An Introduction to Analysis / Edition 2

by Gerald G. Bilodeau, Paul R Thie, G. E. Keough

Overview

Part of the Jones and Bartlett International Series in Advanced Mathematics

Completely revised and update, the second edition of An Introduction to Analysis presents a concise and sharply focused introdution to the basic concepts of analysis from the development of the real numbers through uniform convergences of a sequence of functions, and includes supplementary material on the calculus of functions of several variables and differential equations. This student-friendly text maintains a cautious and deliberate pace, and examples and figures are used extensively to assist the reader in understanding the concepts and then applying them. Students will become actively engaged in learning process with a broad and comprehensive collection of problems found at the end of each section.

Product Details

ISBN-13:	9780763774929
Publisher:	Jones & Bartlett Learning
Publication date:	08/11/2009
Series:	International Series in Mathematics
Edition description:	2E
Pages:	334
Product dimensions:	7.20(w) x 9.20(h) x 0.90(d)

Table of Contents

Preface xi

1 The Real Numbers 1

1.1 Sets 2

1.2 Functions 7

1.3 Algebraic and order properties 12

1.4 The positive integers 18

1.5 The least upper bound axiom 24

2 Sequences 33

2.1 Sequences and limits 33

2.2 Limit theorems 39

2.3 Monotonic sequences 47

2.4 Sequences defined inductively 51

2.5 Subsequences 55

2.6 Cauchy sequences 65

2.7 Infinite limits 68

3 Functions and Continuity 71

3.1 Limit of a function 71

3.2 Limit theorems 76

3.3 Other limits 81

3.4 Continuity 86

3.5 Intermediate values, extreme values 95

3.6 Uniform continuity 103

3.7 Functions of two variables 109

4 The Derivative 119

4.1 Definition of the derivative 119

4.2 Rules for differentiation 127

4.3 The Mean Value Theorem 132

4.4 Inverse functions 143

4.5 Differentiability in R2 148

5 The Integral 163

5.1 The definition of the integral 163

5.2 Properties of the integral 173

5.3 Existence theory 182

5.4 The Fundamental Theorem of Calculus 190

5.5 Improper integrals 195

5.6 Double integrals 200

6 Infinite Series 213

6.1 Basic theory 213

6.2 Absolute convergence 225

6.3 Power series 233

6.4 Taylor series 247

7 Sequences and Series of Functions 257

7.1 Uniform convergence 258

7.2 Consequences of uniform convergence 270

7.3 Two examples 281

8 Introduction to Differential Equations 289

8.1 Elementary first order differential equations 290

8.2 Existence and uniqueness 298

8.3 Power series solutions 307

Appendix: Solutions and Hints for Selected Problems 315

List of Symbols 325

Index 329