# Mathematical Methods for Physicists

### A Comprehensive Guide

## Description

## Key Features

- 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

## Readership

## Table of Contents

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 ℝ
^{3} - 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

## Product details

- No. of pages: 1220
- Language: English
- Copyright: © Academic Press 2012
- Published: December 26, 2011
- Imprint: Academic Press
- Hardcover ISBN: 9780123846549
- eBook ISBN: 9780123846556

## About the Authors

### George Arfken

#### Affiliations and Expertise

### Hans Weber

#### Affiliations and Expertise

### Frank E. Harris

#### Affiliations and Expertise

## Ratings and Reviews

### Latest reviews

(Total rating for all reviews)

**EricDavis**Tue Jan 29 2019**Review of Mathematical Methods of Physics**This title is excellent for its comprehensive coverage of mathematical physics topics.