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

Contents





 * Chapter 1. VECTOR ANALYSIS


 * 1.1 Definitions, elementary approach
 * 1.2 Rotation of coordinates
 * 1.3 Scalar or dot product
 * 1.4 Vector or cross product
 * 1.5 Triple scalar product, triple vector product
 * 1.6 Gradient, $\nabla$
 * 1.7 Divergence, $\nabla \cdot$
 * 1.8 Curl, $\nabla \times$
 * 1.9 Successive applications of $\nabla$
 * 1.10 Vector integration
 * 1.11 Gauss's theorem
 * 1.12 Stokes's theorem
 * 1.13 Potential theory
 * 1.14 Gauss's law, Poisson's equation
 * 1.15 Helmholtz's theorem
 * References


 * Chapter 2. COORDINATE SYSTEMS


 * 2.1 Curvilinear coordinates
 * 2.2 Differential vector operations
 * 2.3 Special coordinate systems -- rectangular cartesian coordinates
 * 2.4 Spherical polar coordinates $\tuple {r, \theta, \varphi}$
 * 2.5 Separation of variables
 * 2.6 Circular cylindrical coordinates $\tuple {\rho, \varphi, z}$
 * 2.7 Elliptic cylindrical coordinates $\tuple {u, v, z}$
 * 2.8 Parabolic cylindrical coordinates $\tuple {\xi, \eta, z}$
 * 2.9 Bipolar coordinates $\tuple {\xi, \eta, z}$
 * 2.10 Prolate spheroidal coordinates $\tuple {u, v, \varphi}$
 * 2.11 Oblate spheroidal coordinates $\tuple {u, v, \varphi}$
 * 2.12 Parabolic coordinates $\tuple {\xi, \eta, \varphi}$
 * 2.13 Toroidal coordinates $\tuple {\xi, \eta, \varphi}$
 * 2.14 Bispherical coordinates $\tuple {\xi, \eta, \varphi}$
 * 2.15 Confocal ellipsoidal coordinates $\tuple {\xi_1, \xi_2, \xi_3}$
 * 2.16 Confocal coordinates $\tuple {\xi_1, \xi_2, \xi_3}$
 * 2.17 Confocal parabolic coordinates $\tuple {\xi_1, \xi_2, \xi_3}$
 * References


 * Chapter 3. TENSOR ANALYSIS


 * 3.1 Introduction, definitions
 * 3.2 Contraction, direct product
 * 3.3 Quotient rule
 * 3.4 Pseudotensors, dual tensors
 * 3.5 Dyadics
 * 3.6 Theory of elasticity
 * 3.7 Lorentz covariance of Maxwell's equations
 * References


 * Chapter 4. DETERMINANTS, MATRICES, AND GROUP THEORY


 * 4.1 Determinants
 * 4.2 Matrices
 * 4.3 Orthogonal matrices
 * 4.4 Oblique coordinates
 * 4.5 Hermitian matrices, unitary matrices
 * 4.6 Diagonalization of matrices
 * 4.7 Introduction to group theory
 * 4.8 Discrete groups
 * 4.9 Continuous groups
 * 4.10 Generators
 * 4.11 $\map {\mathrm {SU} } 2$, $\map {\mathrm {SU} } 3$ and nuclear particles
 * 4.12 Homogeneous Lorentz Group
 * References


 * 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
 * 5.10 Infinite products
 * 5.11 Asymptotic or semiconvergent series
 * References


 * Chapter 6. FUNCTIONS OF A COMPLEX VARIABLE I. ANALYTIC PROPERTIES, CONFORMAL 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 Mapping
 * 6.7 Conformal mapping
 * 6.8 Schwarz-Christoffel transformation
 * References


 * Chapter 7. FUNCTIONS OF A COMPLEX VARIABLE II. CALCULUS OF RESIDUES


 * 7.1 Singularities
 * 7.2 Calculus of residues
 * 7.3 Applications of the calculus of residues
 * 7.4 The method of steepest descents


 * Chapter 8. SECOND-ORDER DIFFERENTIAL EQUATIONS


 * 8.1 Partial differential equations of theoretical physics
 * 8.2 Separation of variables -- ordinary differential equations
 * 8.3 Singular points
 * 8.4 Series solutions -- Frobenius' method
 * 8.5 A second solution
 * 8.6 Nonhomogeneous equation -- Green's function
 * 8.7 Numerical solutions
 * References


 * Chapter 9. STURM-LIOUVILLE THEORY -- ORTHOGONAL FUNCTIONS


 * 9.1 Self-adjoint differential equations
 * 9.2 Hermitian (self-adjoint) operators
 * 9.3 Schmidt Orthogonalization
 * 9.4 Completeness of eigenfunctions
 * References


 * Chapter 10. THE GAMMA FUNCTION (FACTORIAL FUNCTION)


 * 10.1 Definitions, simple properties
 * 10.2 Digamma and polygamma functions
 * 10.3 Stirling's series
 * 10.4 The beta function
 * 10.5 The incomplete gamma functions and related functions
 * References


 * Chapter 11. BESSEL FUNCTIONS


 * 11.1 Bessel functions of the first kind $\map {J_\nu} x$
 * 11.2 Orthogonality
 * 11.3 Neumann functions, Bessel functions of the second kind, $\map {N_\nu} x$
 * 11.4 Hankel functions
 * 11.5 Modified Bessel functions, $\map {I_\nu} x$ and $\map {K_\nu} x$
 * 11.6 Asymptotic expansions
 * 11.7 Spherical Bessel functions
 * References


 * 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 function
 * 12.6 Spherical harmonics
 * 12.7 Angular momentum and ladder operators
 * 12.8 The addition theorem for spherical harmonics
 * 12.9 Integrals of the product of three spherical harmonics
 * 12.10 Legendre functions of the second kind, $\map {Q_n} x$
 * 12.11 Application to spheroidal coordinate systems
 * 12.12 Vector spherical harmonics
 * References


 * Chapter 13. SPECIAL FUNCTIONS


 * 13.1 Hermite functions
 * 13.2 Laguerre functions
 * 13.3 Chebyshev (Tschebyscheff) polynomials
 * 13.4 Hypergeometric functions
 * 13.5 Confluent hypergeometric functions
 * References


 * 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
 * References


 * Chapter 15. INTEGRAL TRANSFORMS


 * 15.1 Integral transforms
 * 15.2 Development of the Fourier integral
 * 15.3 Fourier transforms -- inversion theorem
 * 15.4 Fourier transforms of derivatives
 * 15.5 Convolution theorem
 * 15.6 Momentum representation
 * 15.7 Elementary Laplace transforms
 * 15.8 Laplace transform of derivatives
 * 15.9 Other properties
 * 15.10 Convolution or Faltung theorem
 * 15.11 Inverse Laplace transformation
 * References


 * 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
 * 16.5 Green's function -- one dimension
 * 16.6 Green's functions -- two and three dimensions
 * References


 * Chapter 17. CALCULUS OF VARIATIONS


 * 17.1 One dependent and one independent variable
 * 17.2 Applications of the Euler equation
 * 17.3 Generalizations, several dependent variables
 * 17.4 Several independent variables
 * 17.5 More than one dependent, more than one independent variable
 * 17.6 Lagrangian multipliers
 * 17.7 Variation subject to constraints
 * 17.8 Rayleigh-Ritz variational technique
 * References


 * GENERAL REFERENCES





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* : Introduction: Electromagnetic Theory