Preface 1 Mathematical Preliminaries 1.1 Infinite Series 1.2 Series of Functions 1.3 Binomial Theorem 1.4 Mathematical Induction 1.5 Operations on Series Expansions of Functions 1.6 Some Important Series 1.7 Vectors 1.8 Complex Numbers and Functions 1.9 Derivatives and Extrema 1.10 Evaluation of Integrals 1.11 Dirac Delta Function Additional Readings 2Determinants and Matrices 2.1 Determinants 2.2 Matrices Additional Readings 3 Vector Analysis 3.1 Review of Basic Properties 3.2 Vectors in 3-D Space 3.3 Coordinate Transformations 3.4 Rotations in R3 3.5 Differential Vector Operators 3.6 Differential Vector Operators: Further Properties 3.7 Vector Integration 3.8 Integral Theorems 3.9 Potential Theory 3.10 Curvilinear Coordinates Additional Readings 4 Tensors and Differential Forms 4.1 Tensor Analysis 4.2 Pseudotensors, Dual Tensors 4.3 Tensors in General Coordinates 4.4 Jacobians 4.5 Differential Forms 4.6 Differentiating Forms 4.7 Integrating Forms Additional Readings 5 Vector Spaces 5.1 Vectors in Function Spaces 5.2 Gram-Schmidt Orthogonalization 5.3 Operators 5.4 Self-Adjoint Operators 5.5 Unitary Operators 5.6 Transformations of Operators 5.7 Invariants 5.8 Summary-Vector Space Notation Additional Readings 6 Eigenvalue Problems 6.1 Eigenvalue Equations 6.2 Matrix Eigenvalue Problems 6.3 Hermitian Eigenvalue Problems 6.4 Hermitian Matrix Diagonalization 6.5 Normal Matrices Additional Readings 7 Ordinary Differential Equations 7.1 Introduction 7.2 First-Order Equations 7.3 ODEs with Constant Coefficients 7.4 Second-Order Linear OD. Es 7.5 Series Solutions--Frobenius' Method 7.6 Other Solutions 7.7 Inhomogeneous Linear ODEs 7.8 Nonlinear Differential Equations Additional Readings 8 Sturm-Liouville Theory 8.1 Introduction 8.2 Hermitian Operators 8.3 ODE Eigenvalue Problems 8.4 Variation Method 8.5 Summary, Eigenvalue Problems Additional Readings 9 Partial Differential Equations 9.1 Introduction 9.2 First-Order Equations 9.3 Second-Order Equations 9.4 Separation of Variables 9.5 Laplace and Poisson Equations 9.6 Wave Equation 9.7 Heat-Flow, or Diffusion PDE 9.8 Summary Additional Readings 10 Green's Functions 10.1 One-Dimensional Problems 10.2 Problems in Two and Three Dimensions Additional Readings 11 Complex Variable Theory 11.1 Complex Variables and Functions 11.2 Cauchy-Riemann Conditions 11.3 Cauchy's Integral Theorem 11.4 Cauchy ' s Integral Formula 11.5 Laurent Expansion 11.6 Singularities 11.7 Calculus of Residues 11.8 Evaluation of Definite Integrals 11.9 Evaluation of Sums 11.10 Miscellaneous Topics Additional Readings 12 Further Topics in Analysis 12.1 Orthogonal Polynomials 12.2 Bernoulli Numbers 12.3 Euler-Maclaurin Integration Formula 12.4 Dirichlet Series 12.5 Infinite Products 12.6 Asymptotic Series 12.7 Method of Steepest Descents 12.8 Dispersion Relations Additional Readings 13 Gamma Function 13.1 Definitions, Properties 13.2 Digamma and Polygamma Functions 13,3 The Beta Function 13.4 Stirling's Series 13.5 Riemann Zeta Function 13.6 Other Related Functions Additional Readings 14 Bessel Functions 14.1 Bessel Functions of the First Kind, Jv (x) 14.2 Orthogonality 14.3 Neumann Functions, Bessel Functions of the Second Kind 14.4 HankeI Functions 14.5 Modified Bessel Functions, Ir(x) and Ky(x) 14.6 Asymptotic Expansions 14.7 Spherical Bessel Functions Additional Readings 15 Legendre Functions 15.1 Legendre Polynomials 15.2 Orthogonality 15.3 Physical Interpretation of Generating Function 15.4 Associated Legendre Equation 15.5 Spherical Harmonics 15.6 Legendre Functions of the Second Kind Additional Readings 16 Angular Momentum 16.1 Angular Momentum Operators 16.2 Angular Momentum Coupling 16.3 Spherical Tensors 16.4 Vector Spherical Harmonics Additional Readings 17 Group Theory 17.1 Introduction to Group Theory 17.2 Representation of Groups 17.3 Symmetry and Physics 17.4 Discrete Groups 17.5 Direct Products 17.6 Symmetric Group 17.7 Continuous Groups 17.8 Lorentz Group 17.9 Lorentz Covariance of Maxwell's Equations 17.10 Space Groups Additional Readings 18 More Special Functions 18.1 Hermite Functions 18.2 Applications of Hermite Functions 18.3 Laguerre Functions 18.4 Chebyshev Polynomials 18.5 Hypergeometric Functions 18.6 Confluent Hypergeometric Functions 18,7 Dilogarithm 18.8 Elliptic Integrals Additional Readings 19 Fourier Series 19.1 General Properties 19.2 Applications of Fourier Series 19.3 Gibbs Phenomenon Additional Readings 20 Integral Transforms 20.1 Introduction 20.2 Fourier Transform 20.3 Properties of Fourier Transforms 20.4 Fourier Convolution Theorem 20.5 Signal-Processing Applications 20.6 Discrete Fourier Transform 20.7 Laplace Transforms 20.8 Properties of Laplace Transforms 20.9 Laplace Convolution Theorem 20.10 Inverse Laplace Transform Additional Readings 21 Integral Equations 21.1 Introduction 21.2 Some Special Methods 21.3 Neumann Series 21.4 Hilbert-Schmidt Theory Additional Readings 22 Calculus of Variations 22.1 Euler Equation 22.2 More General Variations 22.3 Constrained Minima/Maxima 22.4 Variation with Constraints Additional Readings 23 Probability and Statistics 23.1 Probability: Definitions, Simple Properties 23.2 Random Variables 23.3 Binomial Distribution 23.4 Poisson Distribution 23.5 Gauss' Normal Distribution 23.6 Transformations of Random Variables 23.7 Statistics Additional Readings Index