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TitleMathematical Methods for Physicists (7th Ed)(gnv64)
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Table of Contents
	Mathematical Methods for Physicists: A Comprehensive Guide
		To the Student
		What's New
		Pathways Through the Material
	1 Mathematical Preliminaries
		1.1 Infinite Series
			Fundamental Concepts
			Comparison Test
			Cauchy Root Test
			D'Alembert (or Cauchy) Ratio Test
			Cauchy (or Maclaurin) Integral Test
			More Sensitive Tests
			Alternating Series
			Absolute and Conditional Convergence
			Operations on Series
			Improvement of Convergence
			Rearrangement of Double Series
		1.2 Series of Functions
			Uniform Convergence
			Weierstrass M (Majorant) Test
			Abel's Test
			Properties of Uniformly Convergent Series
			Taylor's Expansion
			Power Series
			Properties of Power Series
			Uniqueness Theorem
			Indeterminate Forms
			Inversion of Power Series
		1.3 Binomial Theorem
		1.4 Mathematical Induction
		1.5 Operations on Series Expansions of Functions
		1.6 Some Important Series
		1.7 Vectors
			Basic Properties
			Dot (Scalar) Product
		1.8 Complex Numbers and Functions
			Basic Properties
			Functions in the Complex Domain
			Polar Representation
			Complex Numbers of Unit Magnitude
			Circular and Hyperbolic Functions
			Powers and Roots
		1.9 Derivatives and Extrema
			Stationary Points
		1.10 Evaluation of Integrals
			Integration by Parts
			Special Functions
			Other Methods
			Multiple Integrals
			Remarks: Changes of Integration Variables
		1.11 Dirac Delta Function
			Properties of δ(x)
			Kronecker Delta
		Additional Readings
	2 Determinants and Matrices
		2.1 Determinants
			Homogeneous Linear Equations
			Inhomogeneous Linear Equations
			Properties of Determinants
			Laplacian Development by Minors
			Linear Equation Systems
			Determinants and Linear Dependence
			Linearly Dependent Equations
			Numerical Evaluation
		2.2 Matrices
			Basic Definitions
			Addition, Subtraction
			Multiplication (by a Scalar)
			Matrix Multiplication (Inner Product)
			Unit Matrix
			Diagonal Matrices
			Matrix Inverse
			Derivatives of Determinants
			Systems of Linear Equations
			Determinant Product Theorem
			Rank of a Matrix
			Transpose, Adjoint, Trace
			Operations on Matrix Products
			Matrix Representation of Vectors
			Orthogonal Matrices
			Unitary Matrices
			Hermitian Matrices
			Extraction of a Row or Column
			Direct Product
			Functions of Matrices
		Additional Readings
	Vector Analysis
		3.1 Review of Basic Properties
		3.2 Vectors in 3-D Space
			Vector or Cross Product
			Scalar Triple Product
			Vector Triple Product
		3.3 Coordinate Transformations
			Orthogonal Transformations
			Successive Operations
		3.4 Rotations in R3
		3.5 Differential Vector Operators
			Gradient, ∇
			Divergence, ∇·
			Curl, ∇×
		3.6 Differential Vector Operators: Further Properties
			Successive Applications of ∇
			Irrotational and Solenoidal Vector Fields
			Maxwell's Equations
			Vector Laplacian
			Miscellaneous Vector Identities
		3.7 Vector Integration
			Line Integrals
			Surface Integrals
			Volume Integrals
		3.8 Integral Theorems
			Gauss' Theorem
			Green's Theorem
			Stokes' Theorem
		3.9 Potential Theory
			Scalar Potential
			Vector Potential
			Gauss' Law
			Poisson's Equation
			Helmholtz's Theorem
		3.10 Curvilinear Coordinates
			Orthogonal Coordinates in R3
			Integrals in Curvilinear Coordinates
			Differential Operators in Curvilinear Coordinates
			Circular Cylindrical Coordinates
			Spherical Polar Coordinates
			Rotation and Reflection in Spherical Coordinates
		Additional Readings
	4 Tensors and Differential Forms
		4.1 Tensor Analysis
			Introduction, Properties
			Covariant and Contravariant Tensors
			Tensors of Rank 2
			Addition and Subtraction of Tensors
			Isotropic Tensors
			Direct Product
			Inverse Transformation
			Quotient Rule
		4.2 Pseudsotensors, Dual Tensors
			Dual Tensors
		4.3 Tensors in General Coordinates
			Metric Tensor
			Covariant and Contravariant Bases
			Covariant Derivatives
			Evaluating Christoffel Symbols
			Tensor Derivative Operators
		4.4 Jacobians
			Inverse of Jacobian
		4.5 Differential Forms
			Exterior Algebra
			Complementary Differential Forms
		4.6 Differentiating Forms
