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

Contents



 * 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



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