# Mathematical Methods for Physicists

## 7th Edition

### A Comprehensive Guide

**Authors:**George Arfken Hans Weber Frank E. Harris

**Hardcover ISBN:**9780123846549

**eBook ISBN:**9780123846556

**Imprint:**Academic Press

**Published Date:**17th January 2012

**Page Count:**1220

## Description

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.

## 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

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

## 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 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

## Details

- No. of pages:
- 1220

- Language:
- English

- Copyright:
- © Academic Press 2012

- Published:
- 17th January 2012

- Imprint:
- Academic Press

- Hardcover ISBN:
- 9780123846549

- eBook ISBN:
- 9780123846556

## About the Authors

### George Arfken

### Affiliations and Expertise

Miami University, Oxford, Ohio, USA

### Hans Weber

### Affiliations and Expertise

University of Virginia, USA

### Frank E. Harris

Frank E. Harris was awarded his A. B. (Chemistry) from Harvard University in 1951 and his Ph.D. (Physical Chemistry) from University of California in 1954. The author of 244 research publications and multiple books, Dr. Harris has been a Professor of Physics and Chemistry, University of Utah and Resident Adjunct Professor of Chemistry, Quantum Theory Project, University of Florida. He served on the Editorial Board of the International Journal of Quantum Chemistry, and has been named a Fellow for both the American Institute of Chemists and the American Physical Society.

### Affiliations and Expertise

University of Florida, USA

## Reviews

"...a thorough handbook about mathematics that is useful in physics."--**MAA.org**

"This volume is a great collection of essential mathematical tools and techniques used to solve problems in physics, very useful to any student of physics or research professional in the field. It is concentrated to problem-solving art and offers a large amount of problems and exercises."--**Zentralblatt MATH 1239**