			Exterior Derivatives
		4.7 Integrating Forms
			Stokes' Theorem
		Additional Readings
	5 Vector Spaces
		5.1 Vectors in Function Spaces
			Scalar Product
			Hilbert Space
			Schwarz Inequality
			Orthogonal Expansions
			Expansions and Scalar Products
			Bessel's Inequality
			Expansions of Dirac Delta Function
			Dirac Notation
		5.2 Gram-Schmidt Orthogonalization
			Orthonormalizing Physical Vectors
		5.3 Operators
			Commutation of Operators
			Identity, Inverse, Adjoint
			Basis Expansions of Operators
			Basis Expansion of Adjoint
			Functions of Operators
		5.4 Self-Adjoint Operators
		5.5 Unitary Operators
			Unitary Transformations
			Successive Transformations
		5.6 Transformations of Operators
			Nonunitary Transformations
		5.7 Invariants
		5.8 Summary—Vector Space Notation
		Additional Readings
	6 Eigenvalue Problems
		6.1 Eigenvalue Equations
			Basis Expansions
			Equivalence of Operator and Matrix Forms
		6.2 Matrix Eigenvalue Problems
			A Preliminary Example
			Another Eigenproblem
		6.3 Hermitian Eigenvalue Problems
		6.4 Hermitian Matrix Diagonalization
			Finding a Diagonalizing Transformation
			Simultaneous Diagonalization
			Spectral Decomposition
			Expectation Values
			Positive Definite and Singular Operators
		6.5 Normal Matrices
			Nonnormal Matrices
			Defective Matrices
		Additional Readings
	7 Ordinary Differential Equations
		7.1 Introduction
		7.2 First-Order Equations
			Separable Equations
			Exact Differentials
			Equations Homogeneous in x and y
			Isobaric Equations
			Linear First-Order ODEs
		7.3 ODEs with Constant Coefficients
		7.4 Second-Order Linear ODEs
			Singular Points
		7.5 Series Solutions—Frobenius' Method
			First Example—Linear Oscillator
			Expansion about x0
			Symmetry of Solutions
			A Second Example—Bessel's Equation
			Regular and Irregular Singularities
			Fuchs' Theorem
		7.6 Other Solutions
			Linear Independence of Solutions
			Number of Solutions
			Finding a Second Solution
			Series Form of the Second Solution
		7.7 Inhomogeneous Linear ODEs
			Variation of Parameters
		7.8 Nonlinear Differential Equations
			Bernoulli and Riccati Equations
			Fixed and Movable Singularities, Special Solutions
		Additional Readings
	8 Sturm-Liouville Theory
		8.1 Introduction
		8.2 Hermitian Operators
			Self-Adjoint ODEs
			Making an ODE Self-Adjoint
		8.3 ODE Eigenvalue Problems
		8.4 Variation Method
		8.5 Summary, Eigenvalue Problems
		Additional Readings
	9 Partial Differential Equations
		9.1 Introduction
			Examples of PDEs
		9.2 First-Order Equations
			More General PDEs
			More Than Two Independent Variables
		9.3 Second-Order Equations
			Classes of PDEs
			More than Two Independent Variables
			Boundary Conditions
			Nonlinear PDEs
		9.4 Separation of Variables
			Cartesian Coordinates
			Circular Cylindrical Coordinates
			Spherical Polar Coordinates
			Summary: Separated-Variable Solutions
		9.5 Laplace and Poisson Equations
		9.6 Wave Equation
			d'Alembert's Solution
		9.7 Heat-Flow, or Diffusion PDE
			Alternate Solutions
		9.8 Summary
		Additional Readings
	10 Green's Functions
		10.1 One-Dimensional Problems
			General Properties
			Form of Green's Function
			Other Boundary Conditions
			Relation to Integral Equations
		10.2 Problems in Two and Three Dimensions
			Basic Features
			Self-Adjoint Problems
			Eigenfunction Expansions
			Form of Green's Functions
		Additional Readings
	11 Complex Variable Theory
		11.1 Complex Variables and Functions
		11.2 Cauchy-Riemann Conditions
			Analytic Functions
			Derivatives of Analytic Functions
			Point at Infinity
		11.3 Cauchy's Integral Theorem
			Contour Integrals
			Statement of Theorem
			Cauchy's Theorem: Proof
			Multiply Connected Regions
		11.4 Cauchy's Integral Formula
			Morera's Theorem
			Further Applications
		11.5 Laurent Expansion
			Taylor Expansion
			Laurent Series
		11.6 Singularities
			Branch Points
			Analytic Continuation
		11.7 Calculus of Residues
			Residue Theorem
			Computing Residues
			Cauchy Principal Value
			Pole Expansion of Meromorphic Functions
			Counting Poles and Zeros
			Product Expansion of Entire Functions
		11.8 Evaluation of Definite Integrals
			Trigonometric Integrals, Range (0,2π)
			Integrals, Range -∞ to ∞
			Integrals with Complex Exponentials
			Another Integration Technique
