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