阿夫肯著的《物理学家用的数学方法（第7版）（精）》是为具有研究生水平的读者编写的一部入门性工具书，语言简练，结构流畅，可读性很强，很受读者欢迎，本书是第7版。本版全面介绍了物理学中常用数学方法，内容涉及物理学中用到的数学内容，包括矢量／张量分析，矩阵，群论，数列与复变函数，各种特殊函数，微分方程，傅里叶分析与积分变换，非线性方法，变分法和概率论等诸多领域，是从事物理学研究和教学人员的案头必备书。

读者对象：物理、数学及相关专业的研究生和科教工作者。

Preface

1 Mathematical Preliminaries

 1.1 InfiniteSeries

 1.2 Series ofFunctions

 1.3 Binomial Theorem

 1.4 Mathematical Induction

 1.5 Operations on Series Expansions of Functions

 1.6 Some Important Series

 1.7 Vectors

 1.8 Complex Numbers and Functions

 1.9 Derivatives andExtrema

 1.10 Evaluation oflntegrals

 1.1 I Dirac Delta Function

 AdditionaIReadings

2 Determinants and Matrices

 2.1 Determinants

 2.2 Matrices

 AdditionaI Readings

3 Vector Analysis

 3.1 Review ofBasic Properties

 3.2 Vectors in 3-D Space 

 3.3 Coordinate Transformations

 3.4 Rotations in IR3

 3.5 Differential Vector Operators

 3.6 Differential Vector Operators: Further Properties

 3.7 Vectorlntegration

 3.8 Integral Theorems

 3.9 PotentiaITheory

 3.10 Curvilinear Coordinates

 AdditionaIReadings

4 Tensors and Differential Forms

 4.1 TensorAnalysis

 4.2 Pseudotensors, Dual Tensors

 4.3 Tensors in General Coordinates

 4.4 Jacobians

 4.5 DifferentialForms

 4.6 DifferentiatingForms

 4.7 IntegratingForms

 AdditionalReadings

5 Vector Spaces

 5.1 Vectors in Function Spaces

 5.2 Gram-Schmidt Orthogonalization

 5.3 Operators

 5.4 SelfAdjointOperators

 5.5 Unitaty Operators

 5.6 Transformations of Operators

 5.7 Invariants

 5.8 Summary-Vector Space Notation

 AdditionaIReadings

6 Eigenvalue Problems

 6.1 EigenvalueEquations

 6.2 Matrix Eigenvalue Problems

 6.3 Hermitian Eigenvalue Problems

 6.4 Hermitian Matrix Diagonalization

 6.5 NormaIMatrices

 AdditionalReadings

7 Ordinary DifTerential Equations

 7.1 Introduction

 7.2 First-OrderEquations

 7.3 ODEs with Constant Coefficients

 7.4 Second-Order Linear ODEs

 7.5 Series Solutions-Frobenius ' Method

 7.6 OtherSolutions

 7.7 Inhomogeneous Linear ODEs

 7.8 Nonlinear Differential Equations

 Additional Readings

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

 Additional Readings

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 Equation

 9.7 Heat-Flow, or Diffusion PDE

 9.8 Summary

 Additional Readings

10 Green's Functions

 10.1 One-Dimensional Problems

 10.2 Problems in Two and Three Dimensions

 Additional Readings

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 

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 Dispersion Relations

 Additional Readings

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 Related Functions

 Additional Readings

14 Bessel Functions

 14.1 Bessel Functions of the First Kind, ,Iv (x)

 14.2 Orthogonality

 14.3 Neumann Functions, Bessel Functions of the Second Kind

 14.4 Hankel Functions

 14.5 Modified Bessel Functions, Iv (x) and Kv (x)

 14.6 Asymptotic Expansions

 14.7 Spherical Bessel Functions

 Additional Readings

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

16 Angular Momentum

 16.1 Angular Momentum Operators

 16.2 Angular Momentum Coupling

 16.3 Spherical Tensors

 16.4 Vector Spherical Harmonics

 Additional Readings 

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 Symmetric Group

 17.7 Continuous Groups

 17.8 Lorentz Group

 17.9 Lorentz Covariance of Maxwell's Equations

 17.10 Space Groups

 Additional Readings

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

19 Fourier Series

 19.1 General Properties

 19.2 Applications of Fourier Series

 19.3 Gibbs Phenomenon 

 Additional Readings

20 Integral Transforms

 20.1 Introduction

 20.2 Fourier Transform

 20.3 Properties of Fourier Transforms

 20.4 Fourier Convolution Theorem

 20.5 Signal-Processing Applications

 20.6 Discrete Fourier Transform

 20.7 Laplace Transforms

 20.8 Properties of Laplace Transforms

 20.9 Laplace Convolution Theorem

 20.10 Inverse Laplace Transform

 Additional Readings

21 Integral Equations

 21.1 Introduction

 21.2 Some Special Methods

 21.3 Neumann Series

 21.4 Hilbert-Schmidt Theory

 Additional Readings

 17.4 Discrete Groups

 17.5 Direct Products

 17.6 Symmetric Group

 17.7 Continuous Groups

 17.8 Lorentz Group

 17.9 Lorentz Covariance of Maxwell's Equations

 17.10 Space Groups

 Additional Readings

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

19 Fourier Series

 19.1 General Properties

 19.2 Applications of Fourier Series

 19.3 Gibbs Phenomenon 

 Additional Readings

20 Integral Transforms

 20.1 Introduction

 20.2 Fourier Transform

 20.3 Properties of Fourier Transforms

 20.4 Fourier Convolution Theorem

 20.5 Signal-Processing Applications

 20.6 Discrete Fourier Transform

 20.7 Laplace Transforms

 20.8 Properties of Laplace Transforms

 20.9 Laplace Convolution Theorem

 20.10 Inverse Laplace Transform

 Additional Readings

21 Integral Equations

 21.1 Introduction

 21.2 Some Special Methods

 21.3 Neumann Series

 21.4 Hilbert-Schmidt Theory

 Additional Readings

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

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' Normal Distribution

 23.6 Transformations of Random Variables 

 23.7 Statistics

 Additional Readings

Index