Analysis: An Introduction
Beals (Richard)
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Postscript Product 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 rigorous treatments 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. The book provides a large number of examples and nearly 500 exercises.

Contents

  1. Introduction - 1
  2. The Real and Complex Numbers - 15
  3. Real and Complex Sequences - 30
  4. Series - 45
  5. Power Series - 61
  6. Metric Spaces - 73
  7. Continuous Functions - 86
  8. Calculus - 99
  9. Some Special Functions - 119
  10. Lebesgue Measure on the Line - 131
  11. Lebesgue Integration on the Line - 144
  12. Function Spaces - 158
  13. Fourier Series - 173
  14. Applications of Fourier Series - 197
  15. Ordinary Differential Equations - 218


"Beals (Richard) - Analysis: An Introduction"

Source: Beals (Richard) - Analysis: An Introduction


Contents
  • Preface - ix
  • 1. Introduction - 1
    … 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 ±infinity - 40
    … 3G. Sizes of Things: The Logarithm - 42
    … Additional Exercises for Chapter 3 - 43
  • 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
    … Additional Exercises for Chapter 4 - 59
  • 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
    … Additional Exercises for Chapter 8 - 116
  • 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
    … 10A. Introduction - 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
    … 11B*. Two Examples - 148
    … 11C. Integration: Simple Functions - 149
    … 11D. Integration: Measurable Functions - 151
    … 11E. Convergence Theorems - 155
  • 12. Function Spaces
    … 12A. Null Sets and the Notion of "Almost Everywhere"
    … 12B*. Riemann Integration and Lebesgue Integration
    … 12C. The Space L1
    … 12D. The Space L2
    … 12E*. Differentiating the Integral
    … Additional Exercises for Chapter 12
  • 13. Fourier Series
    … 13A. Periodic Functions and Fourier Expansions
    … 13B. Fourier Coefficients of Integrable and Square-Integrable Periodic Functions
    … 13C. Dirichlet's Theorem
    … 13D. Fejer's Theorem
    … 13E. The Weierstrass Approximation Theorem
    … 13F. L2-Periodic Functions: The Riesz-Fischer Theorem
    … 13G. More Convergence
    … 13H*. Convolution
  • 14*. Applications of Fourier Series
    … 14A*. The Gibbs Phenomenon
    … 14B*. A Continuous, Nowhere Differentiable Function
    … 14C*. The Isoperimetric Inequality
    … 14D*. Weyl's Equidistribution Theorem
    … 14E*. Strings
    … 14F*. Woodwinds
    … 14G*. Signals and the Fast Fourier Transform
    … 14H*. The Fourier Integral
    … 14I*. Position, Momentum, and the Uncertainty Principle
  • 15. Ordinary Differential Equations
    … 15A. Introduction
    … 15B. Homogeneous Linear Equations
    … 15C. Constant Coefficient First-Order Systems
    … 15D. Nonuniqueness and Existence
    … 15E. Existence and Uniqueness
    … 15F. Linear Equations and Systems, Revisited
  • Appendix: The Banach-Tarski Paradox
  • Hints for Some Exercises
  • Notation Index
  • General Index
.


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