Mathematical Physics

CONTENTS

Chapter 1 Vectors, Matrices, and Coordinates
1.1 Introduction . . . . . . . . . . . . . . . . . 1
1.2 Vectors in Cartesian Coordinate Systems . . . . . . . . 1
1.3 Changes of Axes. Rotation Matrices . . . . . . . . . 4
1.4 Repeated Rotations. Matrix Multiplication . . . . . . . . 8
1.5 Skew Cartesian Systems. Matrices in General . . . . . . . 11
1.6 Scalar and Vector Fields . . . . . . . . . . . . . 14
1.7 Vector Fields in Plane . . . . . . . . . . . . . . 20
1.8 Vector Fields in Space . . . . . . . . . . . . . . 26
1.9 Curvilinear Coordinates . . . . . . . . . . . . . 34

Chapter 2 Functions of a Complex Variable
2.1 Complex Numbers . . . . . . . . . . . . . . . 44
2.2 Basic Algebra and Geometry of Complex Numbers . . . . . 45
2.3 De Moivre Formula and the Calculation of Roots . . . . . . 48
2.4 Complex Functions. Euler’s Formula . . . . . . . . . 49
2.5 Applications of Euler’s Formula . . . . . . . . . . . 51
2.6 Multivalued Functions and Riemann Surfaces . . . . . . . 54
2.7 Analytic Functions. Cauchy Theorem . . . . . . . . . 58
2.8 Other Integral Theorems. Cauchy Integral Formula . . . . . 62
2.9 Complex Sequences and Series . . . . . . . . . . . 66
2.10 Taylor and Laurent Series . . . . . . . . . . . . . 71
2.11 Zeros and Singularities . . . . . . . . . . . . . . 78
2.12 The Residue Theorem and its Applications . . . . . . . . 33
2.13 Conformal Mapping by Analytic Functions . . . . . . . . 97
2.14 Complex Sphere and Point at Infinity . . . . . . . . . 102
2.15 Integral Representations . . . . . . . . . . . . . 104

Chapter 3 Linear Differential Equations of Second Order
3.] General Introduction. The Wronskian . . . . . . . . . 123
3.2 General Solution of The Homogeneous Equation . . . . . . 125
3.3 The Non-homogeneous Equation. Variation of Constants . . . . 126
3.4 Power Series Solutions . . . . . . . . . . . . . . 128
3.5 The Frobenius Method . . . . . . . . . . . . . . 130
3.6 Some other Methods of Solution . . . . . . . . . . . 147

Chapter 4 Fourier Series
4.1 Trigonometric Series . . . . . . . . . . . . . . 154
4.2 Definition of Fourier Series . . . . . . . . . . . . 155
4.3 Examples of Fourier Series . . . . . . . . . . . . 157
4.4 Parity Properties. Sine and Cosine Series . . . . . . . . 161
4.5 Complex Form of Fourier Series . . . . . . . . . . . 165
4.6 Point-wise Convergence of Fourier Series . . . . . . . . 167
4.7 Convergence in the Mean . . . . . . . . . . . . . 168
4.8 Applications of Fourier Series . . . . . . . . . . . . 172

Chapter 5 The Laplace Transformation
5.1 Operational Calculus . . . . . . . . . . . . . . 179
5.2 The Laplace Integral . . . . . . . . . . . . . . 180
5.3 Basic Properties of Laplace Transform . . . . . . . . . 184
5.4 The Inversion Problem . . . . . . . . . . . . . . 187
5.5 The Rational Fraction Decomposition . . . . . . . . . 189
5.6 The Convolution The0rem . . . . . . . . . . . . . 194
5.7 Additional Properties of Laplace Transform . . . . . . . 200
5.8 Periodic Functions. Rectification . . . . . . . . . . . 204
5.9 The Mellin Inversion Integral . . . . . ‘. . . . . . . 206
5.10 Applications of Laplace Transforms . . .. . . . . . . . 210

