Real Analysis

Saul Stahl 지음

Wiley 2012년 01월 10일 출간

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A provocative look at the tools and history of real analysisThis 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 continuityExplicit coverage of the epsilon-delta convergenceA discussion of the modern preference for the viewpoint of sequences over that of seriesThroughout 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.

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Preface to the Second EditionAcknowledgments1. Archimedes and the Parabola1.1 The Area of the Parabolic Segment1.2 The Geometry of the Parabola2. Fermat, Differentiation, and Integration2.1 Fermat&-s Calculus3. Newton&-s Calculus (Part 1)3.1 The Fractional Binomial Theorem3.2 Areas and Infinite Series3.3 Newton&-s Proofs4. Newton&-s Calculus (Part 2)4.1 The Solution of Differential Equations4.2 The Solution of Algebraic EquationsChapter Appendix. Mathematica implementations of Newton&-s algorithm5. Euler5.1 Trigonometric Series6. The Real Numbers6.1 An Informal Introduction6.2 Ordered Fields6.3 Completeness and Irrational Numbers6.4 The Euclidean Process6.5 Functions7. Sequences and Their Limits7.1 The Definitions7.2 Limit Theorems8. The Cauchy Property8.1 Limits of Monotone Sequences8.2 The Cauchy Property9. The Convergence of Infinite Series9.1 Stock Series9.2 Series of Positive Terms9.3 Series of Arbitrary Terms9.4 The Most Celebrated Problem10. Series of Functions10.1 Power Series10.2 Trigonometric Series11. Continuity11.1 An Informal Introduction11.2 The Limit of a Function11.3 Continuity11.4 Properties of Continuous Functions12. Differentiability12.1 An Informal Introduction to Differentiation12.2 The Derivative12.3 The Consequences of Differentiability12.4　　 Integrability13. Uniform Convergence13.1 Uniform and Non-Uniform Convergence13.2 Consequences of Uniform Convergence14. The Vindication14.1 Trigonometric Series14.2 Power Series15. The Riemann Integral15.1 Continuity Revisited15.2 Lower and Upper Sums15.3 IntegrabilityAppendix A. Excerpts from "Quadrature of the Parabola" by ArchimedesAppendix B. On a Method for Evaluation of Maxima and Minima by Pie

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저자(글) Saul Stahl

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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.