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Analysis : An Introduction - Richard Beals


An Introduction

Paperback Published: 27th January 2005
ISBN: 9780521600477
Number Of Pages: 261

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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.

Industry Reviews

'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
"Analysis: An Introduction is most appropriate for a undergraduate who has already grappled with the main ideas from real analysis, and who is looking for a succinct, well-written treatise that connects these concepts to some of their most powerful applications. Beals' book has the potential to serve this audience very well indeed." MAA Reviews, Christopher Hammond, Connecticut College

Prefacep. ix
Introductionp. 1
Notation and Motivationp. 1
The Algebra of Various Number Systemsp. 5
The Line and Cutsp. 9
Proofs, Generalizations, Abstractions, and Purposesp. 12
The Real and Complex Numbersp. 15
The Real Numbersp. 15
Decimal and Other Expansions; Countabilityp. 21
Algebraic and Transcendental Numbersp. 24
The Complex Numbersp. 26
Real and Complex Sequencesp. 30
Boundedness and Convergencep. 30
Upper and Lower Limitsp. 33
The Cauchy Criterionp. 35
Algebraic Properties of Limitsp. 37
Subsequencesp. 39
The Extended Reals and Convergence to [plus or minus infinity]p. 40
Sizes of Things: The Logarithmp. 42
Additional Exercises for Chapter 3p. 43
Seriesp. 45
Convergence and Absolute Convergencep. 45
Tests for (Absolute) Convergencep. 48
Conditional Convergencep. 54
Euler's Constant and Summationp. 57
Conditional Convergence: Summation by Partsp. 58
Additional Exercises for Chapter 4p. 59
Power Seriesp. 61
Power Series, Radius of Convergencep. 61
Differentiation of Power Seriesp. 63
Products and the Exponential Functionp. 66
Abel's Theorem and Summationp. 70
Metric Spacesp. 73
Metricsp. 73
Interior Points, Limit Points, Open and Closed Setsp. 75
Coverings and Compactnessp. 79
Sequences, Completeness, Sequential Compactnessp. 81
The Cantor Setp. 84
Continuous Functionsp. 86
Definitions and General Propertiesp. 86
Real- and Complex-Valued Functionsp. 90
The Space C(I)p. 91
Proof of the Weierstrass Polynomial Approximation Theoremp. 95
Calculusp. 99
Differential Calculusp. 99
Inverse Functionsp. 105
Integral Calculusp. 107
Riemann Sumsp. 112
Two Versions of Taylor's Theoremp. 113
Additional Exercises for Chapter 8p. 116
Some Special Functionsp. 119
The Complex Exponential Function and Related Functionsp. 119
The Fundamental Theorem of Algebrap. 124
Infinite Products and Euler's Formula for Sinep. 125
Lebesgue Measure on the Linep. 131
Introductionp. 131
Outer Measurep. 133
Measurable Setsp. 136
Fundamental Properties of Measurable Setsp. 139
A Nonmeasurable Setp. 142
Lebesgue Integration on the Linep. 144
Measurable Functionsp. 144
Two Examplesp. 148
Integration: Simple Functionsp. 149
Integration: Measurable Functionsp. 151
Convergence Theoremsp. 155
Function Spacesp. 158
Null Sets and the Notion of "Almost Everywhere"p. 158
Riemann Integration and Lebesgue Integrationp. 159
The Space L[superscript 1]p. 162
The Space L[superscript 2]p. 166
Differentiating the Integralp. 168
Additional Exercises for Chapter 12p. 172
Fourier Seriesp. 173
Periodic Functions and Fourier Expansionsp. 173
Fourier Coefficients of Integrable and Square-Integrable Periodic Functionsp. 176
Dirichlet's Theoremp. 180
Fejer's Theoremp. 184
The Weierstrass Approximation Theoremp. 187
L[superscript 2]-Periodic Functions: The Riesz-Fischer Theoremp. 189
More Convergencep. 192
Convolutionp. 195
Applications of Fourier Seriesp. 197
The Gibbs Phenomenonp. 197
A Continuous, Nowhere Differentiable Functionp. 199
The Isoperimetric Inequalityp. 200
Weyl's Equidistribution Theoremp. 202
Stringsp. 203
Woodwindsp. 207
Signals and the Fast Fourier Transformp. 209
The Fourier Integralp. 211
Position, Momentum, and the Uncertainty Principlep. 215
Ordinary Differential Equationsp. 218
Introductionp. 218
Homogeneous Linear Equationsp. 219
Constant Coefficient First-Order Systemsp. 223
Nonuniqueness and Existencep. 227
Existence and Uniquenessp. 230
Linear Equations and Systems, Revisitedp. 234
The Banach-Tarski Paradoxp. 237
Hints for Some Exercisesp. 241
Notation Indexp. 255
General Indexp. 257
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521600477
ISBN-10: 0521600472
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 261
Published: 27th January 2005
Country of Publication: GB
Dimensions (cm): 25.2 x 17.73  x 1.68
Weight (kg): 0.49