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| Preface to the Third Edition | |
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| Preface to the Second Edition | |
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| Preface to the First Edition | |
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| The Dirac Delta Function and Delta Sequences | |
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| The Heaviside Function | |
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| The Dirac Delta Function | |
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| The Delta Sequences | |
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| A Unit Dipole | |
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| The Heaviside Sequences | |
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| Exercises | |
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| The Schwartz-Sobolev Theory of Distributions | |
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| Some Introductory Definitions | |
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| Test Functions | |
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| Linear Functionals and the Schwartz Sobolev Theory of Distributions | |
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| Examples | |
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| Algebraic Operations on Distributions | |
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| Analytic Operations on Distributions | |
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| Examples | |
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| The Support and Singular Support of a Distribution Exercises | |
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| Additional Properties of Distributions | |
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| Transformation Properties of the Delta Distributions | |
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| Convergence of Distributions | |
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| Delta Sequences with Parametric Dependence | |
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| Fourier Series | |
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| Examples | |
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| The Delta Function as a Stieltjes Integral Exercises | |
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| Distributions Defined by Divergent Integrals | |
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| Introduction | |
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| The Pseudofunction H(x)/x n , n = 1, 2,3 | |
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| Functions with Algebraic Singularity of Order m | |
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| Examples | |
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| Exercises | |
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| Distributional Derivatives of Functions with Jump Discontinuities | |
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| Distributional Derivatives in R 1 | |
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| Moving Surfaces of Discontinuity in R n , n 2 | |
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| Surface Distributions | |
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| Various Other Representations | |
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| First-Order Distributional Derivatives | |
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| Second Order Distributional Derivatives | |
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| Higher-Order Distributional Derivatives | |
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| The Two-Dimensional Case | |
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| Examples | |
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| The Function Pf ( l/r ) and its Derivatives | |
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| Tempered Distributions and the Fourier Transforms | |
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| Preliminary Concepts | |
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| Distributions of Slow Growth (Tempered Distributions) | |
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| The Fourier Transform | |
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| Examples | |
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| Exercises | |
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| Direct Products and Convolutions of Distributions | |
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| Definition of the Direct Product | |
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| The Direct Product of Tempered Distributions | |
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| The Fourier Transform of the Direct Product of Tempered Distributions | |
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| The Convolution | |
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| The Role of Convolution in the Regularization of the Distributions | |
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| The Dual Spaces E and E'' | |
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| Examples | |
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| The Fourier Transform of the Convolution | |
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| Distributional Solutions of Integral Equations | |
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| Exercises | |
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| The Laplace Transform | |
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| A Brief Discussion of the Classical Results | |
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| The Laplace Transform of the Distributions | |
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| The Laplace Transform of the Distributional Derivatives and Vice Versa | |
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| Examples | |
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| Exercises | |
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| Applications to Ordinary Differential Equations | |
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| Ordinary Differential Operators | |
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| Homogeneous Differential Equations | |
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| Inhomogeneous Differentational Equations: The Integral of a Distribution | |
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| Examples | |
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| Fundamental Solutions and Green''s Functions | |
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| Second Order Differential Equations with Constant Coefficients | |
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| Eigenvalue Problems | |
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| Second Order Differential Equations with Variable Coefficients | |
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| Fourth Order Differential Equations | |
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| Differential Equations of n th Order | |
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| Ordinary Differential Equations with Singular Coefficients | |
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| Exercises | |
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| Applications to Partial Differential Equations | |
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| Introduction | |
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| Classical and Generalized Solutions | |
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| Fundamental Solutions | |
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| The Cauchy Riemann Operator | |
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| The Transport Operator | |
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| The Laplace Operator | |
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| The Heat Operator | |
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| The Schroedinger Operator | |
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| The Helmholtz Operator | |
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| The Wave Operator | |
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| The Inhomogeneous Wave Equation | |
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| The Klein Gordon Operator | |
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| Exercises | |
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| Applications to Boundary Value Problems | |
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| Poisson''s Equation | |
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| Dumbbell-Shaped Bodies | |
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| Uniform Axial Distributions | |
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| Linear Axial Distributions | |
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| Parabolic Axial Distributions | |
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| The Four-Order Polynomial Distribution, n = 7 | |
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| Spheroidal Cavities | |
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| The Polarization Tensor for a Spheroid | |
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| The Virtual Mass Tensor for a Sp | |