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1 Vector Analysis 1

1.1 Definitions,Elementary Approach 1

1.2 Rotation ofthe Coordinate Axes 7

1.3 Scalar or Dot Product 12

1.4 Vector or Cross Product 18

1.5 Triple Scalar Product,Triple Vector Product 25

1.6 Gradient,？ 32

1.7 Divergence,？ 38

1.8 Curl,？× 43

1.9 SuccessiveApplications of？ 49

1.10 VectorIntegration 54

1.11 Gauss'Theorem 60

1.12 Stokes'Theorem 64

1.13 Potential Theory 68

1.14 Gauss'Law, Poisson's Equation 79

1.15 Dirac Delta Function 83

1.16 Helmholtz's Theorem 95

Additional Readings 101

2 Vector Analysis in Curved Coordinates and Tensors 103

2.1 Orthogonal Coordinates in R3 103

2.2 Differential Vector Operators 110

2.3 Special Coordinate Systems：Introduction 114

2.4 Circular Cylinder Coordinates 115

2.5 Spherical Polar Coordinates 123

2.6 Tensor Analysis 133

2.7 Contraction,Direct Product 139

2.8 Quotient Rule 141

2.9 Pseudotensors,Dual Tensors 142

2.10 General Tensors 151

2.11 Tensor Derivative Operators 160

Additional Readings 163

3 Determinants and Matrices 165

3.1 Determinants 165

3.2 Matrices 176

3.3 Orthogonal Matrices 195

3.4 Hermitian Matrices,Unitary Matrices 208

3.5 Diagonalization of Matrices 215

3.6 Normal Matrices 231

Additional Readings 239

4 Group Theory 241

4.1 Introduction to Group Theory 241

4.2 Generators of Continuous Groups 246

4.3 Orbital Angular Momentum 261

4.4 Angular Momentum Coupling 266

4.5 Homogeneous Lorentz Group 278

4.6 Lorentz Covariance ofMaxwell's Equations 283

4.7 Discrete Groups 291

4.8 Differential Forms 304

Additional Readings 319

5 Infinite Series 321

5.1 Fundamental Concepts 321

5.2 Convergence Tests 325

5.3 Alternating Series 339

5.4 Algebra ofSeries 342

5.5 Series ofFunctions 348

5.6 Taylor's Expansion 352

5.7 Power Series 363

5.8 Elliptic Integrals 370

5.9 Bernoulli Numbers,Euler-Maclaurin Formula 376

5.10 Asymptotic Series 389

5.11 Infinite Products 396

Additional Readings 401

6 Functions of a Complex Variable I Analytic Properties,Mapping 403

6.1 Complex Algebra 404

6.2 Cauchy-Riemann Conditions 413

6.3 Cauchy's Integral Theorem 418

6.4 Cauchy's Integral Formula 425

6.5 Laurent Expansion 430

6.6 Singularities 438

6.7 Mapping 443

6.8 Conformal Mapping 451

Additional Readings 453

7 Functions of a Complex Variable II 455

7.1 Calculus of Residues 455

7.2 Dispersion Relations 482

7.3 Method of Steepest Descents 489

Additional Readings 497

8 The Gamma Function（Factorial Function） 499

8.1 Definitions,Simple Properties 499

8.2 Digamma and Polygamma Functions 510

8.3 Stirling's Series 516

8.4 The Beta Function 520

8.5 Incomplete Gamma Function 527

Additional Readings 533

9 Differential Equations 535

9.1 Partial Differential Equations 535

9.2 First-Order Differential Equations 543

9.3 Separation of Variables 554

9.4 Singular Points 562

9.5 Series Solutions—Frobenius'Method 565

9.6 A Second Solution 578

9.7 Nonhomogeneous Equation—Green's Function 592

9.8 Heat Flow,or Diffusion,PDE 611

Additional Readings 618

10 Sturm-Liouville Theory—Orthogonal Functions 621

10.1 Self-Adjoint ODEs 622

10.2 Hermitian Operators 634

10.3 Gram-Schmidt Orthogonalization 642

10.4 Completehess of Eigenfunctions 649

10.5 Green's Function—Eigenfunction Expansion 662

Additional Readings 674

11 Bessel Functions 675

11.1 Bessel Functions of the First Kind,Jv(x) 675

11.2 Orthogonality 694

11.3 Neumann Functions 699

11.4 Hankel Functions 707

11.5 Modified Bessel Functions,Iv(x)and Kv(x) 713

11.6 Asymptotic Expansions 719

11.7 Spherical Bessel Functions 725

Additional Readings 739

12 Legendre Functions 741

12.1 Generating Function 741

12.2 Recurrence Relations 749

12.3 Orthogonality 756

12.4 Alternate Definitions 767

12.5 Associated Legendre Functions 771

12.6 Spherical Harmonics 786

12.7 Orbital Angular Momentum Operators 793

12.8 Addition Theoremfor Spherical Harmonics 797

12.9 Integrals of Three Y's 803

12.10 Legendre Functions ofthe SecondKind 806

12.11 Vector Spherical Harmonics 813

Additional Readings 816

13 More Special Functions 817

13.1Hermite Functions 817

13.2 Laguerre Functions 837

13.3 Chebyshev Polynomials 848

13.4 Hypergeometric Functions 859

13.5 Confluent Hypergeometric Functions 863

13.6 Mathieu Functions 869

Additional Readings 879

14 Fourier Series 881

14.1 General Properties 881

142 Advantages,Uses of Fourier Series 888

14.3 Applications of Fourier Series 892

14.4 Properties of Fourier Series 903

14.5 Gibbs Phenomenon 910

14.6 Discrete Fourier Transform 914

14.7 Fourier Expansions of Mathieu Functions 919

Additional Readings 929

15 Integral Transforms 931

15.1 Integral Transforms 931

15.2 Development of the Fourier Integral 936

15.3 Fourier Transforms—Inversion Theorem 938

15.4 Fourier Transform of Derivatives 946

15.5 Convolution Theorem 951

15.6 Momentum Representation 955

15.7 Transfer Functions 961

15.8 Laplace Transforms 965

15.9 Laplace Transform of Derivatives 971

15.10 Other Properties 979

15.11 Convolution（Faltungs）Theorem 990

15.12 Inverse Laplace Transform 994

Additional Readings 1003

16 Integral Equations 1005

16.1 Introduction 1005

16.2 Integral Transforms,Generating Functions 1012

16.3 Neumann Series,Separable（Degenerate）Kernels 1018

16.4 Hilbert-Schmidt Theory 1029

Additional Readings 1036

17 Calculus of Variations 1037

17.1 A Dependent and an Independent Variable 1038

17.2 Applications of the Euler Equation 1044

17.3 Several Dependent Variables 1052

17.4 Several Independent Variables 1056

17.5 Several Dependent and Independent Variables 1058

17.6 Lagrangian Multipliers 1060

17.7 Variation with Constraints 1065

17.8 Rayleigh-Ritz Variational Technique 1072

Additional Readings 1076

18 Nonlinear Methods and Chaos 1079

18.1 Introduction 1079

18.2 The Logistic Map 1080

18.3 Sensitivity to Initial Conditions and Parameters 1085

18.4 NonlinearDifferentialEquations 1088

Additional Readings 1107

19 Probability 1109

19.1 Definitions,Simple Properties 1109

19.2 Random Variables 1116

19.3 Binomial Distribution 1128

19.4 Poisson Distribution 1130

19.5 Gauss'Normal Distribution 1134

19.6 Statistics 1138

Additional Readings 1150

General References 1150

Index 1153