Analysis

An Introduction

By: Richard Beals

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Paperback | 27 January 2005

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272 Pages

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24.61 x 18.9 x 1.45

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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.

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'The self-contained text, suitable for advanced undergraduates, provides an extensive introduction into mathematical analysis, from the fundamentals to more advanced material.' Zentralblatt fur Didaktik der Mathematik

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Preface	p. ix
Introduction	p. 1
Notation and Motivation	p. 1
The Algebra of Various Number Systems	p. 5
The Line and Cuts	p. 9
Proofs, Generalizations, Abstractions, and Purposes	p. 12
The Real and Complex Numbers	p. 15
The Real Numbers	p. 15
Decimal and Other Expansions; Countability	p. 21
Algebraic and Transcendental Numbers	p. 24
The Complex Numbers	p. 26
Real and Complex Sequences	p. 30
Boundedness and Convergence	p. 30
Upper and Lower Limits	p. 33
The Cauchy Criterion	p. 35
Algebraic Properties of Limits	p. 37
Subsequences	p. 39
The Extended Reals and Convergence to [plus or minus infinity]	p. 40
Sizes of Things: The Logarithm	p. 42
Additional Exercises for Chapter 3	p. 43
Series	p. 45
Convergence and Absolute Convergence	p. 45
Tests for (Absolute) Convergence	p. 48
Conditional Convergence	p. 54
Euler's Constant and Summation	p. 57
Conditional Convergence: Summation by Parts	p. 58
Additional Exercises for Chapter 4	p. 59
Power Series	p. 61
Power Series, Radius of Convergence	p. 61
Differentiation of Power Series	p. 63
Products and the Exponential Function	p. 66
Abel's Theorem and Summation	p. 70
Metric Spaces	p. 73
Metrics	p. 73
Interior Points, Limit Points, Open and Closed Sets	p. 75
Coverings and Compactness	p. 79
Sequences, Completeness, Sequential Compactness	p. 81
The Cantor Set	p. 84
Continuous Functions	p. 86
Definitions and General Properties	p. 86
Real- and Complex-Valued Functions	p. 90
The Space C(I)	p. 91
Proof of the Weierstrass Polynomial Approximation Theorem	p. 95
Calculus	p. 99
Differential Calculus	p. 99
Inverse Functions	p. 105
Integral Calculus	p. 107
Riemann Sums	p. 112
Two Versions of Taylor's Theorem	p. 113
Additional Exercises for Chapter 8	p. 116
Some Special Functions	p. 119
The Complex Exponential Function and Related Functions	p. 119
The Fundamental Theorem of Algebra	p. 124
Infinite Products and Euler's Formula for Sine	p. 125
Lebesgue Measure on the Line	p. 131
Introduction	p. 131
Outer Measure	p. 133
Measurable Sets	p. 136
Fundamental Properties of Measurable Sets	p. 139
A Nonmeasurable Set	p. 142
Lebesgue Integration on the Line	p. 144
Measurable Functions	p. 144
Two Examples	p. 148
Integration: Simple Functions	p. 149
Integration: Measurable Functions	p. 151
Convergence Theorems	p. 155
Function Spaces	p. 158
Null Sets and the Notion of "Almost Everywhere"	p. 158
Riemann Integration and Lebesgue Integration	p. 159
The Space L[superscript 1]	p. 162
The Space L[superscript 2]	p. 166
Differentiating the Integral	p. 168
Additional Exercises for Chapter 12	p. 172
Fourier Series	p. 173
Periodic Functions and Fourier Expansions	p. 173
Fourier Coefficients of Integrable and Square-Integrable Periodic Functions	p. 176
Dirichlet's Theorem	p. 180
Fejer's Theorem	p. 184
The Weierstrass Approximation Theorem	p. 187
L[superscript 2]-Periodic Functions: The Riesz-Fischer Theorem	p. 189
More Convergence	p. 192
Convolution	p. 195
Applications of Fourier Series	p. 197
The Gibbs Phenomenon	p. 197
A Continuous, Nowhere Differentiable Function	p. 199
The Isoperimetric Inequality	p. 200
Weyl's Equidistribution Theorem	p. 202
Strings	p. 203
Woodwinds	p. 207
Signals and the Fast Fourier Transform	p. 209
The Fourier Integral	p. 211
Position, Momentum, and the Uncertainty Principle	p. 215
Ordinary Differential Equations	p. 218
Introduction	p. 218
Homogeneous Linear Equations	p. 219
Constant Coefficient First-Order Systems	p. 223
Nonuniqueness and Existence	p. 227
Existence and Uniqueness	p. 230
Linear Equations and Systems, Revisited	p. 234
The Banach-Tarski Paradox	p. 237
Hints for Some Exercises	p. 241
Notation Index	p. 255
General Index	p. 257
Table of Contents provided by Ingram. All Rights Reserved.

Analysis

An Introduction

At a Glance

Paperback

Industry Reviews

Shipping

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