Real Analysis (ePub)

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'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 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" 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

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.

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