Principles of applied mathematics : transformation and approximation

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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 Solutionsâ#x80 #x94 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 Operatorsâ#x80 #x94 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 Greenâ#x80 #x99 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 Hamiltonâ#x80 #x99 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 Differentiationâ#x80 #x94 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 Laplaceâ#x80 #x99 s Equation 6.3.2 Conformal Mappings 6.3.3 Free Boundary Problemsâ#x80 #x94 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 Poissonâ#x80 #x99 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