A chapter on
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“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)
Table of Contents
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 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
