MATHEMATICAL METHODS FOR PHYSICISTS A COMPREHENSIVE GUIDE SEVENTH EDITIONpdf电子书版本下载

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1 Mathematical Preliminaries 1

1.1 Infinite Series 1

1.2 Series of Functions 21

1.3 Binomial Theorem 33

1.4 Mathematical Induction 40

1.5 Operations on Series Expansions of Functions 41

1.6 Some Important Series 45

1.7 Vectors 46

1.8 Complex Numbers and Functions 53

1.9 Derivatives and Extrema 62

1.10 Evaluation of Integrals 65

1.11 Dirac Delta Function 75

Additional Readings 82

2 Determinants and Matrices 83

2.1 Determinants 83

2.2 Matrices 95

Additional Readings 121

3 Vector Analysis 123

3.1 Review of Basic Properties 124

3.2 Vectors in 3-D Space 126

3.3 Coordinate Transformations 133

3.4 Rotations in IR 3 139

3.5 Differential Vector Operators 143

3.6 Differential Vector Operators： Further Properties 153

3.7 Vector Integration 159

3.8 Integral Theorems 164

3.9 Potential Theory 170

3.10 Curvilinear Coordinates 182

Additional Readings 203

4 Tensors and Differential Forms 205

4.1 Tensor Analysis 205

4.2 Pseudotensors， Dual Tensors 215

4.3 Tensors in General Coordinates 218

4.4 Jacobians 227

4.5 Differential Forms 232

4.6 Differentiating Forms 238

4.7 Integrating Forms 243

Additional Readings 249

5 Vector Spaces 251

5.1 Vectors in Function Spaces 251

5.2 Gram-Schmidt Orthogonalization 269

5.3 Operators 275

5.4 Self-Adjoint Operators 283

5.5 Unitary Operators 287

5.6 Transformations of Operators 292

5.7 Invariants 294

5.8 Summary—Vector Space Notation 296

Additional Readings 297

6 Eigenvalue Problems 299

6.1 Eigenvalue Equations 299

6.2 Matrix Eigenvalue Problems 301

6.3 Hermitian Eigenvalue Problems 310

6.4 Hermitian Matrix Diagonalization 311

6.5 Normal Matrices 319

Additional Readings 328

7 Ordinary Differential Equations 329

7.1 Introduction 329

7.2 First-Order Equations 331

7.3 ODEs with Constant Coefficients 342

7.4 Second-Order Linear ODEs 343

7.5 Series Solutions Frobenius’ Method 346

7.6 Other Solutions 358

7.7 Inhomogeneous Linear ODEs 375

7.8 Nonlinear Differential Equations 377

Additional Readings 380

8 Sturm-Liouville Theory 381

8.1 Introduction 381

8.2 Hermitian Operators 384

8.3 ODE Eigenvalue Problems 389

8.4 Variation Method 395

8.5 Summary， Eigenvalue Problems 398

Additional Readings 399

9 Partial Differential Equations 401

9.1 Introduction 401

9.2 First-Order Equations 403

9.3 Second-Order Equations 409

9.4 Separation of Variables 414

9.5 Laplace and Poisson Equations 433

9.6 Wave Equation 435

9.7 Heat-Flow， or Diffusion PDE 437

9.8 Summary 444

Additional Readings 445

10 Green’s Functions 447

10.1 One-Dimensional Problems 448

10.2 Problems in Two and Three Dimensions 459

Additional Readings 467

11 Complex Variable Theory 469

11.1 Complex Variables and Functions 470

11.2 Cauchy-Riemann Conditions 471

11.3 Cauchy’s Integral Theorem 477

11.4 Cauchy’s Integral Formula 486

11.5 Laurent Expansion 492

11.6 Singularities 497

11.7 Calculus of Residues 509

11.8 Evaluation of Deffinite Integrals 522

11.9 Evaluation of Sums 544

11.10 Miscellaneous Topics 547

Additional Readings 550

12 Further Topics in Analysis 551

12.1 Orthogonal Polynomials 551

