Book:George Arfken/Mathematical Methods for Physicists/Seventh Edition
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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
- Additional Readings
- Chapter 2. DETERMINANTS AND MATRICES
- 2.1 Determinants
- 2.2 Matrices
- Additional Readings
- 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
- Additional Readings
- 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
- Additional Readings
- Chapter 5. VECTOR SPACES
- 5.1 Vector 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 Notations
- 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 Methods
- 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 Equations
- 9.7 Heat – Flow, or Diffution PDE
- 9.8 Summary
- Additional Readings
- Chapter 10. GREEN’ 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 Dispertion 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 Ralated Function
- Additional Readings
- 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
- 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 Simmetric Group
- 17.7 Continous Groups
- 17.8 Lorentz Group
- 17.9 Lorentz Covariance of Maxwell’s Equantions
- 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 Application of Fourier Series
- 19.3 Gibbs Phenomenon
- Additional Readings
- 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
- 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’ Nomal Distribution
- 23.6 Transformation of Random Variables
- 23.7 Statistics
- Additional Readings
- index
Further Editions
- 1966: George Arfken: Mathematical Methods for Physicists
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.)
- 2003: George Arfken and Hans J. Weber: Mathematical Methods for Physicists (5th ed.)