Mathematical Methods for Physicists

Now in its 7th edition, Mathematical Methods for Physicists continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. This bestselling text provides mathematical relations and their proofs essential to the study of physics and related fields. While retaining the key features of the 6th edition, the new edition provides a more careful balance of explanation, theory, and examples. Taking a problem-solving-skills approach to incorporating theorems with applications, the book's improved focus will help students succeed throughout their academic careers and well into their professions. Some notable enhancements include more refined and focused content in important topics, improved organization, updated notations, extensive explanations and intuitive exercise sets, a wider range of problem solutions, improvement in the placement, and a wider range of difficulty of exercises.

• Revised and updated version of the leading text in mathematical physics
• Focuses on problem-solving skills and active learning, offering numerous chapter problems
• Clearly identified definitions, theorems, and proofs promote clarity and understanding

New to this edition:

• Improved modular chapters
• New up-to-date examples
• More intuitive explanations

Graduate students and advanced undergraduates in Physics, Engineering, Applied Mathematics, Chemistry, and Environmental Science/Geophysics; also practitioners and researchers in these fields

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, Jν(x)
• 14.2 Orthogonality
• 14.3 Neumann Functions, Bessel Functions of the Second Kind
• 14.4 Hankel Functions
• 14.5 Modified Bessel Functions, Iν(x) and Kν(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