Real analysis: a historical approach

Abstract

Preface to the Second Edition Acknowledgments 1. Archimedes and the 1.1 The Area of the Parabolic Segment 1.2 The Geometry of the 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 s algorithm 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 9.3 Series of Arbitrary 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 by Archimedes Appendix B. On a Method for Evaluation of Maxima and Minima by Pierre de Fermat Appendix C. From a Letter to Henry Oldenburg on the Binomial Series (June 13, 1676) by Isaac Newton Appendix D. From a Letter to Henry Oldenburg on the Binomial Series (October 24, 1676) by Isaac Newton Appendix E. Excerpts from Of Analysis by Equations of an Infinite Number of Terms by Isaac Newton Appendix F. Excerpts from Subsiduum Calculi Sinuum by Leonhard Euler) Solutions to Selected Exercises Bibliography Index