Real Analysis

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.

“Stahl’s book, though relatively modest in its historical ambit, is a workmanlike and very readable introduction to real analysis with a distinctive flavour provided by a plethora of accessible exercises, many of which are historically motivated.”  (The Mathematical Gazette, 1 March 2014)

• Preface to the Second Edition Acknowledgments1. 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 Pierre de FermatAppendix C. From a Letter to Henry Oldenburg on the Binomial Series (June 13, 1676) by Isaac NewtonAppendix D. From a Letter to Henry Oldenburg on the Binomial Series (October 24, 1676) by Isaac NewtonAppendix E. Excerpts from "Of Analysis by Equations of an Infinite Number of Terms" by Isaac NewtonAppendix F. Excerpts from "Subsiduum Calculi Sinuum" by Leonhard Euler)Solutions to Selected ExercisesBibliographyIndex