Mathematical Methods for Physicists

Preface

• To the Student
• What’s New
• Pathways through the Material
• Acknowledgments

Chapter 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

Chapter 2. Determinants and Matrices

• 2.1 Determinants
• 2.2 Matrices
• Additional Readings

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 7. Ordinary Differential Equations

• 7.1 Introduction
• 7.2 First-Order Equations
• 7.3 ODEs with Constant Coefficients
• 7.4 Second-Order Linear ODEs
• 7.5 Series Solutions—Frobenius’ Method
• 7.6 Other Solutions
• 7.7 Inhomogeneous Linear ODEs
• 7.8 Nonlinear Differential Equations
• Additional Readings

Chapter 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

Chapter 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

Chapter 10. Green’s Functions

• 10.1 One-Dimensional Problems
• 10.2 Problems in Two and Three Dimensions
• Additional Readings

Chapter 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

Chapter 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

Chapter 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

Chapter 14. Bessel Functions

• 14.1 Bessel Functions of the First Kind, J_ν(x)
• 14.2 Orthogonality
• 14.3 Neumann Functions, Bessel Functions of the Second Kind
• 14.4 Hankel Functions
• 14.5 Modified Bessel Functions, I_ν(x) and K_ν(x)
• 14.6 Asymptotic Expansions
• 14.7 Spherical Bessel Functions
• Additional Readings

Chapter 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

Chapter 16. Angular Momentum

• 16.1 Angular Momentum Operators
• 16.2 Angular Momentum Coupling
• 16.3 Spherical Tensors
• 16.4 Vector Spherical Harmonics
• Additional Readings

Chapter 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

Chapter 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

Chapter 19. Fourier Series

• 19.1 General Properties
• 19.2 Applications of Fourier Series
• 19.3 Gibbs Phenomenon
• Additional Readings

Chapter 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

Chapter 21. Integral Equations

• 21.1 Introduction
• 21.2 Some Special Methods
• 21.3 Neumann Series
• 21.4 Hilbert-Schmidt Theory
• Additional Readings

Chapter 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

Chapter 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