Chapter 6 Concepts of the Theory of Distributions
6.1 Strongly Peaked Functions and The Dirac Delta Function . . . 221
6.2 Delta Sequences . . . . . . . . . . . . . . . . 223
6.3 The δ-Calculus . . . . . . . . . . . . . . . . 226
6.4 Representations of Delta Functions . . . . . . . . . . 229
6.5 Applications of The δ-Calculus . . . . . . . . . . . 232
6.6 Weak Convergence . . . . . . . . . . . . . . . 236
6.? Correspondence of Functions and Distributions . . . . . . 240
6.8 Properties of Distributions . . . . . . . . . . . .. . 245
6.9 Sequences and Series of Distributions . . . . . . . . . 250
6.10 Distributions in N dimensions . . . . . . . . . . . . 257

Chapter 7 Fourier Transforms '
7.1 Representations of a Function . . . . . . . . . . . 260
7.2 Examples of Fourier Transformations . . . . . . . . . 262
7.3 Properties of Fourier Transforms . . . . . . . . . . . 266
7.4 Fourier Integral Theorem . . . . . . . . . . . . . . 269
7.5 Fourier Transforms of Distributions . . . . . . . . . . 271
7.6 Fourier Sine and Cosine Transforms . . . . . . . . . . 223
7.7 Applications of Fourier Transforms. The Principle of Causality . . 276

Chapter 8 Partial Differential Equations
8.1 The Stretched String. Wave Equation . . . . . . . . . 287
8.2 The Method of Separation of Variables . . . . . . . . . 291
8.3 Laplace and Poisson Equations . . . . . . . . . . . 295
8.4 The Diffusion Equation . . . . . . . . . . . . . 297
8.5 Use of Fourier and Laplace Transforms . . . . . . . . . 299
8.6 The Method of Eigenfunction Expansions and Finite Transforms . 304
8.7 Continuous Eigenvalue Spectrum . . . . . . . . . . . 308
8.8 Vibrations of a Membrane. Degeneracy . . . . . . . . . 313
8.9 Propagation of Sound. Helmholtz Equation . . . . . . . 319

Chapter 9 Special Functions
9.1 Cylindrical and Spherical Coordinates . . . . . . . . . 332
9.2 The Common Boundary-Value Problems . . . . . . . . 334
9.3 The Sturm-Liouville Problem . . . . . . . . . . . . 337
9.4 Self-Adjoint Operators . . . . . . . . . . . . . . 340
9.5 Legendre Polynomials . . . . . . . . . . . . . . 342
9.6 Fourier-Legendre Series . . . . . . . . . . . . . 350
9.7 Bessel Functions . . . . . . . . . . . . . . . 355
9.8 Associated Legendre Functions and Spherical Harmonics . . . . 372
9.9 Spherical Bessel functions . . . . . . . . . . . . . 381
9.10 Neumann Functions . . . . . . . . . . . . . . 388
9.11 Modified Bessel Functions . . . . . . . . . . . . . 394

Chapter 10 Finite-Dimensional Linear Spaces
10.1 Oscillations of Systems with Two Degrees of Freedom . . . . 405
10.2 Normal Coordinates and Linear Transformation . . . . . . 411
10.3 Vector Spaces, Bases, Coordinates . . . . . . . . . . 419
10.4 Linear Operators, Matrices, Inverses . . . . . . . . . . 424
10.5 Changes of Basis . . . . . . . . . . . . . . . 433
10.6 Inner Product. Orthogonality. Unitary Operators . . . . . . 437
10.7 The Metric. Generalized Orthogonality. . . . . . . . . . 441
10.8 Eigenvalue Problems. Diagonalization . . . . . . . . . 443
10.9 Simultaneous Diagonalization . . . . . . . . . . . . 451

Chapter 11 Infinite-Dimensional Vector Spaces
11.1 Spaces of Functions . . . . . . . . . . . . . . 463
11.2 The Postulates of Quantum Mechanics . . . . . . . . . 467
11.3 The Harmonics Oscillator . . . . . . . . . . . . . 471
11.4 Matrix Representations of Linear Operators . . . . . . . 476
11.5 Algebraic Methods of Solution . . . . . . . . . . . 483
11.6 Bases with Generalized Orthogonality . . . . . . . . . 488
11.7 Stretched String with a Discrete Mass in the Middle . . . . . 492
11.8 Applications of Eigenfunctions . . . . . . . . . . . 495

