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Analysis An Introduction

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ISBN-10: 0521600472

ISBN-13: 9780521600477

Edition: 2004

Authors: Richard Beals

List price: $86.99
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Description:

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…    
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Book details

List price: $86.99
Copyright year: 2004
Publisher: Cambridge University Press
Publication date: 9/13/2004
Binding: Paperback
Pages: 272
Size: 6.97" wide x 9.92" long x 0.67" tall
Weight: 1.254
Language: English

Preface
Introduction
Notation and Motivation
The Algebra of Various Number Systems
The Line and Cuts
Proofs, Generalizations, Abstractions, and Purposes
The Real and Complex Numbers
The Real Numbers
Decimal and Other Expansions; Countability
Algebraic and Transcendental Numbers
The Complex Numbers
Real and Complex Sequences
Boundedness and Convergence
Upper and Lower Limits
The Cauchy Criterion
Algebraic Properties of Limits
Subsequences
The Extended Reals and Convergence to [plus or minus infinity]
Sizes of Things: The Logarithm
Additional Exercises for Chapter 3
Series
Convergence and Absolute Convergence
Tests for (Absolute) Convergence
Conditional Convergence
Euler's Constant and Summation
Conditional Convergence: Summation by Parts
Additional Exercises for Chapter 4
Power Series
Power Series, Radius of Convergence
Differentiation of Power Series
Products and the Exponential Function
Abel's Theorem and Summation
Metric Spaces
Metrics
Interior Points, Limit Points, Open and Closed Sets
Coverings and Compactness
Sequences, Completeness, Sequential Compactness
The Cantor Set
Continuous Functions
Definitions and General Properties
Real- and Complex-Valued Functions
The Space C(I)
Proof of the Weierstrass Polynomial Approximation Theorem
Calculus
Differential Calculus
Inverse Functions
Integral Calculus
Riemann Sums
Two Versions of Taylor's Theorem
Additional Exercises for Chapter 8
Some Special Functions
The Complex Exponential Function and Related Functions
The Fundamental Theorem of Algebra
Infinite Products and Euler's Formula for Sine
Lebesgue Measure on the Line
Introduction
Outer Measure
Measurable Sets
Fundamental Properties of Measurable Sets
A Nonmeasurable Set
Lebesgue Integration on the Line
Measurable Functions
Two Examples
Integration: Simple Functions
Integration: Measurable Functions
Convergence Theorems
Function Spaces
Null Sets and the Notion of "Almost Everywhere"
Riemann Integration and Lebesgue Integration
The Space L[superscript 1]
The Space L[superscript 2]
Differentiating the Integral
Additional Exercises for Chapter 12
Fourier Series
Periodic Functions and Fourier Expansions
Fourier Coefficients of Integrable and Square-Integrable Periodic Functions
Dirichlet's Theorem
Fejer's Theorem
The Weierstrass Approximation Theorem
L[superscript 2]-Periodic Functions: The Riesz-Fischer Theorem
More Convergence
Convolution
Applications of Fourier Series
The Gibbs Phenomenon
A Continuous, Nowhere Differentiable Function
The Isoperimetric Inequality
Weyl's Equidistribution Theorem
Strings
Woodwinds
Signals and the Fast Fourier Transform
The Fourier Integral
Position, Momentum, and the Uncertainty Principle
Ordinary Differential Equations
Introduction
Homogeneous Linear Equations
Constant Coefficient First-Order Systems
Nonuniqueness and Existence
Existence and Uniqueness
Linear Equations and Systems, Revisited
The Banach-Tarski Paradox
Hints for Some Exercises
Notation Index
General Index