			Avoidance of Branch Points
			Exploiting Branch Cuts
			Exploiting Periodicity
		11.9 Evaluation of Sums
		11.10 Miscellaneous Topics
			Schwarz Reflection Principle
		Additional Readings
	12 Further Topics in Analysis
		12.1 Orthogonal Polynomials
			Rodrigues Formulas
			Schlaefli Integral
			Generating Functions
			Finding Generating Functions
			Summary—Orthogonal Polynomials
		12.2 Bernoulli Numbers
			Bernoulli Polynomials
		12.3 Euler-Maclaurin Integration Formula
		12.4 Dirichlet Series
		12.5 Infinite Products
		12.6 Asymptotic Series
			Exponential Integral
			Cosine and Sine Integrals
			Definition of Asymptotic Series
		12.7 Method of Steepest Descents
			Saddle Points
			Saddle Point Method
		12.8 Dispersion Relations
			Symmetry Relations
			Optical Dispersion
			The Parseval Relation
		Additional Readings
	13 Gamma Function
		13.1 Definitions, Properties
			Infinite Limit (Euler)
			Definite Integral (Euler)
			Infinite Product (Weierstrass)
			Functional Relations
			Analytic Properties
			Schlaefli Integral
			Factorial Notation
		13.2 Digamma and Polygamma Functions
			Digamma Function
			Polygamma Function
			Maclaurin Expansion
			Series Summation
		13.3 The Beta Function
			Alternate Forms, Definite Integrals
			Derivation of Legendre Duplication Formula
		13.4 Stirling's Series
			Derivation from Euler-Maclaurin Integration Formula
			Stirling’s Formula
		13.5 Riemann Zeta Function
		13.6 Other Related Functions
			Incomplete Gamma Functions
			Incomplete Beta Function
			Exponential Integral
			Error Function
		Additional Readings
	14 Bessel Functions
		14.1 Bessel Functions of the First Kind, Jν(x)
			Generating Function for Integral Order
			Recurrence Relations
			Bessel's Differential Equation
			Integral Representation
			Zeros of Bessel Functions
			Bessel Functions of Nonintegral Order
			Schlaefli Integral
		14.2 Orthogonality
			Bessel Series
		14.3 Neumann Functions, Bessel Functions of the Second Kind
			Definition and Series Form
			Integral Representations
			Recurrence Relations
			Wronskian Formulas
			Uses of Neumann Functions
		14.4 Hankel Functions
			Contour Integral Representation of the Hankel Functions
		14.5 Modified Bessel Functions, Iν(x) and Kν(x)
			Series Solution
			Recurrence Relations for Iν
			Second Solution Kν
			Integral Representations
		14.6 Asymptotic Expansions
			Asymptotic Forms of Hankel Functions
			Expansion of an Integral Representation for Kν
			Additional Asymptotic Forms
			Properties of the Asymptotic Forms
		14.7 Spherical Bessel Functions
			Recurrence Relations
			Limiting Values
			Orthogonality and Zeros
			Modifed Spherical Bessel Functions
		Additional Readings
	15 Legendre Functions
		15.1 Legendre Polynomials
			Recurrence Formulas
			Upper and Lower Bounds for Pn(cosθ)
			Rodrigues Formula
		15.2 Orthogonality
			Legendre Series
		15.3 Physical Interpretation of Generating Function
			Expansion of 1/|r1-r2|
			Electric Multipoles
		15.4 Associated Legendre Equation
			Associated Legendre Polynomials
			Associated Legendre Functions
			Parity and Special Values
		15.5 Spherical Harmonics
			Cartesian Representations
			Overall Solutions
			Laplace Expansion
			Symmetry of Solutions
			Further Properties
		15.6 Legendre Functions of the Second Kind
			Alternate Formulations
		Additional Readings
	16 Angular Momentum
		16.1 Angular Momentum Operators
			Ladder Operators
			Summary, Angular Momentum Formulas
		16.2 Angular Momentum Coupling
			Vector Model
			Ladder Operator Construction
		16.3 Spherical Tensors
			Addition Theorem
			Spherical Wave Expansion
			Laplace Spherical Harmonic Expansion
			General Multipoles
			Integrals of Three Spherical Harmonics
		16.4 Vector Spherical Harmonics
			A Spherical Tensor
			Vector Coupling
		Additional Readings
	17 Group Theory
		17.1 Introduction to Group Theory
			Definition of a Group
			Examples of Groups
		17.2 Representation of Groups
		17.3 Symmetry and Physics
		17.4 Discrete Groups
			Other Discrete Groups
		17.5 Direct Products
		17.6 Symmetric Group
		17.7 Continuous Groups
			Lie Groups and Their Generators
			Groups SO(2) and SO(3)
			Group SU(2) and SU(2)–SO(3) Homomorphism
			Group SU(3)
		17.8 Lorentz Group
			Homogeneous Lorentz Group
			Minkowski Space
		17.9 Lorentz Covariance of Maxwell's Equations
