Real Analysis: A Constructive Approach Through Interval Arithmetic (Pure and Applied Undergraduate Texts)

Real Analysis: A Constructive Approach Through Interval Arithmetic presents a careful treatment of calculus and its theoretical underpinnings from the constructivist point of view. This leads to an important and unique feature of this book: All existence proofs are direct, so showing that the numbers or functions in question exist means exactly that they can be explicitly calculated. For example, at the very beginning, the real numbers are shown to exist because they are constructed from the rationals using interval arithmetic. This approach, with its clear analogy to scientific measurement with tolerances, is taken throughout the book and makes the subject especially relevant and appealing to students with an interest in computing, applied mathematics, the sciences, and engineering. The first part of the book contains all the usual material in a standard one-semester course in analysis of functions of a single real variable: continuity (uniform, not pointwise), derivatives, integrals, and convergence. The second part contains enough more technical material--including an introduction to complex variables and Fourier series--to fill out a full-year course. Throughout the book the emphasis on rigorous and direct proofs is supported by an abundance of examples, exercises, and projects--many with hints--at the end of every section. The exposition is informal but exceptionally clear and well motivated throughout. Contents 6 0. Preliminaries 18 0.1 The Natural Numbers 18 0.2 The Rationals 20 1. The Real Numbers And Completeness 28 1.0 Introduction 28 1.1 Interval Arithmetic 29 1.2 Families of Intersecting Intervals 39 1.3 Fine Families 49 1.4 Defnition of the Reals 57 1.5 Real Number Arithmetic 60 1.6 Rational Approximations 72 1.7 Real Intervals and Completeness 75 1.8 Limits and Limiting Families 80 Appendix: The Goldbach Number and Trichotomy 84 2. An Inverse Function Theoremand Its Application 86 2.0 Introduction 86 2.1 Functions and Inverses 87 2.2 An Inverse Function Theorem 91 2.3 The Exponential Function 100 2.4 Natural Logs and the Euler Number 111 3. Limits, Sequences And Series 116 3.1 Sequences and Convergence 116 3.2 Limits of Functions 125 3.3 Series of Numbers 129 Appendix I: Some Properties of Exp and Log 148 Appendix II: Rearrangements of Series 151 4. Uniform Continuity 156 4.1 Definitions and Elementary Properties 156 4.2 Limits and Extensions 164 Appendix I: Are There Non-Continuous Functions? 174 Appendix II: Continuity of Double-Sided Inverses 178 Appendix III: The Goldbach Function 180 5. The Riemann Integral 182 5.1 Definition and Existence 182 5.2 Elementary Properties 189 5.3 Extensions and Improper Integrals 193 6. Differentiation 202 6.1 Definitions and Basic Properties 202 6.2 The Arithmetic of Differentiability 208 6.3 Two Important Theorems 213 6.4 Derivative Tools 221 6.5 Integral Tools 228 7. Sequences And Series Of Functions 240 7.1 Sequences of Functions 240 7.2 Integrals and Derivatives of Sequences 250 7.3 Power Series 256 7.4 Taylor Series 270 7.5 The Periodic Functions 278 Appendix: Raabe’s Test and Binomial Series 286 8. The Complex Numbers And Fourier Series 288 8.0 Introduction 288 8.1 The Complex Numbers 292 8.2 Complex Functions and Vectors 295 8.3 Fourier Series Theory 301 Index 314 Real,Analysis,A,Constructive,Approach,Through,Interval,Arithmetic Real A Constructive Approach Through Interval Arithmetic presents a careful treatment of calculus and its theoretical underpinnings from the constructivist point of view. This leads to an important and unique feature of this All existence proofs are direct, so showing that the numbers or functions in question exist means exactly that they can be explicitly calculated. For example, at the very beginning, the real numbers are shown to exist because they are constructed from the rationals using interval arithmetic. This approach, with its clear analogy to scientific measurement with tolerances, is taken throughout the book and makes the subject especially relevant and appealing to students with an interest in computing, applied mathematics, the sciences, and engineering. The first part of the book contains all the usual material in a standard one-semester course in analysis of functions of a single real continuity (uniform, not pointwise), derivatives, integrals, and convergence. The second part contains enough more technical material including an introduction to complex variables and Fourier series to fill out a full-year course. Throughout the book the emphasis on rigorous and direct proofs is supported by an abundance of examples, exercises, and projects many with hints at the end of every section. The exposition is informal but exceptionally clear and well motivated throughout.