Mathematics for Physicists

CHAPTER I THE THEORY OF ANALYTIC FUNCTIONS 1

1. Elementary Notions of Set Theory and Analysis, 1
1.1. Sets, 1
1.2 Some Notations of Set Theory, 1
1.3 Sets of Geometrical Points, 4
1.4 The Complex Plane, 5
1.5 Functions, 8
2. Functions of a Complex Argument, 11
3. The Differential Calculus of Functions of a Complex Variable, 12
4, The Cauchy-Riemann Conditions, 14
5. The Integral Calculus of Functions of a Complex Variable, 18
6. The Darboux Inequality, 21
7. Some Definitions, 21
8. Examples of Analytic Functions, 22
8.1 Polynomials, 22
8.2 Power Series, 23
8.3 Exponential and Related Functions, 23
9. Conformal Transformations, 25
9.1 Conformal Mapping, 25
9.2 Homographic Transformations, 27
9.3 Change of Integration Variable, 29
10. A Simple Application of Conformal Mapping, 30
11, The Cauchy Theorem, 33
12. Cauchy's Integral Representation, 37
13, The Derivatives of an Analytic function, 39 .
14, Local Behavior of an Analytic Function, 42
15. The Cauchy-Liouville Theorem, 42
16. The Theorem of Morera, 43
17. Manipulations with Series of Analytic Functions, 44
18. The Taylor Series, 45
19. Poisson's Integral Representation, 47
20. The Laurent Series, 48
21. Zeros and Isolated Singular Points of Analytic Functions, 50
21.1 Zeros, 50
21.2 Isolated Singular Points, 51
22. The Calculus of Residues, 53
22.1 Theorem of Residues, 53
22.2 Evaluation of Integrals, 56
23. The Principal Value of an Integral, 60
24. Multivalued Functions; Riemann Surfaces, 65
24.1 Preliminaries, 65
24.2 The Logarithmic Function and Its Riemann Surface, 66
24.3 The Functions f(z) = 2‘ and Their Riemann Surfaces, 70
24.4 The Function f(z) = (z^2- 1)^{1/2} and Hts Riemann Surface, 71
24.5 Concluding Remarks, 73
25. Example of the Evaluation of an Integral Involving a Multivalued Function, 74
26. Analytic Continuation, 76
27. The Schwarz Reflection Principle, 80
28. Dispersion Relations, 82
29. Meromorphic Functions, 94
29.1 The Mittag-Leffler Expansion, 84
29.2 A Theorem on Meromorphic Functions, 85
30. The Fundamental) Theorem of Algebra, 86
31. The Method of Steepest Descent; Asymptotic Expansions, 87
32. The Gamma Function, 94
33. Functions of Several Complex Variables. Analytic Completion, 98


CHAPTER 2 LINEAR VECTOR SPACES 103
1. Introduction, 103
2. Definition of a Linear Vector Space, 103
3. The Scalar Product, 106 .
4. Duel Vectors and the Cauchy-Schwarz Inequality, 106
5. Real and Complex Vector Spaces, 108
6. Metric Spaces, 109
7. Linear Operators, 111
8. The Algebra of Linear Operators, 113
9. Some Special Operators, 114
10. Linear Independence of Vectors, 118
11. Eigenvalues and Eigenvectors, 119
11.1 .Ordinary Eigenvectors, 119
11.2. Generalized Eigenvectors, 121
12. Orthogonalization Theorem, 124
13. N-Dimensional Vector Space, 126
13.1 Preliminaries, 126
13.2 Representations, 127
13.3 The Representation of a Linear Operator in an N-Dimensional Space, 128
14. Matrix Algebra, 129
15. The Inverse of a Matrix, 132

16. Change of Basis in an N-Dimensional Space, 134
17. Scalars and Tensors, 135
18. Orthogonal Bases and Some Special Matrices, 139
19, Introduction to Tensor Calculus, 143
19.1 Tensors in a Real Vector Space, 143
19.2 Tensor Functions, 148
19.3 Rotations, 150
19.4 Vector Analysis in a Three-dimensional Real Space, 1$2
20. Invariant Subspaces, 154
21. The Characteristic Equation and the Hamilton-Cayley Theorem, 158
22. The Decomposition of an N-Dimensional Space, 159
23. The Canonical Form of a Matrix, 162
24. Hermitian Matrices end Quadratic Forms, 170
24.1 Diagonalization of Hermitian Matrices, £70
24.2 Quadratic Forms, 175
24.3 Simultaneous Diagonalization of Two Hermitian Matrices, 177


