Analysis: An Introduction: Richard Beals: 9780521600477: Book: Computers & Technology: eCampus.com

This self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book includes a number of interesting applications of the subject matter to areas both within and outside the field of mathematics. The aim throughout is to strike a balance between being too austere or too sketchy, and being so detailed as to obscure the essential ideas. A large number of examples and 500 exercises allow the reader to test understanding, practise mathematical exposition and provide a window into further topics.

This book is an extensive introductory text to mathematical analysis for graduate students and advanced undergraduates, complete with 500 exercises and numerous examples.

Preface

1 Introduction

(14)

1A. Notation and Motivation

(4)

1B. The Algebra of Various Number Systems

(4)

1C. The Line and Cuts

(3)

1D. Proofs, Generalizations, Abstractions, and Purposes

(3)

2 The Real and Complex Numbers

(15)

2A. The Real Numbers

(6)

2B. Decimal and Other Expansions; Countability

(3)

2C. Algebraic and Transcendental Numbers

(2)

2D. The Complex Numbers

(4)

3 Real and Complex Sequences

(15)

3A. Boundedness and Convergence

(3)

3B. Upper and Lower Limits

(2)

3C. The Cauchy Criterion

(2)

3D. Algebraic Properties of Limits

(2)

3E. Subsequences

(1)

3F. The Extended Reals and Convergence to � infinity

(2)

3G. Sizes of Things: The Logarithm

(1)

Additional Exercises for Chapter 3

(2)

4 Series

(16)

4A. Convergence and Absolute Convergence

(3)

4B. Tests for (Absolute) Convergence

(6)

4C. Conditional Convergence

(3)

4D. Euler's Constant and Summation

(1)

4E. Conditional Convergence: Summation by Parts

(1)

Additional Exercises for Chapter 4

(2)

5 Power Series

(12)

5A. Power Series, Radius of Convergence

(2)

5B. Differentiation of Power Series

(3)

5C. Products and the Exponential Function

(4)

5D. Abel's Theorem and Summation

(3)

6 Metric Spaces

(13)

6A. Metrics

(2)

6B. Interior Points, Limit Points, Open and Closed Sets

(4)

6C. Coverings and Compactness

(2)

6D. Sequences, Completeness, Sequential Compactness

(3)

6E. The Cantor Set

(2)

7 Continuous Functions

(13)

7A. Definitions and General Properties

(4)

7B. Real- and Complex-Valued Functions

(1)

7C. The Space C(I)

(4)

7D. Proof of the Weierstrass Polynomial Approximation Theorem

(4)

8 Calculus

(20)

8A. Differential Calculus

(6)

8B. Inverse Functions

105

(2)

8C. Integral Calculus

107

(5)

8D. Riemann Sums

112

(1)

8E. Two Versions of Taylor's Theorem

113

(3)

Additional Exercises for Chapter 8

116

(3)

9 Some Special Functions

119

(12)

9A. The Complex Exponential Function and Related Functions

119

(5)

9B. The Fundamental Theorem of Algebra

124

(1)

9C. Infinite Products and Euler's Formula for Sine

125

(6)

10 Lebesgue Measure on the Line

131

(13)

10A. Introduction

131

(2)

10B. Outer Measure

133

(3)

1OC. Measurable Sets

136

(3)

10D. Fundamental Properties of Measurable Sets

139

(3)

10E. A Nonmeasurable Set

142

(2)

11 Lebesgue Integration on the Line

144

(14)

11A. Measurable Functions

144

(4)

11B. Two Examples

148

(1)

11C. Integration: Simple Functions

149

(2)

11D. Integration: Measurable Functions

151

(4)

11E. Convergence Theorems

155

(3)

12 Function Spaces

158

(15)

12A. Null Sets and the Notion of "Almost Everywhere"

158

(1)

12B. Riemann Integration and Lebesgue Integration

159

(3)

12C. The Space L�

162

(4)

12D. The Space L�

166

(2)

12E. Differentiating the Integral

168

(4)

Additional Exercises for Chapter 12

172

(1)

13 Fourier Series

173

(24)

13A. Periodic Functions and Fourier Expansions

173

(3)

13B. Fourier Coefficients of Integrable and Square-Integrable Periodic Functions

176

(4)

13C. Dirichlet's Theorem

180

(4)

13D. Fej�r's Theorem

184

(3)

13E. The Weierstrass Approximation Theorem

187

(2)

13F. L�-Periodic Functions: The Riesz-Fischer Theorem

189

(3)

13G. More Convergence

192

(3)

13H. Convolution

195

(2)

14 Applications of Fourier Series

197

(21)

14A. The Gibbs Phenomenon

197

(2)

14B. A Continuous, Nowhere Differentiable Function

199

(1)

14C. The Isoperimetric Inequality

200

(2)

14D. Weyl's Equidistribution Theorem

202

(1)

14E. Strings

203

(4)

14F. Woodwinds

207

(2)

14G. Signals and the Fast Fourier Transform

209

(2)

14H. The Fourier Integral

211

(4)

14I. Position, Momentum, and the Uncertainty Principle

215

(3)

15 Ordinary Differential Equations

218

(19)

15A. Introduction

218

(1)

15B. Homogeneous Linear Equations

219

(4)

15C. Constant Coefficient First-Order Systems

223

(4)

15D. Nonuniqueness and Existence

227

(3)

15E. Existence and Uniqueness

230

(4)

15F. Linear Equations and Systems, Revisited

234

(3)

Appendix: The Banach-Tarski Paradox

237

(4)

Hints for Some Exercises

241

(14)

Notation Index

255

(2)

General Index

257




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