| Preface to the First Edition | p. xi |
| Preface to the Second Edition | p. xiv |
| The Concept of a Green's Function | p. 1 |
| Vector Spaces and Linear Transformations | p. 9 |
| Vector Spaces | p. 9 |
| Linearly Independent Vectors | p. 16 |
| Orthonormal Vectors | p. 20 |
| Linear Transformations | p. 24 |
| Systems of Finite Dimension | p. 31 |
| Matrices and Linear Transformations | p. 31 |
| Change of Basis | p. 36 |
| Eigenvalues and Eigenvectors | p. 38 |
| Symmetric Operators | p. 51 |
| Bounded Operators | p. 55 |
| Positive Definite Operators | p. 59 |
| Continuous Functions | p. 61 |
| Limiting Processes | p. 61 |
| Continuous Functions | p. 65 |
| Integral Operators | p. 79 |
| The Kernel of an Integral Operator | p. 79 |
| Symmetric Integral Transformations | p. 83 |
| Separable Kernels | p. 85 |
| Eigenvalues of a Symmetric Integral Operator | p. 91 |
| Expansion Theorems for Integral Transformations | p. 99 |
| Generalized Fourier Series and Complete Vector Spaces | p. 112 |
| Generalized Fourier Series | p. 112 |
| Approximation Theorem | p. 121 |
| Complete Vector Spaces | p. 127 |
| Differential Operators | p. 141 |
| Introduction | p. 141 |
| Inverse Operators and the [delta]-function | p. 141 |
| The Domain of a Linear Differential Operator | p. 152 |
| Adjoint Differential Operators | p. 154 |
| Self-Adjoint Second-Order Differential Operators | p. 157 |
| Non-Homogeneous Problems and Symbolic Operators | p. 159 |
| Green's Functions and Second-Order Differential Operators | p. 163 |
| The Problem of Eigenfunctions | p. 177 |
| Green's Functions and the Adjoint Operator | p. 181 |
| Spectral Representation and Green's Functions | p. 182 |
| Integral Equations | p. 187 |
| Classification of Integral Equations | p. 187 |
| Method of Successive Approximations | p. 188 |
| The Fredholm Alternative | p. 195 |
| Symmetric Integral Equations | p. 206 |
| Equivalence of Integral and Differential Equations | p. 210 |
| Green's Functions in Higher-Dimensional Spaces | p. 213 |
| Introduction | p. 213 |
| Partial Differential Operators and [delta]-functions | p. 215 |
| Green's Identities | p. 224 |
| Fundamental Solutions | p. 227 |
| Self-Adjoint Elliptic Equations (The Dirichlet Problem) | p. 237 |
| Self-Adjoint Elliptic Equations (The Neumann Problem) | p. 243 |
| Parabolic Equations | p. 248 |
| Hyperbolic Equations | p. 251 |
| Worked Examples | p. 256 |
| Calculation of Particular Green's Functions | p. 274 |
| Method of Images | p. 274 |
| Generalized Green's Functions | p. 278 |
| Mixed Problems | p. 287 |
| Approximate Green's Functions | p. 291 |
| Introduction | p. 291 |
| Fundamental Solutions | p. 292 |
| Generalized Potentials | p. 295 |
| A Representation Theorem | p. 300 |
| Choice of Approximate Kernal | p. 302 |
| Summary of the Green's Function Method | p. 304 |
| Green's Function Method for Ordinary Differential Equations | p. 304 |
| Green's Function Method for Partial Differential Equations | p. 305 |
| Operators and Expressions | p. 307 |
| The Lebesgue Integral | p. 312 |
| Distributions | p. 316 |
| Bibliography | p. 319 |
| Chapter References | p. 321 |
| Index | p. 323 |
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