Green's Functions

Preface to the First Edition 

Preface to the Second Edition 

CHAPTER 1 THE CONCEPT OF A GREEN\'S FUNCTION  

CHAPTER 2 VECTOR SPACES AND LINEAR TRANSFORMATIONS 
2.1 Vector Spaces 
2.2 Linearly Independent Vectors 
2.3 Orthonormal Vectors 
2.4 Linear Transformations 

CHAPTER 3 SYSTEMS OF FINITE DIMENSION 31 
3.1 Matrices and Linear Transformations 31 
3.2 Change of Basis 36 
3.3 Eigenvalues and Eigenvectors 38 
3.4 Symmetric Operators 51 
3.5 Bounded Operators 55 
3.6 Positive Definite Operators 59 

CHAPTER 4 CONTINUOUS FUNCTIONS 61 
4.1 Limiting Processes 61 
4.2 Continuous Functions 65 

CHAPTER 5 INTEGRAL OPERATORS 79 
5.1 The Kernel of an Integral Operator 79 
5.2 Symmetric Integral Transformations 83 
5.3 Separable Kernels 85 
5.4 Eigenvalues of a Symmetric Integral Operator 91 
5.5 Expansion Theorems for Integral Transformations 99 

CHAPTER 6 GENERALIZED FOURIER SERIES AND COMPLETE VECTOR SPACES 
6.1 Generalized Fourier Series 
6.2 Approximation Theorem 
6.3 Complete Vector Spaces

CHAPTER 7 DIFFERENTIAL OPERATORS 141 
7.1 Introduction 141 
7.2 Inverse Operators and the delta-function 141 
7.3 The Domain of a Linear Differential Operator 152 
7.4 Adjoint Differential Operators 154 
7.5 Self-Adjoint Second-Order Differential Operators 157 
7.6 Non-Homogeneous Problems and Symbolic Operators 159 
7.7 Green\'s Functions and Second-Order Differential Operators 163 
7.8 rrhe Problem of Eigenfunctions 177 
7.9 Green\'s Functions and the Adjoint Operator 181 
7.10 Spectral Representation and Green\'s Functions 182 

CHAPTER 8 INTEGRAL EQUATIONS 187 
8.1 Classification of Integral Equations 187 
8.2 Method of Successive Approximations 188 
8.3 The Fredholm Alternative 195 
8.4 Symmetric Integral Equations 206 
8.5 Equivalence of Integral and Differential Equations 210 

CHAPTER 9 GREEN\'S FUNCTIONS IN HIGHER-DIMENSIONAL SPACES 213 
9.1 Introduction 213 
9.2 Partial Differential Operators and delta-functions 215 
9.3 Green\'s Identities 224 
9.4 Fundamental Solutions 227 
9.5 Self-Adjoint Elliptic Equations (The Dirichlet Problem) 237 
9.6 Self-Adjoint Elliptic Equations (The Neumann Problem) 243 
9.7 Parabolic Equations 248 
9.8 Hyperbolic Equations 251 
9.9 Worked Examples 256 

CHAPTER 10 CALCULATION OF PARTICULAR GREEN\'S FUNCTIONS 274 
10.1 Method of Images 274 
10.2 Generalized Green\'s Function 278 
10.3 Mixed Problems 287 

CHAPTER 11 APPROXIMATE GREEN\'S FUNCTIONS 291 
11.1 Introduction 291 
11.2 Fundamental Solutions 292 
11.3 Generalized Potentials 295 
11.4 A Representation Theorem 300 
11.5 Choice of Approximate Kernal 302 

APPENDIX A SUMMARY OF THE GREEN\'S FUNCTION METHOD 304 
Al Green\'s Function Method for Ordinary Differential Equations 304 
A2 Green\'s Function Method for Partial Differential Equations 305 

APPENDIX B OPERATORS AND EXPRESSIONS 

APPENDIX C THE LEBESGUE INTEGRAL 

APPENDIX D DISTRIBUTIONS 

BIBLIOGRAPHY 

Chapter References 

INDEX