12.2 Bernoulli Numbers 560

12.3 Euler-Maclaurin Integration Formula 567

12.4 Dirichlet Series 571

12.5 Infinite Products 574

12.6 Asymptotic Series 577

12.7 Method of Steepest Descents 585

12.8 Dispersion Relations 591

Additional Readings 598

13 Gamma Function 599

13.1 Definitions， Properties 599

13.2 Digamma and Polygamma Functions 610

13.3 The Beta Function 617

13.4 Stirling’s Series 622

13.5 Riemann Zeta Function 626

13.6 Other Related Functions 633

Additional Readings 641

14 Bessel Functions 643

14.1 Bessel Functions of the First Kind， Jv （x） 643

14.2 Orthogonality 661

14.3 Neumann Functions， Bessel Functions of the Second Kind 667

14.4 Hankel Functions 674

14.5 Modified Bessel Functions， Iv （x） and Kv （x） 680

14.6 Asymptotic Expansions 688

14.7 Spherical Bessel Functions 698

Additional Readings 713

15 Legendre Functions 715

15.1 Legendre Polynomials 716

15.2 Orthogonality 724

15.3 Physical Interpretation of Generating Function 736

15.4 Associated Legendre Equation 741

15.5 Spherical Harmonics 756

15.6 Legendre Functions of the Second Kind 766

Additional Readings 771

16 Angular Momentum 773

16.1 Angular Momentum Operators 774

16.2 Angular Momentum Coupling 784

16.3 Spherical Tensors 796

16.4 Vector Spherical Harmonics 809

Additional Readings 814

17 Group Theory 815

17.1 Introduction to Group Theory 815

17.2 Representation of Groups 821

17.3 Symmetry and Physics 826

17.4 Discrete Groups 830

17.5 Direct Products 837

17.6 Symmetric Group 840

17.7 Continuous Groups 845

17.8 Lorentz Group 862

17.9 Lorentz Covariance of Maxwell’s Equations 866

17.10 Space Groups 869

Additional Readings 870

18 More Special Functions 871

18.1 Hermite Functions 871

18.2 Applications ofHermite Functions 878

18.3 Laguerre Functions 889

18.4 Chebyshev Polynomials 899

18.5 Hypergeometric Functions 911

18.6 Confluent Hypergeometric Functions 917

18.7 Dilogarithm 923

18.8 Elliptic Integrals 927

Additional Readings 932

19 Fourier Series 935

19.1 General Properties 935

19.2 Applications of Fourier Series 949

19.3 Gibbs Phenomenon 957

Additional Readings 962

20 Integral Transforms 963

20.1 Introduction 963

20.2 Fourier Transform 966

20.3 Properties of Fourier Transforms 980

20.4 Fourier Convolution Theorem 985

20.5 Signal-Processing Applications 997

20.6 Discrete Fourier Transform 1002

20.7 Laplace Transforms 1008

20.8 Properties of Laplace Transforms 1016

20.9 Laplace Convolution Theorem 1034

20.10 Inverse Laplace Transform 1038

Additional Readings 1045

21 Integral Equations 1047

21.1 Introduction 1047

21.2 Some Special Methods 1053

21.3 Neumann Series 1064

21.4 Hilbert-Schmidt Theory 1069

Additional Readings 1079

22 Calculus of Variations 1081

22.1 Euler Equation 1081

22.2 More General Variations 1096

22.3 Constrained Minima/Maxima 1107

22.4 Variation with Constraints 1111

Additional Readings 1124

23 Probability and Statistics 1125

23.1 Probability： Definitions， Simple Properties 1126

23.2 Random Variables 1134

23.3 Binomial Distribution 1148

23.4 Poisson Distribution 1151

23.5 Gauss’ Normal Distribution 1155

23.6 Transformations ofRandom Variables 1159

23.7 Statistics 1165

Additional Readings 1179

Index 1181