Content: Cover
Title Page
Copyright Page
Dedication
Table of Contents
1 Finite Dimensional Vector Spaces
1.1 Linear Vector Spaces
1.2 Spectral Theory for Matrices
1.3 Geometrical Significance of Eigenvalues
1.4 Fredholm Alternative Theorem
1.5 Least Squares Solutionsaฬ#x80
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Pseudo Inverses
1.6 Numerical Considerations
Further Reading
Problems for Chapter 1
1.7 Appendix: Jordan Canonical Form
2 Function Spaces
2.1 Complete Metric Spaces
2.1.1 Sobolev Spaces
2.2 Approximation in Hilbert Spaces
2.2.1 Fourier Series and Completeness
2.2.2 Orthogonal Polynomials
2.2.3 Trigonometric Series 2.2.4 Discrete Fourier Transforms2.2.5 Walsh Functions and Walsh Transforms
2.2.6 Finite Elements
2.2.7 Sine Functions
Further Reading
Problems for Chapter 2
3 Integral Equations
3.1 Introduction
3.2 The Fredholm Alternative
3.3 Compact Operatorsaฬ#x80
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Hilbert Schmidt Kernels
3.4 Spectral Theory for Compact Operators
3.5 Resolvent and Pseudo-Resolvent Kernels
3.6 Approximate Solutions
3.7 Singular Integral Equations
Further Reading
Problems for Chapter 3
4 Differential Operators
4.1 Distributions and the Delta Function
4.2 Greenaฬ#x80
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s Functions
4.3 Differential Operators 4.3.1 Domain of an Operator4.3.2 Adjoint of an Operator
4.3.3 The Extended Definition of an Operator
4.3.4 Inhomogeneous Boundary Data
4.3.5 The Fredholm Alternative
4.4 Least Squares Solutions
4.5 Eigenfunction Expansions
4.5.1 Trigonometric Functions
4.5.2 Orthogonal Polynomials
4.5.3 Special Functions
4.5.4 Discretized Operators
Further Reading
Problems for Chapter 4
5 Calculus of Variations
5.1 Euler-Lagrange Equations
5.1.1 Constrained Problems
5.1.2 Several Unknown Functions
5.1.3 Higher Order Derivatives
5.1.4 Variable Endpoints
5.1.5 Several Independent Variables 5.2 Hamiltonaฬ#x80
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s Principle5.3 Approximate Methods
5.4 Eigenvalue Problems
Further Reading
Problems for Chapter 5
6 Complex Variable Theory
6.1 Complex Valued Functions
6.2 The Calculus of Complex Functions
6.2.1 Differentiationaฬ#x80
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Analytic Functions
6.2.2 Integration
6.2.3 Cauchy Integral Formula
6.2.4 Taylor and Laurent Series
6.3 Fluid Flow and Conformal Mappings
6.3.1 Laplaceaฬ#x80
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s Equation
6.3.2 Conformal Mappings
6.3.3 Free Boundary Problemsaฬ#x80
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Hodograph Transformation
6.4 Contour Integration
6.5 Special Functions
6.5.1 The Gamma Function
6.5.2 Bessel Functions 6.5.3 Legendre Functions6.5.4 Sine Functions
Further Reading
Problems for Chapter 6
7 Transform and Spectral Theory
7.1 Spectrum of an Operator
7.2 Fourier Transforms
7.2.1 Transform Pairs
7.2.2 Completeness of Hermite and Laguerre Polynomials
7.2.3 Sine Functions
7.3 Laplace, Mellin and Hankel Transforms
7.4 Z Transforms
7.5 Scattering Theory
Further Reading
Problems for Chapter 7
8 Partial Differential Equations
8.1 Poissonaฬ#x80
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s Equation
8.1.1 Fundamental solutions
8.1.2 The Method of Images
8.1.3 Transform Methods
8.1.4 Eigenfunctions
8.2 The Wave Equation