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 Theorem

Chapter 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,φ,z)


2.5 Spherical Polar Coordinates (r,0,φ)


2.6 Separation of Variables

Chapter 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 Operations

Chapter 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 Group

Chapter 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 Products

Chapter 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 Mapping

Chapter 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 Descents

Chapter 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 Solutions

Chapter 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 Eigenfunctions

Chapter 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 Functions

Chapter 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 Functions

Chapter 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)


12.11 Vector Spherical Harmonics

Chapter 13 Special Functions


13.1 Hermite Functions


13.2 Laguerre Functions


13.3 Chebyshev (Tschebyscheff) Polynomials


13.4 Chebyshev Polynomials—Numerical Applications


13.5 Hypergeometric Functions


13.6 Confluent Hypergeometric Functions

Chapter 14 Fourier Series


14.1 General Properties


14.2 Advantages, Uses of Fourier Series


14.3 Applications of Fourier Series


14.4 Properties of Fourier Series


14.5 Gibbs Phenomenon


14.6 Discrete Orthogonality—Discrete Fourier Transform

Chapter 15 Integral Transforms


15.1 Integral Transforms


15.2 Development of the Fourier Integral


15.3 Fourier Transforms—Inversion Theorem


15.4 Fourier Transform of Derivatives


15.5 Convolution Theorem


15.6 Momentum Representation


15.7 Transfer Functions


15.8 Elementary Laplace Transforms


15.9 Laplace Transform of Derivatives


15.10 Other Properties


15.11 Convolution or Faltung Theorem


15.12 Inverse Laplace Transformation

Chapter 16 Integral Equations


16.1 Introduction


16.2 Integral Transforms, Generating Functions


16.3 Neumann Series, Separable (Degenerate) Kernels


16.4 Hilbert-Schmidt Theory


16.5 Green's Functions—One Dimension


16.6 Green's Functions—Two and Three Dimensions

Chapter 17 Calculus of Variations


17.1 One-Dependent and One-Independent Variable


17.2 Applications of the Euler Equation


17.3 Generalizations, Several Dependent Variables


17.4 Several Independent Variables


17.5 More Than One Dependent, More than One Independent Variable


17.6 Lagrangian Multipliers


17.7 Variation Subject to Constraints


17.8 Rayleigh-Ritz Variational Technique

Appendix 1 Real Zeros of a Function

Appendix 2 Gaussian Quadrature

General References

Index