# Mathematical Methods For Physicists International Student Edition

## 6th Edition

**Authors:**George Arfken Hans Weber

**eBook ISBN:**9780080916729

**eBook ISBN:**9780080470696

**Paperback ISBN:**9780120885848

**Imprint:**Academic Press

**Published Date:**3rd June 2005

**Page Count:**1200

## Description

This best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital addition to the bookshelf of any serious student of physics or research professional in the field. The authors have put considerable effort into revamping this new edition.

## Key Features

- Updates the leading graduate-level text in mathematical physics
- Provides comprehensive coverage of the mathematics necessary for advanced study in physics and engineering
- Focuses on problem-solving skills and offers a vast array of exercises
- Clearly illustrates and proves mathematical relations

**New in the Sixth Edition:**

- Updated content throughout, based on users' feedback
- More advanced sections, including differential forms and the elegant forms of Maxwell's equations
- A new chapter on probability and statistics
- More elementary sections have been deleted

## Readership

A reference and as a textbook for aspiring and practicing physicists. As a text it functions best at the graduate level, although it’s recommended often as a reference to an undergrad course, almost always in physics departments

## Table of Contents

PREFACE

CHAPTER 1. VECTOR ANALYSIS

1.1 DEFINITIONS, ELEMENTARY APPROACH

1.2 ROTATION OF THE COORDINATE AXES3

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 ∇

1.10 VECTOR INTEGRATION

1.11 Gauss’ THEOREM

1.12 STOKES’ THEOREM

1.13 POTENTIAL THEORY

1.14 Gauss’ LAW, POISSON’s EQUATION

1.15 DIRAC DELTA FUNCTION

1.16 HELMHOLTZ’S THEOREM

Additional Readings

CHAPTER 2. VECTOR ANALYSIS IN CURVED COORDINATES AND TENSORS

2.1 ORTHOGONAL COORDINATES IN 3

2.2 DIFFERENTIAL VECTOR OPERATORS

2.3 SPECIAL COORDINATE SYSTEMS: INTRODUCTION

2.4 CIRCULAR CYLINDER COORDINATES

2.5 SPHERICAL POLAR COORDINATES

2.6 TENSOR ANALYSIS

2.7 CONTRACTION, DIRECT PRODUCT

2.8 QUOTIENT RULE

2.9 PSEUDOTENSORS, DUAL TENSORS

2.10 GENERAL TENSORS

2.11 TENSOR DERIVATIVE OPERATORS

Additional Readings

CHAPTER 3. DETERMINANTS AND MATRICES

3.1 DETERMINANTS

3.2 MATRICES

3.3 ORTHOGONAL MATRICES

3.4 HERMITIAN MATRICES, UNITARY MATRICES

3.5 DIAGONALIZATION OF MATRICES

3.6 NORMAL MATRICES

Additional Readings

CHAPTER 4. GROUP THEORY

4.1 INTRODUCTION TO GROUP THEORY

4.2 GENERATORS OF CONTINUOUS GROUPS

4.3 ORBITAL ANGULAR MOMENTUM

4.4 ANGULAR MOMENTUM COUPLING

4.5 HOMOGENEOUS LORENTZ GROUP

4.6 LORENTZ COVARIANCE OF MAXWELL’S EQUATIONS

4.7 DISCRETE GROUPS

4.8 DIFFERENTIAL FORMS

Additional Readings

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 SERIES

5.11 INFINITE PRODUCTS

Additional Readings

CHAPTER 6. FUNCTIONS OF A COMPLEX VARIABLE I ANALYTIC PROPERTIES, MAPPING

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 SINGULARITIES

6.7 MAPPING

6.8 CONFORMAL MAPPING

Additional Readings

CHAPTER 7. FUNCTIONS OF A COMPLEX VARIABLE II

7.1 CALCULUS OF RESIDUES

7.2 DISPERSION RELATIONS

7.3 METHOD OF STEEPEST DESCENTS

Additional Readings

CHAPTER 8. THE GAMMA FUNCTION (FACTORIAL FUNCTION)

