence, 190,194 Descartes, Rene (1596-1650), 29, 127, 128, absolute, 45-47, 53, 59 135, 146, 213 pointwise, 75 Description of Egypt, 137 unifonn,47, 74-79, 112,136,143 diary, 138 convergence tests, 83, 125, 138, 155-163, Diderot, Denis (1713-1784), 134 172 differentiable function, 20, 36, 37, 143 alternating series test, 45, 54, 58, 59, 73, differential, 134, 144, 177 162 , 143 comparison test, 103, 137 differentiation, 125, 129 direct comparision test, 157, 159, 162 implicit, 11 integral test, 16, 51, 85, 157, 160, 161 logarithmic, 11 limit fonn of the comparison test, 16, 84, Dini, Ulisse (1845-1918), 199 156,159 , 157, 162 Raabe's test, 29, 42, 45-47 direct proof, 164 ratio fonn of the comparison test, 46, 159, Dirichlet, Lejeune (1805-1859), 37, 138, 160 139,146,208,209,212 ratio test, 16, 28, 29, 42-47, 76, 88, 157, discontinuities, 36-38, 76, 103, 166 159,160,195 discontinuous function, 37-39, 78, 137, root test, 42-47, 76, 161, 162, 195 198 converse statement, 47, 155, 167 Disquisitiones arithmeticae, 138 Copernican revolution, 187 divergence, 116 Copernicus, Nicolas (1473-1543), 99, 128 divergentsequence,37, 120,167 cotangentfunction,204,205 divergent series, 37, 115-121, 144, 155, countably additive set function, 201 214 countably infinite, 122, 123, 142, 170 division of series, 26-27 counterexample, 36, 37, 41, 42, 136, 167 double integral, 14,214 Cours d' analyse, 135, 198, 209 Du Bois-Reymond, Paul (1831-1889), 200 Crelle, August(1780-1855), 141 Crelle's Journal, 141 e,42,81-83, 112,153,158,171,172 Cromwell, Oliver (1599-1658), 129 Ecole Normal, 135, 198 cubic equation, 50, 189,208 Ecole Polytechnique, 135, 139, 192, 193 , radius of, 177 Edinburgh, 182,183 cycloid, 132 Egypt, 130, 137 224 Index

Einsteinian revolution, 187 Fourier series, 14,21,49,52, 75, 78, 79, Elements, 163 112-115, 137,142,199,208,212,213, Encyclopedie, 134 215 Enlightenment, 134 fractals, 39 epsilon,60, 194 Franco-Prussian War, 139 equivalence relation, 170 Frederick the Great (1712-1786), 134 erro~52,53,55,58,60,65,68, 73,194, Frege, Gottlob (1848-1925), 139, 143 196,213 French Academy, see Academie des Euclid (ca. 295 B.C.), 95, 146, 163, 164 Sciences Euclidean plane, 170 French Revolution, 135, 136, 192 Eudoxus (408-355 B.C.), 126, 146 Fubini, Guido (1879-1943), 200 Euler constant, 33, 82, 83, 101, 146,209 function, 37-39 Euler, Leonhard (1707-1783), 27, 37, bounded,36,38,41, 199 48-52,81,82,87, 100--103, 112, continuous,36-39, 136,154,160,164, 131-137, 139,144,146,175, 179-188, 165, 171, 194 190,194,196,202,206,209,210, decreasing, 160, 161 212-216 defined by an integral, 11, 102, 103, 105, Euler-Maclaurin formula, 31, 52, 101, 102, 147,154 111, 145, 213 even, 19-21, 113,202 Euler-Stirling letters, 52, 179-187 integrable, 36 Eulerian numbers, 94 inverse, 148, 149 even function, 19-21, 113, 202 odd, 19-22, 113,114 everywhere continuous, ,1owhere spaces, 201 differentiable function, 39, 141, 143 special, 103, 107 evolute, 177 uniformly continuous, 37, 40-42, 74, 143, excluded middle, 164 164,167,168,194 , 147, 150, 151 functional analysis, 144 exterior point, 39, 40, 166, 173 fundamental theorem of algebra, 189 , 167, 168, 174 fundamental theorem of calculus, 14, 23, 78, 154,167,168,194,199,200,209 function, 105 Fatou, Pierre (1878-1929), 201 Galilei, Galileo (1564-1642), 29, 99, 129 Faulhaber, Johann (1580-1635), 110 Galois, Evariste (1811-1832), 136, 146 Faulhaber's formula, 31,209 gamma,33 Ferguson, D. F., 61, 65 gamma function, 14, 70-72, 102-108, 154 Fermat, Pierre de (1601-1665), 29, 31, 69, gapsequence,83-87 70, 72,100,128,135,146,177,215, Gauss, Carl Friedrich (1777-1855), 127, 216 133, 138-140, 142,146,189,195,207, Fermat's last theorem, 140, 147 212 