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