Analysis : an introduction / Richard Beals.

Format:

Book

Author/Creator:

Beals, Richard, 1938-

Contributor:

Class of 1953 Fund.

Language:

English

Subjects (All):

Mathematical analysis.

Physical Description:

ix, 261 pages : illustrations ; 27 cm

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2004.

Summary:

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 of 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 nearly 500 exercises allow the reader to test understanding and practice mathematical exposition, and they provide a window into further topics.

Contents:

1A. Notation and Motivation 1

1B. The Algebra of Various Number Systems 5

1C. The Line and Cuts 9

1D. Proofs, Generalizations, Abstractions, and Purposes 12

2 The Real and Complex Numbers 15

2A. The Real Numbers 15

2B. Decimal and Other Expansions; Countability 21

2C. Algebraic and Transcendental Numbers 24

2D. The Complex Numbers 26

3 Real and Complex Sequences 30

3A. Boundedness and Convergence 30

3B. Upper and Lower Limits 33

3C. The Cauchy Criterion 35

3D. Algebraic Properties of Limits 37

3E. Subsequences 39

3F. The Extended Reals and Convergence to [plus or minus infinity] 40

3G. Sizes of Things: The Logarithm 42

4 Series 45

4A. Convergence and Absolute Convergence 45

4B. Tests for (Absolute) Convergence 48

4C. Conditional Convergence 54

4D. Euler's Constant and Summation 57

4E. Conditional Convergence: Summation by Parts 58

5 Power Series 61

5A. Power Series, Radius of Convergence 61

5B. Differentiation of Power Series 63

5C. Products and the Exponential Function 66

5D. Abel's Theorem and Summation 70

6 Metric Spaces 73

6A. Metrics 73

6B. Interior Points, Limit Points, Open and Closed Sets 75

6C. Coverings and Compactness 79

6D. Sequences, Completeness, Sequential Compactness 81

6E. The Cantor Set 84

7 Continuous Functions 86

7A. Definitions and General Properties 86

7B. Real- and Complex-Valued Functions 90

7C. The Space C(I) 91

7D. Proof of the Weierstrass Polynomial Approximation Theorem 95

8 Calculus 99

8A. Differential Calculus 99

8B. Inverse Functions 105

8C. Integral Calculus 107

8D. Riemann Sums 112

8E. Two Versions of Taylor's Theorem 113

9 Some Special Functions 119

9A. The Complex Exponential Function and Related Functions 119

9B. The Fundamental Theorem of Algebra 124

9C. Infinite Products and Euler's Formula for Sine 125

10 Lebesgue Measure on the Line 131

10B. Outer Measure 133

10C. Measurable Sets 136

10D. Fundamental Properties of Measurable Sets 139

10E. A Nonmeasurable Set 142

11 Lebesgue Integration on the Line 144

11A. Measurable Functions 144

11C. Integration: Simple Functions 149

11D. Integration: Measurable Functions 151

11E. Convergence Theorems 155

12 Function Spaces 158

12A. Null Sets and the Notion of "Almost Everywhere" 158

12B. Riemann Integration and Lebesgue Integration 159

12C. The Space L[superscript 1] 162

12D. The Space L[superscript 2] 166

12E. Differentiating the Integral 168

13 Fourier Series 173

13A. Periodic Functions and Fourier Expansions 173

13B. Fourier Coefficients of Integrable and Square-Integrable Periodic Functions 176

13C. Dirichlet's Theorem 180

13D. Fejer's Theorem 184

13E. The Weierstrass Approximation Theorem 187

13F. L[superscript 2]-Periodic Functions: The Riesz-Fischer Theorem 189

13G. More Convergence 192

13H. Convolution 195

14 Applications of Fourier Series 197

14A. The Gibbs Phenomenon 197

14B. A Continuous, Nowhere Differentiable Function 199

14C. The Isoperimetric Inequality 200

14D. Weyl's Equidistribution Theorem 202

14E. Strings 203

14F. Woodwinds 207

14G. Signals and the Fast Fourier Transform 209

14H. The Fourier Integral 211

14I. Position, Momentum, and the Uncertainty Principle 215

15 Ordinary Differential Equations 218

15B. Homogeneous Linear Equations 219

15C. Constant Coefficient First-Order Systems 223

15D. Nonuniqueness and Existence 227

15E. Existence and Uniqueness 230

15F. Linear Equations and Systems, Revisited 234

Appendix The Banach-Tarski Paradox 237.

Notes:

Includes indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1953 Fund.

ISBN:

0521840724

0521600472

OCLC:

54007211

Online:

Publisher description