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| Preface to First Edition | |
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| Preface to Second Edition | |
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| Finite Dimensional Vector Spaces | |
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| Linear Vector Spaces | |
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| Spectral Theory for Matrices | |
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| Geometrical Significance of Eigenvalues | |
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| Fredholm Alternative Theorem | |
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| Least Squares Solutions-Pseudo Inverses | |
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| The Problem of Procrustes | |
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| Applications of Eigenvalues and Eigenfunctions | |
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| Exponentiation of Matrices | |
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| The Power Method and Positive Matrices | |
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| Iteration Methods | |
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| Further Reading | |
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| Problems for Chapter 1 | |
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| Function Spaces | |
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| Complete Vector Spaces | |
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| Sobolev Spaces | |
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| Approximation in Hilbert Spaces | |
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| Fourier Series and Completeness | |
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| Orthogonal Polynomials | |
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| Trigonometric Series | |
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| Discrete Fourier Transforms | |
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| Sinc Functions | |
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| Wavelets | |
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| Finite Elements | |
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| Further Reading | |
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| Problems for Chapter 2 | |
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| Integral Equations | |
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| Introduction | |
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| Bounded Linear Operators in Hilbert Space | |
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| Compact Operators | |
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| Spectral Theory for Compact Operators | |
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| Resolvent and Pseudo-Resolvent Kernels | |
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| Approximate Solutions | |
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| Singular Integral Equations | |
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| Further Reading | |
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| Problems for Chapter 3 | |
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| Differential Operators | |
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| Distributions and the Delta Function | |
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| Green's Functions | |
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| Differential Operators | |
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| Domain of an Operator | |
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| Adjoint of an Operator | |
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| Inhomogeneous Boundary Data | |
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| The Fredholm Alternative | |
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| Least Squares Solutions | |
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| Eigenfunction Expansions | |
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| Trigonometric Functions | |
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| Orthogonal Polynomials | |
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| Special Functions | |
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| Discretized Operators | |
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| Further Reading | |
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| Problems for Chapter 4 | |
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| Calculus of Variations | |
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| The Euler-Lagrange Equations | |
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| Constrained Problems | |
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| Several Unknown Functions | |
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| Higher Order Derivatives | |
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| Variable Endpoints | |
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| Several Independent Variables | |
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| Hamilton's Principle | |
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| The Swinging Pendulum | |
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| The Vibrating String | |
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| The Vibrating Rod | |
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| Nonlinear Deformations of a Thin Beam | |
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| A Vibrating Membrane | |
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| Approximate Methods | |
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| Eigenvalue Problems | |
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| Optimal Design of Structures | |
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| Further Reading | |
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| Problems for Chapter 5 | |
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| Complex Variable Theory | |
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| Complex Valued Functions | |
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| The Calculus of Complex Functions | |
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| Differentiation-Analytic Functions | |
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| Integration | |
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| Cauchy Integral Formula | |
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| Taylor and Laurent Series | |
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| Fluid Flow and Conformal Mappings | |
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| Laplace's Equation | |
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| Conformal Mappings | |
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| Free Boundary Problems | |
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| Contour Integration | |
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| Special Functions | |
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| The Gamma Function | |
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| Bessel Functions | |
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| Legendre Functions | |
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| Sinc Functions | |
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| Further Reading | |
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| Problems for Chapter 6 | |
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| Transform and Spectral Theory | |
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| Spectrum of an Operator | |
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| Fourier Transforms | |
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| Transform Pairs | |
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| Completeness of Hermite and Laguerre Polynomials | |
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| Sinc Functions | |
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| Windowed Fourier Transforms | |
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| Wavelets | |
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| Related Integral Transforms | |
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| Laplace Transform | |
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| Mellin Transform | |
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| Hankel Transform | |
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| Z Transforms | |
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| Scattering Theory | |
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| Scattering Examples | |
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| Spectral Representations | |
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| Further Reading | |
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| Problems for Chapter 7 | |
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| Fourier Transform Pairs | |
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| Partial Differential Equations | |
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| Poisson's Equation | |
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| Fundamental Solutions | |
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| The Method of Images | |
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| Transform Methods | |
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| Hilbert Transforms | |
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| Boundary Integral Equations | |
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| Eigenfunctions | |
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| The Wave Equation | |
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| Derivations | |
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| Fundamental Solutions | |
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| Vibrations | |
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| Diffraction Patterns | |
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| The Heat Equation | |
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| Derivations | |
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| Fundamental Solutions | |
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| Transform Methods | |
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| Differential-Difference Equations | |
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| Transform Methods | |
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| Numerical Methods | |
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| Further Reading | |
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| Problems for Chapter 8 | |
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| Inverse Scattering Transform | |
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| Inverse Scattering | |
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| Isospectral Flows | |
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| Korteweg-deVries Equation | |
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| The Toda Lattice | |
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| Further Reading | |
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| Problems for Chapter 9 | |
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| Asymptotic Expansions | |
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| Definitions and Properties | |
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| Integration by Parts | |
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| Laplace's Method | |
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| Method of Steepest Descents | |
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| Method of Stationary Phase | |
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| Further Reading | |
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| Problems for Chapter 10 | |
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| Regular Perturbation Theory | |
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| The Implicit Function Theorem | |
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| Perturbation of Eigenvalues | |
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| Nonlinear Eigenvalue Problems | |
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| Lyapunov-Schmidt Method | |
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| Oscillations and Periodic Solutions | |
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| Advance of the Perihelion of Mercury | |
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| Van der Pol Oscillator | |
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| Knotted Vortex Filaments | |
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| The Melnikov Function | |
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| Hopf Bifurcations | |
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| Further Reading | |
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| Problems for Chapter 11 | |
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| Singular Perturbation Theory | |
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| Initial Value Problems I | |
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| Van der Pol Equation | |
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| Adiabatic Invariance | |
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| Averaging | |
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| Homogenization Theory | |
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| Initial Value Problems II | |
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| Operational Amplifiers | |
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| Enzyme Kinetics | |
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| Slow Selection in Population Genetics | |
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| Boundary Value Problems | |
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| Matched Asymptotic Expansions | |
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| Flame Fronts | |
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| Relaxation Dynamics | |
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| Exponentially Slow Motion | |
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| Further Reading | |
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| Problems for Chapter 12 | |
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| Bibliography | |
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| Selected Hints and Solutions | |
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| Index | |