Fibonacci, see Leonardo of Pisa generalization, 105,121,123, 142-144, 157, Fibonaccisequence,34,36, 79,208 187,199 field, 171 generalized alternating series test, 53, 55, 58 complete ordered, 171 generalized ratio test, 43, 44 finite subcover, 173 generalized root test, 43, 44 Fischer, Ernst (1875-1945), 201 geometricsequence,50,69, 70,128 , 131, 135 , 14, 22, 24, 44, 47, 50, 76, , 101,131,132,135,144,178,180, 79,81,88,91,96, 156,158,160,162, 182-184,209 167,206 formalist school, 143 geometry,95, 127,138,140,164,191,201, foundations of analysis, 190-193 209,211 Fourier, Joseph (1768-1830), 112, 135-137, George I (1660-1727), 130 146,199,207,212,213,215 Germain, Sophie (1776-1831), 142 Index 225 ghosts of departed quantities, 192 series,31,33, 128,130, 155-163, Girard, Albert (1595-1632), 189 179-188, 190,195,202,209,215 Godel, Kurt (1906-1978), 143 sets, 112,125,142,143,171,200 golden ratio, 34 inflection point, 106, 109 Gottingen, University of, 139, 140, 142 integrable, 21, 36, 74, 164, 167 Great Books of the Western World, 137 Lebesgue,37,38, 116 greatest integer function, 38 Riemann, 37, 38, 116 greatest lower bound, 171 integral, 121, 128, 130, 132 Greece, 187 definite, 32, 123, 194 Greek period, 81, 95-97, 126, 128, 191, 207, improper, 15, 32, 103, 104, 116, 160 210,214 Lebesgue,37, 121,122,144,176,198, Gregory, James (1638-1675), 61 212 Groeningen, 178 lower sum, 69, 70 Grundzuge der Mengenlehre, 144 Riemann, 121,128,140,143,144,199, 200,215 habilitationschrift, 140 integral test, 16, 51, 85, 157, 160, 161 habilitationsvortag, 140 integration,30, 125,129,140,200,201,213 Halley, Edmund (1656-1742), 130 , 14, 70, 71, 104, 108 Hamilton, William (1805-1865), 144 interchange of limit, 79, 112, 168 Hankel, Hermann (1839-1873), 187 interior point, 39, 40, 166, 173 Hardy, Godfrey Harold (1877-1947), 116, intermediate value theorem, 139, 194 142, 144-146 International Congress of Mathematicians, Hardy-Littlewood, 145 142,143 harmonic series, 58, 83, 119, 155, 167,210 interpolation, 128 harmonic triangle, 94, 130 interval of convergence, 29, 42, 45, 78, 142, Harnack, Axel (1851-1888), 199 164 . Hausdorff, Felix (1868-1942), 144 intervals, 121, 122, 171 Hawking, Stephen, 131 lntroductio in a11alysin infinitorum, 134 heat flow, 137 intuition,37,39,40,83, 123,127,164,191, Hermite, Charles (1822-1901), 82 200 hexagon,97 intuitionism, 164 Hilbert, David (1862-1943), 139, 142, 143 irrational number, 34, 37, 38, 81-83, 112, Hippocrates (ca. 460-377 B.C.), 96 123,141,146,147,164,171,172,209 Holmboe, Bernt (1795-1850), 136 Humboldt, Wilhelm von (1767-1835), 139 Jacobi, Carl Gustav (1804--1851), 138, 139 Huygens, Christiaan (1629-1695), 130 Jacobian, 214 hypergeometric series, 138, 195 Jordan, Camille (1838-1921), 198, 199 implicit differentiation, 11 Keill, John (1671-1721), 131 , 15, 32, 103, 104, 116, Kepler, Johann (1571-1630), 99,128,216 160 Klein, Felix (1849-1925), 139 comparison test for, 103 Konigsberg, University of, 141 of first kind, 103,104 Kovalevsky, Sonja (1850-1891), 141 of second kind, 103 Kronecker, Leopold (1823-1891), 139, 140, incompleteness theorem, 143 143 increasing sequence, 53, 80, 157, 165 Kummer, Ernst (1810-1893), 139, 140 indirect proof, 164, 165 induction, see mathematical induction Lagrange, Joseph Louis (1736-1813), 132, Industry Institute in Berlin, 141 134,135, 137-139, 146, 192-195,212 infimum, 121, 168, 171 Lambert,Johann(l728-1777),82 infinite, 190 Laplace, Pierre-Simon (1749-1827), 137, product, 205, 213 138, 146, 188, 213 228 Index

