arfken mathematical methods for physicists

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Contents
	Introduction
	Chapter I Vector analysis
	Definitions, elementary approach
	Advanced definitions
	Scalar or dot product
	Vector or cross product
	Triple scalar product, triple vector product
	Gradient
	Divergence
	Crul
	Successive aplications of grad
	Vector integration
	Gauss´s Theorem
	Stokes´s Theorem
	Potential theory
	Gauss´s law, Poisson´s equation
	Helmhotz´s theorem
	Chapter II Coordinate systems
	Curvilinear coordinates
	Differential vector operations
	Special coordinate systems- Rectangular cartesian coordinates
	Circular cylindrical coordinates (ro,phi,z)
	Spherical polar coordinates (r,theta,phi)
	Separation of variables
	Chapter III Tensor analysis
	Introduction, definitions
	Contraction, direct product
	Quotient rule
	Pseudotensors, dual tensors
	Dyadics
	Theory of elasticity
	Lorentz covariance of Maxwell´s equations
	Noncartesian tensors, covariant differentiation
	Tensor differential operations
	Chapter IV Determinants, matrices, and group theory
	Determinants
	Matrices
	Orthogonal matrices
	Oblique coordinates
	Hermitian matrices, unitary matrices
	Diagonalization of matrices
	Eigenvectors, eigenvalues
	Introduction to group theory
	Discrete groups
	Continuous groups
	Generators
	SU(2), SU(3), and nuclear particles
	Homogeneus Lorenz group
	Chapter V Infinite series
	Fundamental concepts
	Convergence tests
	Alternating series
	Algebra of series
	Series of functions
	Taylor´s expansion
	Power series
	Elliptic integrals
	Bernoulli numbers, Euler-Maclaurin formula
	Asymptotic or semiconvergent series
	Infinite products
	Chapter VI Functions of a complex variable I
	Complex algebra
	Cauchy-Riemann conditions
	Cauchy´s Integral theorem
	Cauchy´s Integral formula
	Laurent expansion
	Mapping
	Conformal mapping
	Chapter VII Functions of a complex variable II: Calculus of residues
	Singularities
	Calculus of residues
	Dispersion relations
	The method of steepest descents
	Chapter VIII Differential equations
	Partial differential equations fo theoretical physics
	Frist-order differential equations
	Separation of variables- ordinary differential equations
	Singular points
	Series solutions -Frobenius´s Method
	A second solution
	Nonhomogeneus equation- Green´s function
	Numerical solutions
	Chapter IX Strum-Luiville theory-orthogonal functions
	Self-adjoint differential equations
	Hermitian (Self-adjoint) operators
	Gram-Schmidt orthogonalization
	Completeness of eigenfunctions
	Chapter X The gamma function (factorial function)
	Definitions, simple properties
	Digamma and polygamma functions
	Stirling´s series
	The beta function
	The incomplete gamma functions and related functions
	Chapter XI Bessel functions
	Bessel functions of the first kind, Jv(x)
	Orthogonality
	Neumann functions, Bessel functions of the second kind, Nv(x)
	Hankel functions
	Modified Bessel functions, Iv(x) and Kv(x)
	Asymptotic expansions
	Spherical Bessel functions
	Chapter XII Legendre functions
	Generating function
	Recurrence relations and special properties
	Orthogonality
	Alternate definitions of Legendre polynomials
	Associated Legendre functions
	Spherical harmonics
	Angular momentum ladder operators
	The addition theorem for spherical harmonics
	Integrals of the product of three spherical harmonics
	Legendre functions of the second kind, Qn(x)
	Vector spherical harmonics
	Chapter XIII Special functions
	Hermite functions
	Laguerre functions
	Chebyshev (Tschevyscheff) polynomials
	Chebyshev polynomials-numerical applications
	Hypergeometric functions
	Confluent hypergeometric functions
	Chapter XIV Fourier series
	General properties
	Adventages, uses of Fourier series
	Applications of Fourier series
	Properties of Fourier series
	Gibbs phenomenon
	Discrete orthogonality-discrete Fourier transform
	Chapter XV Integral transforms
	Integral transforms
	Development of the Fourier integral
	Fourier transforms-Inversion theorem
	Fourier transform of Derivates
	Convolution theorem
	Momentum representation
	Transfer functions
	Elementary Laplace transforms
	Laplace transform of derivates
	Other properties
	Convolution or Faltung theorem
	Inverse Laplace transformation
	Chapter XVI Integral equations
	Introduction
	Integral transforms, generating functions
	Neumann series, separable (degenerate) kernels
	Hilbert-Schmidt theory
	Green´s functions-one dimension
	Green´s functions-two and three dimensions
	Chapter XVII Calculus of variations
	One-dependent and one-independent variable
	Applications of the Euler equation
	Generalizations, several dependent variables
	Several independent variables
	More than one dependent, more than one independent variable
	Lagrangian multiplers
	Variation subject to constraints
	Rayleigh-Ritz variational technique
	Appendix 1 Real zeros of a function
	Appendix 2 Gaussian quadrature
	General references
	Index