A provocative look at the tools and history of realanalysisThis 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 seriesThroughout 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.

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'salgorithm5. 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" byArchimedesAppendix B. On a Method for Evaluation of Maxima and Minima byPierre de FermatAppendix C. From a Letter to Henry Oldenburg on the BinomialSeries (June 13, 1676) by Isaac NewtonAppendix D. From a Letter to Henry Oldenburg on the BinomialSeries (October 24, 1676) by Isaac NewtonAppendix E. Excerpts from "Of Analysis by Equations of anInfinite Number of Terms" by Isaac NewtonAppendix F. Excerpts from "Subsiduum Calculi Sinuum" by LeonhardEuler)Solutions to Selected ExercisesBibliographyIndex