# Book:George Arfken/Mathematical Methods for Physicists/Seventh Edition

## George Arfken, H.J. Weber and F.E. Harris: Mathematical Methods for Physicists (7th Edition)

Published $\text {2013}$, Academic Press

ISBN 0-12-384654-9

### Contents

preface

Chapter 1. MATHEMATICAL PRELIMINARIES
1.1. Infinite Series
1.2. Series of Functions
1.3. Binomial Theorem
1.4. Mathematical Induction
1.5. Operations of 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 Functions

Chapter 2. DETERMINANTS AND MATRICES
2.1 Determinants
2.2 Matrices

Chapter 3. VECTOR ANALYSIS
3.1 Review of Basics Properties
3.2 Vector in 3 ‐ D Spaces
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 Integrations
3.8 Integral Theorems
3.9 Potential Theory
3.10 Curvilinear Coordinates

Chapter 4. TENSOR AND DIFFERENTIAL FORMS
4.1 Tensor Analysis
4.2 Pseudotensors, Dual Tensors
4.3 Tensor in General Coordinates
4.4 Jacobians
4.5 Differential Forms
4.6 Differentiating Forms
4.7 Integrating Forms

Chapter 5. VECTOR SPACES
5.1 Vector in Function Spaces
5.2 Gram ‐ Schmidt Orthogonalization
5.3 Operators
5.5 Unitary Operators
5.6 Transformations of Operators
5.7 Invariants
5.8 Summary – Vector Space Notations

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

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

Chapter 8. STURM – LIOUVILLE THEORY
8.1 Introduction
8.2 Hermitian Operators
8.3 ODE Eigenvalue Problems
8.4 Variation Methods
8.5 Summary, Eigenvalue Problems

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 Equations
9.7 Heat – Flow, or Diffution PDE
9.8 Summary

Chapter 10. GREEN’ FUNCTIONS
10.1 One – Dimensional Problems
10.2 Problems in Two and Three Dimensions

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

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 Dispertion Relations

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 Ralated Function

Chapter 14. BESSEL FUNCTIONS
14.1 Bessel Functions of the First kind, $\map {J_ν} x$
14.2 Orthogonality
14.3 Neumann Functions, Bessel Functions of the Second kind
14.4 Hankel Functions
14.5 Modified Bessel Functions, $\map {I_ν} x$ and $\map {K_ν} x$
14.6 Asymptotic Expansions
14.7 Spherical Bessel Functions

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

Chapter 16. ANGULAR MOMENTUM
16.1 Angular Momentum Operators
16.2 Angular Momentum Coupling
16.3 Spherical Tensors
16.4 Vector Spherical Harmonics

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 Simmetric Group
17.7 Continous Groups
17.8 Lorentz Group
17.9 Lorentz Covariance of Maxwell’s Equantions
17.10 Space Groups

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

Chapter 19. FOURIER SERIES
19.1 General Properties
19.2 Application of Fourier Series
19.3 Gibbs Phenomenon

Chapter 20. INTEGRAL TRANSFORMS
20.1 Introduction
20.2 Fourier Transforms
20.3 Properties of Fourier Transforms
20.4 Fourier Convolution Theorem
20.5 Signal – Proccesing Applications
20.6 Discrete Fourier Transforms
20.7 Laplace Transforms
20.8 Properties of Laplace Transforms
20.9 Laplace Convolution Transforms
20.10 Inverse Laplace Transforms

Chapter 21. INTEGRAL EQUATIONS
21.1 Introduction
21.2 Some Special Methods
21.3 Neumann Series
21.4 Hilbert – Schmidt Theory

Chapter 22. CALCULUS OF VARIATIONS
22.1 Euler Equation
22.2 More General Variations
22.3 Constrained Minima/Maxima
22.4 Variation with Constraints

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’ Nomal Distribution
23.6 Transformation of Random Variables
23.7 Statistics