Mathematical Methods in Physics and Engineering

Preface CHAPTER 1. Linear Algebra 1.1 Linear Equations. Summation Convention 1.2 Matrices 1.3 Determinants 1.4 Systems of Linear Algebraic Equations. Rank of a Matrix 1.5 Vector Spaces 1.6 Scalar Product 1.7 Orthonormal Basis. Linear Transformations 1.8 Quadratic Forms. Hermitian Forms 1.9 Systems of Ordinary Differential Equations. Vibration Problems 1.10 Linear Programming CHAPTER 2. Hilbert Spaces 2.1 Infinite-dimensional Vector Spaces. Function Spaces 2.2 Fourier Series 2.3 Separable Hilbert Spaces 2.4 The Projection Theorem 2.5 Linear Functionals 2.6 Weak Convergence 2.7 Linear Operators 2.8 Completely Continuous Operators CHAPTER 3. Calculus of Variations 3.1 Maxima and Minima of Functions. Lagrange Multipliers 3.2 Maxima and Minima of Functionals. Euler's Equation 3.3 Hamilton's Principle. Lagrange's Equations 3.4 Theory of Small Vibrations 3.5 The Vibrating String 3.6 Boundary-value Problems of Mathematical Physics 3.7 Eigenvalues and Eigenfunctions 3.8 Eigenfunction Expansions 3.9 Upper and Lower Bounds for Eigenvalues CHAPTER 4. Boundary-value Problems. Separation of Variables 4.1 Orthogonal Coordinate Systems. Separation of Variables 4.2 Sturm-Liouville Problems 4.3 Series Solutions of Ordinary Differential Equations 4.4 Series Solutions of Boundary-value Problems CHAPTER 5. Boundary-value Problems. Green's Functions 5.1 Nonhomogeneous Boundary-value Problems 5.2 One-dimensional Green's Functions 5.3 Generalized Functions 5.4 Green's Functions in Higher Dimensions 5.5 Problems in Unbounded Regions 5.6 A Problem in Diffraction Theory CHAPTER 6. Integral Equations 6.1 Integral-equation Formulation of Boundary-value Problems 6.2 Hilbert-Schmidt Theory 6.3 Fredholm Theory 6.4 Integral Equations of the First Kind CHAPTER 7. Analytic Function Theory 7.1 Introduction 7.2 Analytic Functions 7.3 Elementary Functions 7.4 Complex Integration 7.5 Integral Representations 7.6 Sequences and Series 7.7 Series Representations of Analytic Functions 7.8 Contour Integration 7.9 Conformal Mapping 7.10 Potential Theory CHAPTER 8. Integral Transform Methods 8.1 Fourier Transforms 8.2 Applications of Fourier Transforms. Ordinary Differential Equations 8.3 Applications of Fourier Transforms. Partial Differential Equations 8.4 Applications of Fourier Transforms. Integral Equations 8.5 Laplace Transforms. Applications 8.6 Other Transform Techniques Index