Lavoisier, Antoine (1743-1794), 190 logic, 42 least upper bound. 165, 171 London, 129-131, 182 least upper bound axiom, 165, 171, 172 long division, 27, 28, 110, 111, 176 Lebesgue, Henri (1875-1941), 121, 122, Louis XIV (1638-1715), 130 142,144,176, 198-201,213 lower Lebesgue integrable, 37, 38, 116 bound, 15,53-57,60, 121 Lebesgue integral, 37, 121, 122, 144, 176, integral, 121 198,212 sum, 33, 69, 121, 168 Lebesgue measure, 37, 121-123, 144, 176, Lucasian chair, 131 200,212,213 lunar crescent, 96, 100 Lecons sur ['integration, 201 Luzin, Nikolai (1883-1950), 201 Lectiones Geometricae, 129 Legendre, Adrien Marie (1752-1833), 137, 138 Machin,John(l680-1751),61,65-68 Leibniz, Gottfried (1646-1716), 48, 61, 66, Maclaurin, Colin (1698-1746), 101, 132, 72,94, 125-132, 134,135,139,144, 135,144,183,192,209,211 146,178,188, 190,207-213,215,216 Maclaurin series, 17, 20, 26-29, 37, 45, 47, notation of, 180 48,61,62,68,69, 72, 77, 78,82,87, Leipzig, 131, 177 88,91,99, 110,112,113,135,176,209 University of, 130 for arcsinx, 47 lemniscate, 132 for arctanx, 48, 61, 62, 99 length of interval, 121, 123, 143 forcosx, 50 Leonardo of Pisa (1170-1240), 79 for In (x + 1), 54 Letters to a German Princess, 134 forsinx, 49 levels of mathematical understanding, 169 fortanx, 111,204,215 L'Hospital, Marquis de (1661-1704), 131, fore\ 82 132,134,135,175, i88,211 fore-lfx2, 68 L'Hospital's rule, 9, 10, 13, 68, 71, 74, 75, for e<-:_ 1, 110 175,211 for(l +xi,47 limit, 135, 136, 138, 143, 170, 177, 178, Malebranche, Nicolas (1638-1715), 177 190,212 Mascheroni, Lorenzo (1750-1800), 209 inferior, 43, 47 master of the mint, 131 superior, 43 Mathematical Association of America, 142 limit form of the comparison test, 16, 55, 84, mathematical discovery, 143 156,159 mathematical induction, 7, 34, 70, 84, limit inferior, 43, 47 92-94, 108,164,166,169 limit point, 39, 40, 43, 139, 166, 169, 170, mathematical truth, 187, 191, 196 172,173 Mathematician's Apology, 144 limit superior, 43 maximum, 106 Lindemann, Ferdinard (1852-1939), 82 , 12, 23, 42, 152, 167, linear combination, 87, 93 195 linear vector space, 144 measure, 121-123, 142-144, 198,199,212 Liouville, Joseph (1809-1882), 138 measure-theoretic, 198 Littlewood, John Edensor (1885-1977), 145 measurement, 95, 96 Lobachevsky, Nicolai (1793-1856), 127, Memoirs of a Russian Childhood, 141 138,146 method of differences, 183 logarithm test, 195 method of exhaustion, 97 logarithmic Methodus fluxionum, 131 differentiation, 11 Methodus Incrementorum Directa et Inversa, function, 150-154 112, 135 spiral, 132 middle third, 123 logarithms, 127,147,148,182,209 mixed signs, 52, 53, 57, 162, 163 Index 227

modusponens, 164-167 outline of calculus topics, 1-6 Monge, Gaspard (1746-1818), 138 derivative,2 IDonotone,33,59,80, 155,157,163,166, differential equations, 4-5 200,213 integral, 3 Montmort, Pierre (1676-1719), 177, 178 limits, 1 multiplication of series, 26, 29 multivariate calculus, 4 , 5-6 Napier, John (1550-1617), 127 series, 3 Napoleon (1769-1821), 135, 137, 139 Oxford University, 128, 129, 144 , 11, 32, 102, 125, 151 natural number, 105, 143, 147, 166, 170 parabolic segment, 48, 96 Nazi regixne, 139 Paradoxes of the Infinite, 139 negation, 164 Paris, 130,131,135,137,139,142,178,192 neighborhood, 144, 171-173 University of, 142 nest of closed intervals, 169 Paris Academy, 136, 137 nested interval property, 167, 169, 172 parity New Method/or , 131 opposite, 20, 21 Newton, Isaac (1642-1727), 28, 30, 48, 72, same, 20 101,112, 125-135, 137,144,146,178, , 154 189,191, 192,207,208,210-213,215, partial differential equation, 137, 189 216 partial fraction, 14, 15, 25,204, 205 Newton's method, 11, 12, 32, 35 partition Noether, Emmy (1882-1935), 142 of an interval, 33, 35, 69, 121, 140, 168 non-commutative algebra, 187 ofa set, 40, 53,121,170, 171, 173 non-Euclidean geometry, 127, 138, 187, Pascal, Blaise (1623-1662), 