			Lorentz Transformation of E and B
		17.10 Space Groups
		Additional Readings
	18 More Special Functions
		18.1 Hermite Functions
			Recurrence Relations
			Special Values
			Hermite ODE
			Rodrigues Formula
			Series Expansion
			Orthogonality and Normalization
		18.2 Applications of Hermite Functions
			Simple Harmonic Oscillator
			Operator Approach
			Molecular Vibrations
			Hermite Product Formula
		18.3 Laguerre Functions
			Rodrigues Formula and Generating Function
			Properties of Laguerre Polynomials
			Associated Laguerre Polynomials
		18.4 Chebyshev Polynomials
			Type II Polynomials
			Type I Polynomials
			Recurrence Relations
			Special Values
			Trigonometric Form
			Application to Numerical Analysis
		18.5 Hypergeometric Functions
			Contiguous Function Relations
			Hypergeometric Representations
		18.6 Confluent Hypergeometric Functions
			Integral Representations
			Confluent Hypergeometric Representations
			Further Observations
		18.7 Dilogarithm
			Expansion and Analytic Properties
			Properties and Special Values
		18.8 Elliptic Integrals
			Series Expansions
			Limiting Values
		Additional Readings
	19 Fourier Series
		19.1 General Properties
			Sturm-Liouville Theory
			Discontinuous Functions
			Operations on Fourier Series
			Summing Fourier Series
		19.2 Applications of Fourier Series
		19.3 Gibbs Phenomenon
			Partial Summation of Fourier Series
			Square Wave
			Calculation of Overshoot
		Additional Readings
	20 Integral Transforms
		20.1 Introduction
			Some Important Transforms
		20.2 Fourier Transform
			Fourier Integral
			Inverse Fourier Transform
			Transforms in 3-D Space
		20.3 Properties of Fourier Transforms
			Successes and Limitations
		20.4 Fourier Convolution Theorem
			Parseval Relation
			Multiple Convolutions
			Transform of a Product
			Momentum Space
		20.5 Signal-Processing Applications
			Limitations on Transfer Functions
		20.6 Discrete Fourier Transform
			Orthogonality on Discrete Point Sets
			Discrete Fourier Transform
			Fast Fourier Transform
		20.7 Laplace Transforms
			Elementary Functions
			Heaviside Step Function
			Dirac Delta Function
			Inverse Transform
		20.8 Properties of Laplace Transforms
			Transforms of Derivatives
			Change of Scale
			RLC Analog
			Derivative of a Transform
			Integration of Transforms
		20.9 Laplace Convolution Theorem
		20.10 Inverse Laplace Transform
			Bromwich Integral
		Additional Readings
	21 Integral Equations
		21.1 Introduction
			Transformation of a Differential Equation into an Integral Equation
		21.2 Some Special Methods
			Integral-Transform Methods
			Generating-Function Method
			Separable Kernel
		21.3 Neumann Series
		21.4 Hilbert-Schmidt Theory
			Symmetrization of Kernels
			Orthogonal Eigenfunctions
			Inhomogeneous Integral Equation
		Additional Readings
	22 Calculus of Variations
		22.1 Euler Equation
			Alternate Forms of Euler Equations
			Soap Film: Minimum Area
		22.2 More General Variations
			Several Dependent Variables
			Hamilton's Principle
			Hamilton's Equations
			Several Independent Variables
			Several Dependent and Independent Variables
			Relation to Physics
		22.3 Constrained Minima/Maxima
			Lagrangian Multipliers
		22.4 Variation with Constraints
			Lagrangian Formulation with Constraints
			Rayleigh-Ritz Technique
			Ground State Eigenfunction
		Additional Readings
	23 Probability and Statistics
		23.1 Probability: Definitions, Simple Properties
			Sets, Unions, and Intersections
			Counting Permutations and Combinations
		23.2 Random Variables
			Computing Discrete Probability Distributions
			Mean and Variance
			Moments of Probability Distributions
			Covariance and Correlation
			Marginal Probability Distributions
			Conditional Probability Distributions
		23.3 Binomial Distribution
		23.4 Poisson Distribution
			Relation to Binomial Distribution
		23.5 Gauss' Normal Distribution
			Limits of Poisson and Binomial Distributions
		23.6 Transformations of Random Variables
			Addition of Random Variables
			Multiplication or Division of Random Variables
			Gamma Distribution
		23.7 Statistics
			Error Propagation
			Fitting Curves to Data
			The x2 Distribution
			Student t Distribution
			Confidence Intervals
		Additional Readings
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