Fundamental Ideas of Analysis

The ideas and methods of mathematics, long central to the physical sciences, now play an increasingly important role in a wide variety of disciplines. Analysis provides theorems that prove that results are true and provides techniques to estimate the errors in approximate calculations. The ideas and methods of analysis play a fundamental role in ordinary differential equations, probability theory, differential geometry, numerical analysis, complex analysis, partial differential equations, as well as in most areas of applied mathematics.

Michael C. Reed is a practicing anthropologist who has worked since 1992 for CSR Incorporated, a Washington, DC, research and evaluation firm. He received his PhD in sociocultural anthropology in 1988, following 15 months of Fulbright fellowship-funded fieldwork in francophone Africa. His dissertation was titled An Ethnohistorical Study of the Political Economy of Ndjole', Gabon. Since 1990, Reed has evaluated several U.S. programs, such as youth substance abuse prevention, school-age child care, secondary education, and community development corporations. He reviews articles for the American Evaluation Association journal Evaluation Practice.

Preface

CHAPTER 1 Preliminaries

1

(26)

1.1 The Real Numbers

1

(5)

1.2 Sets and Functions

6

(9)

1.3 Cardinality

15

(5)

1.4 Methods of Proof

20

(7)

CHAPTER 2 Sequences

27

(46)

2.1 Convergence

27

(8)

2.2 Limit Theorems

35

(5)

2.3 Two-state Markov Chains*

40

(4)

2.4 Cauchy Sequences

44

(8)

2.5 Supremum and Infimum

52

(3)

2.6 The Bolzano-Weierstrass Theorem

55

(5)

2.7 The Quadratic Map*

60

(8)

Projects

68

(5)

CHAPTER 3 The Riemann Integral

73

(48)

3.1 Continuity

73

(7)

3.2 Continuous Functions on Closed Intervals

80

(7)

3.3 The Riemann Integral

87

(8)

3.4 Numerical Methods*

95

(8)

3.5 Discontinuities

103

(10)

3.6 Improper Integrals

113

(6)

Projects

119

(2)

CHAPTER 4 Differentiation

121

(42)

4.1 Differentiable Functions

121

(8)

4.2 The Fundamental Theorem of Calculus

129

(5)

4.3 Taylor's Theorem

134

(6)

4.4 Newton's Method*

140

(7)

4.5 Inverse Functions

147

(4)

4.6 Functions of Two Variables*

151

(8)

Projects

159

(4)

CHAPTER 5 Sequences of Functions

163

(60)

5.1 Pointwise and Uniform Convergence

163

(6)

5.2 Limit Theorems

169

(6)

5.3 The Supremum Norm

175

(8)

5.4 Integral Equations*

183

(5)

5.5 The Calculus of Variations*

188

(8)

5.6 Metric Spaces

196

(7)

5.7 The Contraction Mapping Principle

203

(7)

5.8 Normed Linear Spaces*

210

(9)

Projects

219

(4)

CHAPTER 6 Series of Functions

223

(50)

6.1 Lim sup and Lim inf

223

(5)

6.2 Series of Real Constants

228

(10)

6.3 The Weierstrass M-test

238

(7)

6.4 Power Series

245

(7)

6.5 Complex Numbers*

252

(8)

6.6 Infinite Products and Prime Numbers*

260

(10)

Projects

270

(3)

CHAPTER 7 Differential Equations*

273

(26)

7.1 Local Existence

273

(10)

7.2 Global Existence

283

(6)

7.3 The Error Estimate for Euler's Method

289

(7)

Projects

296

(3)

CHAPTER 8 Complex Analysis*

299

(24)

8.1 Analytic Functions

299

(6)

8.2 Integration on Paths

305

(7)

8.3 Cauchy's Theorem

312

(8)

Projects

320

(3)

CHAPTER 9 Fourier Series*

323

(36)

9.1 The Heat Equation

323

(8)

9.2 Definitions and Examples

331

(6)

9.3 Pointwise Convergence

337

(8)

9.4 Mean-square Convergence

345

(10)

Projects

355

(4)

CHAPTER 10 Probability Theory*

359

(44)

10.1 Discrete Random Variables

359

(9)

10.2 Coding Theory

368

(8)

10.3 Continuous Random Variables

376

(10)

10.4 The Variation Metric

386

(12)

Projects

398

(5)

Bibliography

403

(3)

Symbol Index

406

(3)

Index

409

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Fundamental Ideas of Analysis

Summary

Author Biography

Table of Contents

Supplemental Materials