128, 130, 135, 208 216 notebook, 145 Pascal's triangle, 91, 93, 95, 128 nth , 156, 164 patterns,91, 145,164,209,216 n-tuple, 62-64 Peacock, George (1791-1858), 144 number,211 Peano,Giuseppe(1858-1932), 166 algebraic,82, 146,171 pendulum motion, 130 itrational,37,38,81-83, 123,141,146, perfect set, 174 147,164,171,172 Peter the Great (1682-1725), 133 natural, 105,143,147,166,170 philosophies of mathematics, 143 prime, 87, 170 formalism, 143 rational, 34, 37, 38, 81, 82, 109, 122, 123, intuitionism, 143 146,147,170,172,203 logicism, 143 real,39, 126,143,170 ,r,49,51,61-68,82,83,97,99, 101,112, transcendental,82, 146,171 171,209,210 number theory, 133, 138, 209, 216 planetary motion, 127-129 three laws of, 128 odd function, 19-22, 113, 114 Poincare, Henri (1854-1912), 142, 143, 146 Oldenburg, Henry (1615-1677), 130, 131 point of a set one-to-one correspondence, 171 boundary point, 39, 40, 166, 173 open cluster point, 166, 167, 169, 173 cover, 121, 122, 173 exterior point, 39, 40,166, 173 intervai, 121, 122 interior point, 39, 40, 166, 173 set, 171, 173 limit point, 39, 40, 166, 173 order relation, 172 pointwise convergence, 75-77 ordinal number, 142 polar coordinates, 14 oscillation, 117 polygons, 98-100, 138 osculating circle, 177 Poncelet, J. V. (1788-1867), 138, 146 228 Index , 11, 147 Riemann power series, 75, 77, 78, 88, 108, 112, 117, hypothesis, 140, 147 137,138,143,195,202,204,208,215 integrability criterion, 140, 168, 212 power set, 170 integrable, 37, 38, 116, 140, 199 pre-calculus material, 7-9 integral, 121, 128, 140, 143, 144, 199, prime num~rs, 87, 170,210 200,215 Prince Elector of Mainz, 130 sum, 12, 23, 32 Princeton University, 145 zeta function, 140, 213 Principia Mathematica Rietz, Frederic (1880-1956), 201 of Newton, 128, 130, 178 rigor in analysis, 126, 135, 137, 141, 143, of Russell, 165,213 144,170, 189-196,211,212,214 Privatdocent, 140 Robinson, Abraham (1918-1974), 215 probability, 123, 132, 138,208 Rolle's theorem, 167 , 147 root test, 42-47, 76, 161, 162, 195 proof techniques, 126, 163-169, 193 generalized form, 43, 44 Prussia, 139 Royal Society, 100, 129-131, 145, 179, 183, p-series,45,48,51, 101,156,176 184,187 Pythagoras (ca. 585-500 B.C.), 81 Russell, Bertrand (1872-1970), 143, 165, 213 quadrature Russia, 133, 134, 141 of the circle, 96, 100, 181 of the lune, 96 St. Petersburg Academy, 52, 101, 133, 134, of the parabola, 96 175,179,180,183,184 quintic equation, 138 Scheeffer, Ludwig (1859-1885), 200 , 11, 147-149, 168 Scientific Revolution, 189 , 7, 12 Raabe, Joseph Ludwig (1801-1859), 45 Seidel, Paul (1821-1896), 136 Raabe's test, 29, 42, 45-47 sequence,31-36, 170 Radon,Johann(1887-1956),201 bounded,33,35, 155,157,158,163,165, Ramanujan, Srinivasa (1887-1920), 145 166 ratio and proportion, 126 Cauchy, 155, 163 ratio form of comparison test, 46, 159, 160 cluster point, 119 ratio test, 16, 28, 29, 42-47, 76, 88, 157, convergent, 164, 166 159,160,195 decreasing, 33, 35, 53, 80, 119 generalized form, 43, 44 divergent, 37, 120, 167 rational numbers, 34, 37, 38, 40, 81, 82, 109, increasing, 53, 80, 157, 165 122,123,146,147,170,172,203 monotone,33,80, 155,157,163,166 real numbers, 39, 126, 143, 170 of functions, 73-76 rearrangements, 52-60 of integrals, 73 recursion formula, 70, 71, 80, 88-91, 93, of partial sums, 15, 33, 53, 54, 57, 58, 60, 95,98, 100,101,104,105,110, 76,116,117,120,155,157,163 111, 176 series,31,33, 128, 155-163, 179-187, recursively defined, 12, 31, 32, 34, 35, 74, 195 81,203 convergent, 42-44, 53, 84, 85, 118, 156, related function, 10 180 relative maximum, 12 divergent, 37, 43, 45, 84-86, 115, 116, removable discontinuity, 9 118, 155, 214 Renaissance, 99, 127, 189 of functions, 76-79 revolution, 136, 187-189, 196 harmonic, 58, 210 Riemann, Georg Friedrich Bernhard infinite, 31, 33, 128, 130, 155-163, (1826-1866), 128, 139-141, 146,207, 179-188, 190,195,202,209,215 208, 211-213 of mixed signs, 52, 53, 57,162,163 Index 229