CHAPTER III FUNCTION SPACE, ORTHOGONAL POLYNOMIALS, AND FOURIER ANALYSIS 179


1. Introduction, 179
2. Space Of Continuous Functions, 179
3. Metric Properties of the Space of Continuous Functions, 181
4. Elementary Introduction 1o the Lebesgue Integral, 184
5. The Riesz-Fischer Theorem, 189
6. Expansions of Orthogonal Functions, 191
7. Hilbert Space, 196
8. The Generalization of the Notion of a Basis, 197
9. The Weierstrass Theorem, 199
10. The Classical Orthogonal Polynomials, 203
10.1. Introductory Remarks, 203
10.2. The Generalized Rodriguez Formula, 203
10.3. Classification of the Classical Polynomials, 205
10.4. The Recurrence Relations, 208
10.5. Differential Equations Satisfied by the Classical Polynomials, 209
10.6. The Classical Polynomials, 211
11. Trigonometrical Series, 216
11.1 An Orthogonal Basis in L^2[−π,π], 216
11.2 The Convergence Problem, 217
12. The Fourier Transform, 223
13. An Introduction to the Theory of Generalized Functions, 225
13.1 Preliminaries, 225
13.2 Definition of a Generalized Function, 227
13.3 Handling Generalized Functions, 230
13.4 The Fourier Transform of a Generalized Function, 232
13.5 The Dirac  δ Function, 235
14. Linear Operators In Infinite-Dimensional Spaces, 237
14.1 Introduction, 237
14.2 Compact Sets, 238
14.3 The Norm of a Linear Operator. Bounded Operators, 239
14.4 Sequences of Operators, 241
14.5 Completely Continuous Linear Operators, 241
14.6 The Fundamental Theorem on Completely Continuous Hermitian Operators, 244
14.7 A Convenient Notation, 249
14.8 Integral and Differential Operators, 251

CHAPTER IV DIFFERENTIAL EQUATIONS 257
Part I Ordinary Differential Equations, 257
1. Introduction, 257
2. Second-Order Differential Equations; Preliminaries, 260
3. The Transition from Linear Algebraic Systems to Linear Differential Equations—Difference Equations, 264
4. Generalized Green's Identity, 266
5. Green's Identity and Adjoint Boundary Conditions, 268
6. Second-Order Self-Adjoint Operators, 270
7. Green's Functions, 273
8. Properties of Green's Functions, 274
9. Construction and Uniqueness of Green's Functions, 277
10. Generalized Green's Function, 284
11. Second-Order Equations with In homogeneous Boundary Conditions, 285
12. The Sturm-Liouville Problem, 286
13. Eigenfunction Expansion of Green’s Functions, 268
14. Series Solutions of Linear Differential Equations of the Second Order that Depend on a Complex Variable, 291
14.1 Introduction, 291
14.2 Classification of Singularities, 291
14.3 Existence and Uniqueness of the Solution of a Differential Equation in the Neighborhood of an Ordinary Point, 292
14.4 Solution of a Differential Equation in a Neighborhood of a Regular Singular Point, 296
15. Solutions of Differential Equations Using the Method of Integral Representations, 301
15.1 General Theory, 301
15.2. Kernels of Integral Representations, 303
16. Fuchsian Equations with Three Regular Singular Points, 303
17. The Hyper-geometric Function, 306
17.1 Solutions of the Hyper-geometric Equation, 306
17.2 Integral Representations for the Hyper-geometric Function, 308
17.3 Some Further Relations Between. Thermometric Functions, 312
18. Functions Related to the Hyper-geometric Function, 314
18.1 The Jacobi Functions, 314
18.2 The Gegenbauer polynomials, 315
18.3 The Legendre Functions, 316
19, The Confluent Hyper-geometric Function, 316
20. Functions Related to the Confluent Hyper-geometric Function, 321
20.1 Parabolic Cylinder Functions; Hermite and Laguerre Polynomials, 321
20.2 The Error Function, 322
20.3 Bessel Functions, 322

ParII Introduction to Partial Differential Equations, 333
21. Preliminaries, 333
22. The Cauchy-Kovalevska Theorem, 333
23, Classification of Second-Order Quasi-linear Equations, 334
2A, Characteristics, 336
25. Boundary Conditions and Types of Equations, 341
25.1 Que-dimensional Wave Equation, 341
25.2 The One-dimensional Diffusion Equation, 344
25.3 The Two-dimensional Laplace Equation, 345
26. Multidimensional Fourier Transforms and & Function, 346
27. Green's Functions for Partial Differential Equations, 348
28. The Singular Part of the Green’s Function for Partial Differential Equations
with Constant Coefficients, 351
28.1 The General Method, 351
24.2 An Elliptic Equation: Poisson's Equation, 351 ,
28.3 A Parabolic Equation: The Diffusion Equation, 352
28.4 A Hyperbolic Equation: The Time-dependent Wave Equation, 3$3
29. Some Uniqueness Theorems, 355
29.1 Introduction, 355
29.2 The Dirichlet and Neumann Problems for the Three-dimensional La-
place Equation, 355
29.3 The One-dimensional Diffusion Equation, 358
29.4 The Initial Value Problem for the Wave Equation, 360
30. The Method of Images, 362
31. The Method of Separation of Variables, 364
31.1 Introduction, 364
31,2 The Three-dimensional Laplace Equation in Spherical Coordinates, 36S
31.3 Associated Legendre Functions and Spherical Harmonics, 366
31.4 The General Factorized Solution of the Laplace Equation in Spherical
Coordinates, 371
31.5 General Solution of Laplace's Equation with Dirichlet Boundary Conditions on a Sphere, 372
BIBLIOGRAPHY 378
INDEX 377