Title Mathematical Methods for Physicists (7th Ed)(gnv64) English 10.3 MB 1206
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Mathematical Methods for Physicists: A Comprehensive Guide
1.2
Contents
1.3
Preface
To the Student
What's New
Pathways Through the Material
Acknowledgments
1
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
Orthogonality
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
Logarithm
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
2
2 Determinants and Matrices
2.1 Determinants
Homogeneous Linear Equations
Inhomogeneous Linear Equations
Definitions
Properties of Determinants
Laplacian Development by Minors
Linear Equation Systems
Determinants and Linear Dependence
Linearly Dependent Equations
Numerical Evaluation
2.2 Matrices
Basic Definitions
Equality
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
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
3
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
Rotations
Orthogonal Transformations
Reflections
Successive Operations
3.4 Rotations in R3
3.5 Differential Vector Operators
Divergence, ∇·
Curl, ∇×
3.6 Differential Vector Operators: Further Properties
Successive Applications of ∇
Laplacian
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
4
4 Tensors and Differential Forms
4.1 Tensor Analysis
Introduction, Properties
Covariant and Contravariant Tensors
Tensors of Rank 2
Symmetry
Isotropic Tensors
Contraction
Direct Product
Inverse Transformation
Quotient Rule
Spinors
4.2 Pseudsotensors, Dual Tensors
Pseudotensors
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
Introduction
Exterior Algebra
Complementary Differential Forms
4.6 Differentiating Forms
Exterior Derivatives
4.7 Integrating Forms
Stokes' Theorem
5
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
Basis Expansions of Operators
Functions of 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
6
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
Degeneracy
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
7
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
Symmetry of Solutions
A Second Example—Bessel's Equation
Regular and Irregular Singularities
Fuchs' Theorem
Summary
7.6 Other Solutions
Linear Independence of Solutions
Number of Solutions
Finding a Second Solution
Series Form of the Second Solution
Summary
7.7 Inhomogeneous Linear ODEs
Variation of Parameters
7.8 Nonlinear Differential Equations
Bernoulli and Riccati Equations
Fixed and Movable Singularities, Special Solutions
8
8 Sturm-Liouville Theory
8.1 Introduction
8.2 Hermitian Operators
8.3 ODE Eigenvalue Problems
8.4 Variation Method
8.5 Summary, Eigenvalue Problems
9
9 Partial Differential Equations
9.1 Introduction
Examples of PDEs
9.2 First-Order Equations
Characteristics
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
10
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
Eigenfunction Expansions
Form of Green's Functions
11
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
Derivatives
Morera's Theorem
Further Applications
11.5 Laurent Expansion
Taylor Expansion
Laurent Series
11.6 Singularities
Poles
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
Mapping
12
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
12.8 Dispersion Relations
Symmetry Relations
Optical Dispersion
The Parseval Relation
13
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
14
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
Normalization
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
Definitions
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
Summary
14.6 Asymptotic Expansions
Asymptotic Forms of Hankel Functions
Expansion of an Integral Representation for Kν
Properties of the Asymptotic Forms
14.7 Spherical Bessel Functions
Definitions
Recurrence Relations
Limiting Values
Orthogonality and Zeros
Modifed Spherical Bessel Functions
15
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
Orthogonality
15.5 Spherical Harmonics
Cartesian Representations
Overall Solutions
Laplace Expansion
Symmetry of Solutions
Further Properties
15.6 Legendre Functions of the Second Kind
Properties
Alternate Formulations
16
16 Angular Momentum
16.1 Angular Momentum Operators
Spinors
Summary, Angular Momentum Formulas
16.2 Angular Momentum Coupling
Vector Model
16.3 Spherical Tensors
Spherical Wave Expansion
Laplace Spherical Harmonic Expansion
General Multipoles
Integrals of Three Spherical Harmonics
16.4 Vector Spherical Harmonics
A Spherical Tensor
Vector Coupling
17
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
Classes
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
18
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
Orthogonality
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
Definitions
Series Expansions
Limiting Values
19
19 Fourier Series
19.1 General Properties
Sturm-Liouville Theory
Discontinuous Functions
Symmetry
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
20
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
Limitations
Fast Fourier Transform
20.7 Laplace Transforms
Definition
Elementary Functions
Heaviside Step Function
Dirac Delta Function
Inverse Transform
20.8 Properties of Laplace Transforms
Transforms of Derivatives
Change of Scale
Substitution
RLC Analog
Translation
Derivative of a Transform
Integration of Transforms
20.9 Laplace Convolution Theorem
20.10 Inverse Laplace Transform
Bromwich Integral
21
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
22
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
Geodesics
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
23
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
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
24
Index
Numbers
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z
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