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