8.1 DEFINITIONS, SIMPLE PROPERTIES

8.2 DIGAMMA AND POLYGAMMA FUNCTIONS

8.3 STIRLING’S SERIES

8.4 THE BETA FUNCTION

8.5 THE INCOMPLETE GAMMA FUNCTIONS AND RELATED FUNCTIONS

Additional Readings

CHAPTER 9. DIFFERENTIAL EQUATIONS

9.1 Partial Differential Equations

9.2 First-Order Differential Equations

9.3. SEPARATION OF VARIABLES

9.4 Singular Points

9.5 Series Solutions—Frobenius’ Method

9.8 Heat Flow, or Diffusion, PDE

Additional Readings

CHAPTER 10. STURM-LIOUVILLE THEORY—ORTHOGONAL FUNCTIONS

10.1 SELF-ADJOINT ODES

10.2 HERMITIAN OPERATORS

10.3 GRAM–SCHMIDT ORTHOGONALIZATION

10.4 COMPLETENESS OF EIGENFUNCTIONS

10.5 GREEN’S FUNCTION—EIGENFUNCTION EXPANSION

Additional Readings

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

11.4 HANKEL FUNCTIONS

11.5 MODIFIED BESSEL FUNCTIONS, Iv(x) AND Kv(x)

11.6 ASYMPTOTIC EXPANSIONS

11.7 SPHERICAL BESSEL FUNCTIONS

Additional Readings

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 ORBITAL ANGULAR MOMENTUM OPERATORS

12.8 THE ADDITION THEOREM FOR SPHERICAL HARMONICS

12.9 INTEGRALS OF PRODUCTS OF THREE SPHERICAL HARMONICS

12.10 LEGENDRE FUNCTIONS OF THE SECOND KIND

12.11 VECTOR SPHERICAL HARMONICS

Additional Readings

CHAPTER 13. MORE SPECIAL FUNCTIONS

13.1 HERMITE FUNCTIONS

13.2 LAGUERRE FUNCTIONS

13.3 CHEBYSHEV POLYNOMIALS

13.4 HYPERGEOMETRIC FUNCTIONS

13.5 CONFLUENT HYPERGEOMETRIC FUNCTIONS

13.6 MATHIEU FUNCTIONS

Additional Readings

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 FOURIER TRANSFORM

14.7 FOURIER EXPANSIONS OF MATHIEU FUNCTIONS

Additional Readings

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. LAPLACE TRANSFORMS

15.9. LAPLACE TRANSFORM OF DERIVATIVES

15.10. OTHER PROPERTIES

15.11. CONVOLUTION (FALTUNGS) THEOREM

15.12. INVERSE LAPLACE TRANSFORM

Additional Readings

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

Additional Readings

CHAPTER 17. CALCULUS OF VARIATIONS

17.1 A DEPENDENT AND AN INDEPENDENT VARIABLE

17.2 APPLICATIONS OF THE EULER EQUATION

17.3 SEVERAL DEPENDENT VARIABLES

17.4 SEVERAL INDEPENDENT VARIABLES

17.5 SEVERAL DEPENDENT AND INDEPENDENT VARIABLES

17.6 LAGRANGIAN MULTIPLIERS

17.7 VARIATION WITH CONSTRAINTS

17.8 RAYLEIGH–RITZ VARIATIONAL TECHNIQUE

Additional Readings

CHAPTER 18. NONLINEAR METHODS AND CHAOS

18.1 INTRODUCTION

18.2 THE LOGISTIC MAP

18.3 SENSITIVITY TO INITIAL CONDITIONS AND PARAMETERS

18.4 NONLINEAR DIFFERENTIAL EQUATIONS

Additional Readings

CHAPTER 19. PROBABILITY

19.1 DEFINITIONS, SIMPLE PROPERTIES

19.2 RANDOM VARIABLES

19.3 BINOMIAL DISTRIBUTION

19.4 POISSON DISTRIBUTION

19.5 GAUSS’ NORMAL DISTRIBUTION

19.6 STATISTICS

Additional Readings

INDEX

## Details

- No. of pages:
- 1200

- Language:
- English

- Copyright:
- © Academic Press 2005

- Published:
- 3rd June 2005

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080916729

- eBook ISBN:
- 9780080470696

- Paperback ISBN:
- 9780120885848

## About the Author

### George Arfken

### Affiliations and Expertise

Miami University, Oxford, Ohio, USA

### Hans Weber

### Affiliations and Expertise

University of Virginia, USA

## Reviews

"As to a comparison with other books of the same ilk, well, in all honesty, there are none. No other text on methods of mathematical physics is as comprehensive and as complete..."** --Tristan Hubsch, Howard University**