set, 170---174 telescoping principle, 22, 48, 130 bounded,41 term-by-term integration, 28, 199 closed, 41, 171 textbook, 133,134,137,151,190,193 connected, 174 Theorie analytique de la chaleur, 137 compact. 41, 121, 164, 173, 174 Theorie desfonctions analytiques, 137 dense, 167 Thomae, Johannes Karl (1840-1921), 38 derived, 142 topology, 39, 126, 143, 144, 170-174 infinite, 112, 125, 142, 143,171,200 topological spaces, 144 open, 171,173 tractrix, 132 separable, 174 Transactions, 179,182,183,196 simply ordered, 172 transcendental numbers, 82, 146, 171 uncountable, 142,170,171 transitive property, 164, 172 set theory, 142, 144,212, 213 trapezoid rule, 14, 32, 153 sine function, 49, 148-150 Treatise of Fluxions, 135, 209 Shanks, Daniel, 65 triangular numbers, 48, 130 Shanks, William (1812-1882), 61, 68 trichotomy law, 172 simply ordered set, 172 trigonometric functions, 147-150 Sloane, Hans (1660---1753), 184 trigonometricidentities,8,21,66, 115,149 , 127, 129 trigonometric series, 39, 78, 112, 137, 140, Sorbonne, 198 191,199,201,208,215 special functions, 103, 107 Trinity College, 145 squaring the circle, 96 triple, 61-63, 66 squeeze theorem, 9, 150, 151 trisection of an angle, 96 Stieltjes, Thomas (1856-1894), 201 Turin, 134, 193 Stirling, James (1692-1779), 100, 101, 135, University of, 137 175,179,182,183 twin prime conjecture, 87 subsequence,34,37,53,58,80, 162,165, 166 uncountable, 37, 123, 142, 170 convergent, 37, 53 uniform continuity, 37, 40---42, 74, 143, 164, monotone decreasing, 53, 80, 162 167,168,194 monotone increasing, 53, 80, 162 uniform convergence, 47, 74-79, 112, 136, subseries, 83-87 143,163,168,195 summability, 116, 118, 119 uniformly bounded series, 199 summable, 115-121 universities, 129, 139, 141, 192 summation formulas, 100, 185 upper supremum, 43, 121, 165, 171 bound, 15,30,53-57,60, 121,122,157, supremum principle, see least upper bound 158,160,161 axiom integral, 121 Sylvester, James Joseph (1814-1897), 144 sum, 121, 168 symbols, 127, 189, 190 chemical, 190 Veblen, Oswald (1880-1960), 145 Sweden, 142 vector space, 87, 88, 113, 144,171,214 vibrating string, 112, 137, 212 tangentfunction,26,204 Viete, Francoise (1540-1603), 99, 127, 135, tangent line, 12, 127-129 189 Taylor, Brook (1685-1731), 112, 132, 135, Vitali, Giuseppe (1875-1932), 199 188 Volterra, Vito (1860-1940), 38, 199, 209 Taylor polynomials, 17 volume, 126,128,207 , 112, 135, 193, 195, Taylor's theorem with remainder, 45, 78, 87 Wallis, John (1616-1703), 128, 129, 131, teaching, 192 215 telescope, 129 Weggersloff, Friedrich (1702-1763), 179 230 Index

Weierstrass, Karl (1815-1897), 39, 139-143, Woolsthorpe, 129 146,170,193,195, 196,207-209,211, Wrench, J. W., 61, 65 212 Weierstrass M-test, 72, 76, 77, 163 Young, William Henry (1863-1942), Wessel, C. (1745-1818), 138 199 Westminster Abbey, 131 Whitehead, Alfred North (1861-1947), 165, Zeno (ca. 490-425 B.C.), 48 213 zeroes, 49, 50, 82, 140 wine casks, 128 zeta function, 140, 147, 213, 215 About the Author 231 About the Author Robert L. Brabenec was born on January 11, 1939 in Chicago, Illinois. His grandparents came to America from Poland and Czechoslovakia. He majored in mathematics and re­ ceived a BS with highest honors from Wheaton College in 1960, and a PhD in mathe­ matics from Ohio State University in 1964, with a dissertation on the topic of measure and integration theory in a linear vector space. He began teaching mathematics at Wheaton College in June 1964, and after a two-year