Unified Integration
Published on 1. December 1983
606 pages
978-0-08-087426-5 (ISBN)
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Unified Integration
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E.J. McShane
Unified Integration
Book
01/1983
Academic Press
€121.95
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Content
- Front Cover
- Unified Integration
- Copyright Page
- Contents
- Preface
- Acknowledgments
- Chapter 0. Introduction
- 1. Functions, Intervals, and Limits
- 2. Bounds
- Chapter I. Elementary Properties of the Integral in One-Dimensional Space
- 1. A Heuristic Approach to the Definition of the Integral
- 2. Definition of the Integral
- 3. Examples
- 4. Existence of ?-Fine Partitions
- 5. Elementary Computational Formulas
- 6. Order Properties of the Integral
- 7. Comparison with the Riemann Integral
- 8. Additivity of the Integral
- 9. The Fundamental Theorem of the Calculus, First Part
- 10. The Fundamental Theorem of the Calculus, Second Part
- 11. Substitution and Integration by Parts
- 12. Estimates of Integrals
- Chapter II. Integration in One-Dimensional Space: Further Development
- 1. A Condition for Integrability
- 2. Absolute Integrability
- 3. Integration of Composite Functions
- 4. The Monotone Convergence Theorem
- 5. Integrals of Products
- 6. Power Series
- 7. "Improper" Integrals
- 8. Examples
- 9. Continuity of the Indefinite Integral: Integration by Parts and Substitution
- 10. The Dominated Convergence Theorem
- 11. Differentiation under the Integration Sign
- 12. Sets of Measure 0
- 13. Approximation by Step-Functions
- 14. Differentiation of Indefinite Integrals
- 15. Calculus of Variations
- Chapter III. Applications to Differential Equations and to Probability Theory
- 1. Ordinary Differential Equations
- 2. Existence Theorems for Solutions of Differential Equations
- 3. Effects of Change of Data
- 4. Computation of Approximate Solutions
- 5. Linear Differential Equations
- 6. Differentiation of Solutions with Respect to Initial Values and Parameters
- 7. Definition of a More General Integral
- 8. Properties of the Integral
- 9. Densities
- 10. Measurable Functions and Measurable Sets
- 11. Applications to Probability Theory
- 12. Examples
- Chapter IV. Integration in Spaces of More Than One Dimension
- 1. Notation and Definitions
- 2. Elementary Properties of Intervals and Measure
- 3. Generalizations to Integration in Rr of Theorems in Preceding Chapters
- 4. Iterated Integration
- 5. Change of Variables in Multiple Integrals
- 6. Approximation of Sets by Unions of Intervals and of Integrable Functions by Limits of Step-Functions
- 7. A Second Form of Fubini's Theorem
- 8. Second Form of the Substitution Theorem
- 9. Integration with Respect to Other Measures
- 10. Applications to Probability Theory: Multivariate Distributions
- 11. Independence
- 12. Convolutions
- 13. The Central Limit Theorem
- 14. Distributions in Some Infinite-Dimensional Spaces
- Chapter V. Line Integrals and Areas of Surfaces
- 1. Geometry in r-Dimensional Space
- 2. Vectors
- 3. Inner Products and Length-Preserving Maps
- 4. Covectors
- 5. Differentiation and Integration of Vector-Valued Functions
- 6. Curves and Their Lengths
- 7. Line Integrals
- 8. The Behavior of Inscribed Polyhedra
- 9. Areas of Surfaces
- Chapter VI. Vector Spaces, Orthogonal Expansions, and Fourier Transforms
- 1. Complex Vector Spaces
- 2. The Spaces L1 and L2
- 3. Normed Vector Spaces
- 4. Completeness of Spaces L1, L2, L1, and L2
- 5. Hilbert Spaces and Their Geometry
- 6. Approximation by Step-Functions and by Differentiable Functions
- 7. Fourier Series
- 8. Indefinite Integrals and the Weierstrass Approximation Theorem
- 9. Legendre Polynomials
- 10. The Hermite Polynomials and the Hermite Functions
- 11. The Schrödinger Equation for the Harmonic Oscillator
- 12. The Fourier Transform for Certain Smooth Functions
- 13. The Fourier-Plancherel Transform
- 14. The Fourier Transformation and the Fourier-Plancherel Transformation
- 15. Applications to Differential Equations
- Chapter VII. Measure Theory
- 1. s-Algebras and Measurable Functions
- 2. Definition of the Lebesgue Integral
- 3. Bore1 Sets and Borel-Measurable Functions
- 4. Integration with Respect to Other Functions of Sets
- 5. The Radon-Nikodým Theorem
- 6. Conditional Expectations
- 7. Brownian Motion
- Index
- Pure and Applied Mathematics
