| 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 |
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