tour of active duty as an Army captain in missile intelligence, he returned to Wheaton in 1967 to become chair of the newly-formed Depart­ ment of Mathematics, and he has continued in that role for thirty-six years. He received the Teacher of the Year award from Wheaton College in 1970. He has also taught part-time at the University of Alabama at Huntsville and at Northwestern University. He has been a member of the Mathematical Association of America since 1960 and a long-term member of the American Mathematical Society as well. In 1977, he began the organization that is now called the Association of Christians in the Mathematical Sciences and has continued to. serve as the Executive Secretary until the present. This organization of about 400 members sponsors a major conference every two years, and supports other activities to help mathematicians see connections between mathematics and religious faith. Over the years, he has developed a strong interest in the history and foundations of mathematics, integrating these ideas into most courses that he teaches. He is author of a textbook Introduction to Real Analysis that which was published by PWS-Kent in 1990, and has presented several articles, talks, and a MAA minicourse dealing with historical themes in mathematics. He spent sabbaticals in 1988 at the Universities of London and Cambridge, in 1995 at the University of Virginia, and in 2002 back at Cambridge Uni­ versity. In a typical year, his teaching load consists of a calculus course, the mathematics colloquium course, and the required major courses of real analysis, as well as the history and foundations of mathematics. . VOL 4 VOL TEXTBOOKS Robert L Brabenec Robert Resources for the Study Study the for Resources of Analysis Real 243 pages • Spine: 1/2 in • 50lb paper • Softcover • 4-color Process (CMYK) • 4-color Process 243 pages • Spine: 1/2 in • 50lb paper • Softcover / MAA AMS

Resources for the Study of Real Analysis 4 AMS / MAA PRESS

VOL Robert L Brabenec —Paul Zorn, St. Olaf College —William Dunham, Muhlenberg College Trim Size 7 x 10 Size Trim TEXTBOOKS The book offers even more than its title suggests: a true trove of resources of trove true a suggests: title its than more even offers book The rough— who face the exciting—but often their teachers) for students (and calculus to the routine calculations of elementary passage from the for here gems are There analysis. real of insights and arguments deeper students who need for the quickest average students who need review; ideas; and for everyone who need for teachers mathematical challenges; and culture. with a taste for mathematical highlights analysis— of mathematical smorgasbord a rich Brabenec has provided essays, and selected readings— including a host of problems, historical hungry. no one should go away from which by the author while gathered This book is a collection of materials a period of years. real analysis over for use as a It is intended teaching material for textbook, or to provide supplement to a traditional analysis seminars its historical development. or independent study in analysis and biographical information, a wide range The book includes historical and on a variety of topics, and many of problem types, selected readings materials are collected Since all these references for additional study. items most easily choose and students can into a single book, teachers may use the book as a supplement to Teachers suitable for their purpose. of the book on their own. their courses, while students may read much a real for supplement a as specifically written been has book other No analysis course. AMS / MAA

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