Mathematical Methods for Physicists

Chapter 1 Vector Analysis 1.1 Definitions, Elementary Approach 1.2 Advanced Definitions 1.3 Scalar or Dot Product 1.4 Vector or Cross Product 1.5 Triple Scalar Product, Triple Vector Product 1.6 Gradient 1.7 Divergence 1.8 Curl 1.9 Successive Applications of V 1.10 Vector Integration 1.11 Gauss's Theorem 1.12 Stokes's Theorem 1.13 Potential Theory 1.14 Gauss's Law, Poisson's Equation 1.15 Helmholtz's TheoremChapter 2 Coordinate Systems 2.1 Curvilinear Coordinates 2.2 Differential Vector Operations 2.3 Special Coordinate Systems-Rectangular Cartesian Coordinates 2.4 Circular Cylindrical Coordinates (p,f,z) 2.5 Spherical Polar Coordinates (r,0,f) 2.6 Separation of VariablesChapter 3 Tensor Analysis 3.1 Introduction, Definitions 3.2 Contraction, Direct Product 3.3 Quotient Rule 3.4 Pseudotensors, Dual Tensors 3.5 Dyadics 3.6 Theory of Elasticity 3.7 Lorentz Co variance of Maxwell's Equations 3.8 Noncartesian Tensors, Co variant Differentiation 3.9 Tensor Differential OperationsChapter 4 Determinants, Matrices, and Group Theory 4.1 Determinants 4.2 Matrices 4.3 Orthogonal Matrices 4.4 Oblique Coordinates 4.5 Hermitian Matrices, Unitary Matrices 4.6 Diagonalization of Matrices 4.7 Eigenvectors, Eigenvalues 4.8 Introduction to Group Theory 4.9 Discrete Groups 4.10 Continuous Groups 4.11 Generators 4.12 SU(2), SU(3), and Nuclear Particles 4.13 Homogeneous Lorentz GroupChapter 5 Infinite Series 5.1 Fundamental Concepts 5.2 Convergence Tests 5.3 Alternating Series 5.4 Algebra of Series 5.5 Series of Functions 5.6 Taylor's Expansion 5.7 Power Series 5.8 Elliptic Integrals 5.9 Bernoulli Numbers, Euler-Maclaurin Formula 5.10 Asymptotic or Semiconvergent Series 5.11 Infinite ProductsChapter 6 Functions of a Complex Variable I 6.1 Complex Algebra 6.2 Cauchy-Riemann Conditions 6.3 Cauchy's Integral Theorem 6.4 Cauchy's Integral Formula 6.5 Laurent Expansion 6.6 Mapping 6.7 Conformal MappingChapter 7 Functions of a Complex Variable II: Calculus of Residues 396 7.1 Singularities 7.2 Calculus of Residues 7.3 Dispersion Relations 7.4 The Method of Steepest DescentsChapter 8 Differential Equations 8.1 Partial Differential Equations of Theoretical Physics 8.2 First-Order Differential Equations 8.3 Separation of Variables-Ordinary Differential Equations 8.4 Singular Points 8.5 Series Solutions-Frobenius Method 8.6 A Second Solution 8.7 Nonhomogeneous Equation-Green's Function 8.8 Numerical SolutionsChapter 9 Sturm-Liouville Theory - Orthogonal Functions 9.1 Self-Adjoint Differential Equations 9.2 Hermitian (Self-Adjoint) Operators 9.3 Gram-Schmidt Orthogonalization 9.4 Completeness of EigenfunctionsChapter 10 The Gamma Function (Factorial Function) 10.1 Definitions, Simple Properties 10.2 Digamma and Polygamma Functions 10.3 Stirling's Series 10.4 The Beta Function 10.5 The Incomplete Gamma Functions and Related FunctionsChapter 11 Bessel Functions 11.1 Bessel Functions of the First Kind, Jv(x) 11.2 Orthogonality 11.3 Neumann Functions, Bessel Functions of the Second Kind, Nv(x) 11.4 Hankel Functions 11.5 Modified Bessel Functions, Iv(x) and Kv(x) 11.6 Asymptotic Expansions 11.7 Spherical Bessel FunctionsChapter 12 Legendre Functions 12.1 Generating Function 12.2 Recurrence Relations and Special Properties 12.3 Orthogonality 12.4 Alternate Definitions of Legendre Polynomials 12.5 Associated Legendre Functions 12.6 Spherical Harmonics 12.7 Angular Momentum Ladder Operators 12.8 The Addition Theorem for Spherical Harmonics 12.9 Integrals of the Product of Three Spherical Harmonics 12.10 Legendre Functions of the Second Kind, Qn(x) 12.