Real Analysis
A Historical Approach
2nd Edition
Published on 20. September 2011
320 pages
978-1-118-09684-0 (ISBN)
Description
A provocative look at the tools and history of realanalysis
This new edition of Real Analysis: A Historical Approachcontinues 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 toabstract ideas.
The book begins with an exciting sampling of classic and famousproblems first posed by some of the greatest mathematicians of alltime. Archimedes, Fermat, Newton, and Euler are each summoned inturn, illuminating the utility of infinite, power, andtrigonometric series in both pure and applied mathematics. Next,Dr. Stahl develops the basic tools of advanced calculus, whichintroduce the various aspects of the completeness of the realnumber system as well as sequential continuity anddifferentiability and lead to the Intermediate and Mean ValueTheorems. The Second Edition features:
* A chapter on the Riemann integral, including the subject ofuniform continuity
* Explicit coverage of the epsilon-delta convergence
* A discussion of the modern preference for the viewpoint ofsequences over that of series
Throughout the book, numerous applications and examplesreinforce concepts and demonstrate the validity of historicalmethods and results, while appended excerpts from originalhistorical works shed light on the concerns of influentialmathematicians in addition to the difficulties encountered in theirwork. Each chapter concludes with exercises ranging in level ofcomplexity, and partial solutions are provided at the end of thebook.
Real Analysis: A Historical Approach, Second Edition isan ideal book for courses on real analysis and mathematicalanalysis at the undergraduate level. The book is also a valuableresource for secondary mathematics teachers and mathematicians.
Reviews / Votes
"Stahl's book, though relatively modest in itshistorical ambit, is a workmanlike and very readable introductionto real analysis with a distinctive flavour provided by a plethoraof accessible exercises, many of which are historicallymotivated." (The Mathematical Gazette, 1March 2014)More details
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Person
SAUL STAHL, PhD, is Professor in the Department of Mathematics at The University of Kansas. He has published numerous journal articles in his areas of research interest, which include combinatorics, discrete mathematics, and topological graph theory. Dr. Stahl is the author of Introductory Modern Algebra: A Historical Approach and Introduction to Topology and Geometry, both published by Wiley. He was awarded the Carl B. Allendoerfer Award from the Mathematical Association of America for expository articles in both 1986 and 2006.
Content
Preface to the Second Edition
Acknowledgments
1. Archimedes and the Parabola
1.1 The Area of the Parabolic Segment
1.2 The Geometry of the Parabola
2. Fermat, Differentiation, and Integration
2.1 Fermat's Calculus
3. Newton's Calculus (Part 1)
3.1 The Fractional Binomial Theorem
3.2 Areas and Infinite Series
3.3 Newton's Proofs
4. Newton's Calculus (Part 2)
4.1 The Solution of Differential Equations
4.2 The Solution of Algebraic Equations
Chapter Appendix. Mathematica implementations of Newton'salgorithm
5. Euler
5.1 Trigonometric Series
6. The Real Numbers
6.1 An Informal Introduction
6.2 Ordered Fields
6.3 Completeness and Irrational Numbers
6.4 The Euclidean Process
6.5 Functions
7. Sequences and Their Limits
7.1 The Definitions
7.2 Limit Theorems
8. The Cauchy Property
8.1 Limits of Monotone Sequences
8.2 The Cauchy Property
9. The Convergence of Infinite Series
9.1 Stock Series
9.2 Series of Positive Terms
9.3 Series of Arbitrary Terms
9.4 The Most Celebrated Problem
10. Series of Functions
10.1 Power Series
10.2 Trigonometric Series
11. Continuity
11.1 An Informal Introduction
11.2 The Limit of a Function
11.3 Continuity
11.4 Properties of Continuous Functions
12. Differentiability
12.1 An Informal Introduction to Differentiation
12.2 The Derivative
12.3 The Consequences of Differentiability
12.4 Integrability
13. Uniform Convergence
13.1 Uniform and Non-Uniform Convergence
13.2 Consequences of Uniform Convergence
14. The Vindication
14.1 Trigonometric Series
14.2 Power Series
15. The Riemann Integral
15.1 Continuity Revisited
15.2 Lower and Upper Sums
15.3 Integrability
Appendix A. Excerpts from "Quadrature of the Parabola" byArchimedes
Appendix B. On a Method for Evaluation of Maxima and Minima byPierre de Fermat
Appendix C. From a Letter to Henry Oldenburg on the BinomialSeries (June 13, 1676) by Isaac Newton
Appendix D. From a Letter to Henry Oldenburg on the BinomialSeries (October 24, 1676) by Isaac Newton
Appendix E. Excerpts from "Of Analysis by Equations of anInfinite Number of Terms" by Isaac Newton
Appendix F. Excerpts from "Subsiduum Calculi Sinuum" by LeonhardEuler)
Solutions to Selected Exercises
Bibliography
Index
