Analysis - Kopp, P. E. (Elsevier, 1996)
Includes index
Front Cover; Analysis; Copyright Page; Table of Contents; Series Preface; Preface; Acknowledgements; Chapter 1. Introduction: Why We Study Analysis; 1.1 What the computer cannot see ... ; 1.2 From counting to complex numbers; 1.3 From infinitesimals to limits; Chapter 2. Convergent Sequences and Series; 2.1 Convergence and summation; 2.2 Algebraic and order properties of limits; Summary; Further exercises; Chapter 3. Completenessand Convergence; 3.1 Completeness and sequences; 3.2 Completeness and series; 3.3 Alternating series; 3.4 Absolute and conditional convergence of series; Summary
Further exercisesChapter 4. Functions Definedby Power Series; 4.1 Polynomials -- and what Euler did with them!; 4.2 Multiplying power series: Cauchy products; 4.3 The radius of convergence of a power series; 4.4 The elementary transcendental functions; Summary; Further exercises; Chapter 5. Functions and Limits; 5.1 Historical interlude: curves, graphs and functions; 5.2 The modern concept of function: ordered pairs, domainand range; 5.3 Combining real functions; 5.4 Limits of real functions -- what Cauchy meant!; Summary; Further exercises; Chapter 6. Continuous Functions; 6.1 Limits that fit
6.2 Limits that do not fit: types of discontinuity6.3 General power functions; 6.4 Continuity of power series; Summary; Further exercises; Chapter 7. Continuity on Intervals; 7.1 From interval to interval; 7.2 Applications: fixed points, roots and iteration; 7.3 Reaching the maximum: the Boundedness Theorem; 7.4 Uniform continuity -- what Cauchy meant?; Summary; Further exercises; Chapter 8. Differentiable Real Functions; 8.1 Tangents: prime and ultimate ratios; 8.2 The derivative as a limit; 8.3 Differentiation and continuity; 8.4 Combining derivatives; 8.5 Extreme points and curve sketching
11.1 The Fundamental Theorem of the Calculus11.2 Integration by parts and change of variable; 11.3 Improper integrals; 11.4 Convergent integrals and convergent series; Summary; Further exercises; Chapter 12. What Next? Extensions and Developments; 12.1 Generalizations of completeness; 12.2 Approximation of functions; 12.3 Integrals of real functions: yet more completeness; Appendix A: Program Listings; A.I Sequences program; A.2 Another sequence program; A.3 Taylor series; A.4 Newton's method in one dimension; Solutions to exercises; Index