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Analysis: An Introduction / Edition 1
     

Analysis: An Introduction / Edition 1

by Richard Beals
 

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ISBN-10: 0521600472

ISBN-13: 9780521600477

Pub. Date: 09/13/2004

Publisher: Cambridge University Press

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

Overview

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.

Product Details

ISBN-13:
9780521600477
Publisher:
Cambridge University Press
Publication date:
09/13/2004
Edition description:
New Edition
Pages:
272
Product dimensions:
7.40(w) x 9.60(h) x 0.70(d)

Table of Contents

Prefaceix
1Introduction1
1A.Notation and Motivation1
1B.The Algebra of Various Number Systems5
1C.The Line and Cuts9
1D.Proofs, Generalizations, Abstractions, and Purposes12
2The Real and Complex Numbers15
2A.The Real Numbers15
2B.Decimal and Other Expansions; Countability21
2C.Algebraic and Transcendental Numbers24
2D.The Complex Numbers26
3Real and Complex Sequences30
3A.Boundedness and Convergence30
3B.Upper and Lower Limits33
3C.The Cauchy Criterion35
3D.Algebraic Properties of Limits37
3E.Subsequences39
3F.The Extended Reals and Convergence to [plus or minus infinity]40
3G.Sizes of Things: The Logarithm42
Additional Exercises for Chapter 343
4Series45
4A.Convergence and Absolute Convergence45
4B.Tests for (Absolute) Convergence48
4C.Conditional Convergence54
4D.Euler's Constant and Summation57
4E.Conditional Convergence: Summation by Parts58
Additional Exercises for Chapter 459
5Power Series61
5A.Power Series, Radius of Convergence61
5B.Differentiation of Power Series63
5C.Products and the Exponential Function66
5D.Abel's Theorem and Summation70
6Metric Spaces73
6A.Metrics73
6B.Interior Points, Limit Points, Open and Closed Sets75
6C.Coverings and Compactness79
6D.Sequences, Completeness, Sequential Compactness81
6E.The Cantor Set84
7Continuous Functions86
7A.Definitions and General Properties86
7B.Real- and Complex-Valued Functions90
7C.The Space C(I)91
7D.Proof of the Weierstrass Polynomial Approximation Theorem95
8Calculus99
8A.Differential Calculus99
8B.Inverse Functions105
8C.Integral Calculus107
8D.Riemann Sums112
8E.Two Versions of Taylor's Theorem113
Additional Exercises for Chapter 8116
9Some Special Functions119
9A.The Complex Exponential Function and Related Functions119
9B.The Fundamental Theorem of Algebra124
9C.Infinite Products and Euler's Formula for Sine125
10Lebesgue Measure on the Line131
10A.Introduction131
10B.Outer Measure133
10C.Measurable Sets136
10D.Fundamental Properties of Measurable Sets139
10E.A Nonmeasurable Set142
11Lebesgue Integration on the Line144
11A.Measurable Functions144
11B.Two Examples148
11C.Integration: Simple Functions149
11D.Integration: Measurable Functions151
11E.Convergence Theorems155
12Function Spaces158
12A.Null Sets and the Notion of "Almost Everywhere"158
12B.Riemann Integration and Lebesgue Integration159
12C.The Space L[superscript 1]162
12D.The Space L[superscript 2]166
12E.Differentiating the Integral168
Additional Exercises for Chapter 12172
13Fourier Series173
13A.Periodic Functions and Fourier Expansions173
13B.Fourier Coefficients of Integrable and Square-Integrable Periodic Functions176
13C.Dirichlet's Theorem180
13D.Fejer's Theorem184
13E.The Weierstrass Approximation Theorem187
13F.L[superscript 2]-Periodic Functions: The Riesz-Fischer Theorem189
13G.More Convergence192
13H.Convolution195
14Applications of Fourier Series197
14A.The Gibbs Phenomenon197
14B.A Continuous, Nowhere Differentiable Function199
14C.The Isoperimetric Inequality200
14D.Weyl's Equidistribution Theorem202
14E.Strings203
14F.Woodwinds207
14G.Signals and the Fast Fourier Transform209
14H.The Fourier Integral211
14I.Position, Momentum, and the Uncertainty Principle215
15Ordinary Differential Equations218
15A.Introduction218
15B.Homogeneous Linear Equations219
15C.Constant Coefficient First-Order Systems223
15D.Nonuniqueness and Existence227
15E.Existence and Uniqueness230
15F.Linear Equations and Systems, Revisited234
AppendixThe Banach-Tarski Paradox237
Hints for Some Exercises241
Notation Index255
General Index257

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