Chapter 12 Green’s Functions
12.1 Introduction . . . . . . . . . . . . . . . . . 503
12.2 Green’s Function for the Sturm-Liouville Operator . . . . . 508
12.3 Series Expansions for G(X,E) . . . . . , . . . . . . 514
12.4 Green’s Functions in Two Dimensions . . . . . . . . . 520
12.5 Green’s Functions for Initial Conditions . . . . . . . . . 523
12.6 Green’s Functions with Reflection Properties . . . . . . . 527
12.7 Green’s Functions for Boundary Conditions . . . . . . . 531
12.8 The Green’s Function Method . . . . . . . . . . . 536
12.9 A Case of Continuous Spectrum . . . . . . . . . . . 543

Chapter 13 Variational Methods
13.1 The Brachistochrone Problem . . . . . . . . . . . . 553
13.2 The Euler-Lagrange Equation . . . . . . . . . . . . 554
13.3 Hamilton’s Principle . . . . . . . . . . . . . . 560
13.4 Problems involving Sturm-Liouville Operators . . . . . . . 562
13.5 The Rayleigh-Ritz Method . . . . . . . . . . . . 565
13.6 Variational Problems with Constraints . . . . . . . . . 567
13.7 Variational Formulation of Eigenvalue Problems . . . . . . 573
13.3 Variational Problems in Many Dimensions . . . . . . . . 577
13.9 Formulation of Eigenvalue Problems by The Ratio Method . . . 581

Chapter 14 Travelling Waves. Radiation, Scattering
14.1 Motion of Infinite Stretched String . . . . . . . . . . 589
14.2 Propagation of Initial Conditions . . . . . . . . . . . 592
14.3 Semi-infinite String. Use of Symmetry Properties . . . . . . 595
14.4 Energy and Power Flow in a Stretched String . . . . . . . 599
14.5 Generation of Waves in a Stretched String . . . . . . . . 603
14.6 Radiation of Sound from a Pulsating Sphere . . . . . . . 611
14.7 The Retarded Potential . . . . . . . . . . . . . 619
14.8 Travailing Waves in Inhomogeneous Media . . . . . . . 624
14.9 Scattering Amplitudes and Phase Shifts . . . . . . . . . 628
14.10 Scattering in Three Dimensions. Partial Wave Analysis . . . . 633

Chapter 15 Perturbation Methods
15.1 Introduction . . . . . . . . . . . . . . . . . 644
15.2 The Born Approximation . . . . . . . . . . . . . . 647
15.3 Perturbation of Eigenvalue Problems . . . . . . . . . . 650
15.4 First-Order Rayleigh-Schrödinger Theory . . . . . . . . 653
15.5 The Second-Order Non-degenerate Theory . . . . . . . . 658
15.6 The Case of Degenerate Eigenvalues . . . . . . . . . . 665

Chapter 16 Tensors
16.1 Introduction . . . . . . . . . . . . . . . . . 671
16.2 Two-Dimensional Stresses . . . . . . . . . . . . . 672
16.3 Cartesian Tensors . . . . . . . . . . . . . . . 676
16.4 Algebra of Cartesian Tensors . . . . . . . . . . . . 681
16.5 Kronecker and Levi-Civita Tensors. Pseudo-tensors . . . . . 684
16.6 Derivatives of Tensors. Strain Tensor and Hooke’s Law . . . . 637
16.7 Tensors in Skew Cartesian Frames. Covariant and
Contravariant Representations . . . . . . . . . . . 696
16.8 General Tensors . . . . . . . . . . . . . . . . 700
16.9 Algebra of General Tensors. Relative Tensors . . . . . . . 705
16.10 The Covariant Derivative . . . . . . . . . . . . . 711
16.11 Calculus of General Tensors . . . . . . . . . . . . 715

Index. . . . . . . . . . . . . . . . . . . 727