Graduate Mathematical Physics: With MATHEMATICA Supplements Book

Preface     V
Note to the Reader     XV
Analytic Functions     1
Complex Numbers     1
Motivation and Definitions     1
Triangle Inequalities     4
Polar Representation     4
Argument Function     5
Take Care with Multivalued Functions     8
Functions as Mappings     13
Mapping: w = e[superscript z]     14
Mapping: w = Sin[z]     16
Elementary Functions and Their Inverses     17
Exponential and Logarithm     17
Powers     18
Trigonometric and Hyperbolic Functions     19
Standard Branch Cuts     20
Sets, Curves, Regions and Domains     21
Limits and Continuity     22
Differentiability     23
Cauchy-Riemann Equations     23
Differentiation Rules     25
Properties of Analytic Functions     26
Cauchy-Goursat Theorem     28
Simply Connected Regions     28
Proof     28
Example     31
Cauchy Integral Formula     32
Integration Around Nonanalytic Regions     32
Cauchy Integral Formula     33
Example: Yukawa Field     34
Derivatives of Analytic Functions     36
Morera's Theorem     37
Complex Sequences and Series     37
Convergence Tests     37
Uniform Convergence     37
Derivatives and Taylor Series for Analytic Functions     41
Taylor Series     41
Cauchy Inequality     44
Liouville's Theorem     44
Fundamental Theorem of Algebra     44
Zeros of Analytic Functions     45
Laurent Series     45
Derivation     45
Example     47
Classification of Singularities     48
Poles and Residues     49
Meromorphic Functions     51
Pole Expansion     51
Example: Tan[z]     53
Product Expansion     54
Example: Sin[z]     54
Problems for Chapter 1     55
Integration     65
Introduction     65
Good Tricks     65
Parametric Differentiation     65
Convergence Factors     66
Contour Integration     66
Residue Theorem     66
Definite Integrals of the Form [Characters not reproducible] [sin [Theta], cos [Theta]] d[Theta]     67
Definite Integrals of the Form [Characters not reproducible]     69
Fourier Integrals     70
Custom Contours     72
Isolated Singularities on the Contour     73
Removable Singularity     73
Cauchy Principal Value     75
Integration Around a Branch Point     77
Reduction to Tabulated Integrals     79
Example: [Characters not reproducible]     80
Example: The Beta Function     81
Example: [Characters not reproducible]     81
Integral Representations for Analytic Functions     82
Using Mathematica to Evaluate Integrals     86
Symbolic Integration     86
Numerical Integration     88
Further Information     89
Problems for Chapter 2     89
Asymptotic Series     95
Introduction     95
Method of Steepest Descent     96
Example: Gamma Function     99
Partial Integration     101
Example: Complementary Error Function     102
Expansion of an Integrand     104
Example: Modified Bessel Function     105
Problems for Chapter 3     108
Generalized Functions     111
Motivation     111
Properties of the Dirac Delta Function     113
Other Useful Generalized Functions     115
Heaviside Step Function     115
Derivatives of the Dirac Delta Function     116
Green Functions     118
Multidimensional Delta Functions     120
Problems for Chapter 4     122
Integral Transforms     125
Introduction     125
Fourier Transform     126
Motivation     126
Definition and Inversion     128
Basic Properties     130
Parseval's Theorem     131
Convolution Theorem     132
Correlation Theorem     133
Useful Fourier Transforms     134
Fourier Transform of Derivatives     138
Summary     139
Green Functions via Fourier Transform     139
Example: Green Function for One-Dimensional Diffusion     139
Example: Three-Dimensional Green Function for Diffusion Equations     141
Example: Green Function for Damped Oscillator     143
Operator Method     147
Cosine or Sine Transforms for Even or Odd Functions      147
Discrete Fourier Transform     148
Sampling     149
Convolution     153
Temporal Correlation     156
Power Spectrum Estimation     160
Laplace Transform     165
Definition and Inversion     165
Laplace Transforms for Elementary Functions     167
Laplace Transform of Derivatives     170
Convolution Theorem     171
Summary     173
Green Functions via Laplace Transform     173
Example: Series RC Circuit     174
Example: Damped Oscillator     175
Example: Diffusion with Constant Boundary Value     176
Problems for Chapter 5     181
Analytic Continuation and Dispersion Relations     191
Analytic Continuation     191
Motivation     191
Uniqueness     192
Reflection Principle     194
Permanence of Algebraic Form     195
Example: Gamma Function     195
Dispersion Relations     196
Causality     196
Oscillator Model     200
Kramers-Kronig Relations     203
Sum Rules     205
Hilbert Transform      207
Spreading of a Wave Packet     208
Solitons     212
Problems for Chapter 6     216
Sturm-Liouville Theory     223
Introduction: The General String Equation     223
Hilbert Spaces     226
Schwartz Inequality     229
Gram-Schmidt Orthogonalization     230
Properties of Sturm-Liouville Systems     232
Self-Adjointness     232
Reality of Eigenvalues and Orthogonality of Eigenfunctions     233
Discreteness of Eigenvalues     235
Completeness of Eigenfunctions     235
Parseval's Theorem     237
Reality of Eigenfunctions     238
Interleaving of Zeros     238
Comparison Theorems     240
Green Functions     242
Interface Matching     242
Eigenfunction Expansion of Green Function     246
Example: Vibrating String     252
Perturbation Theory     253
Example: Bead at Center of a String     255
Variational Methods     256
Example: Vibrating String     259
Problems for Chapter 7     260
Legendre and Bessel Functions     269
Introduction      269
Legendre Functions     270
Generating Function for Legendre Polynomials     270
Series Representation and Rodrigues' Formula     274
Schlafli's Integral Representation     275
Legendre Expansion     275
Associated Legendre Functions     277
Spherical Harmonics     281
Multipole Expansion     282
Addition Theorem     283
Legendre Functions of the Second Kind     286
Relationship to Hypergeometric Functions     287
Analytic Structure of Legendre Functions     289
Bessel Functions     291
Cylindrical     291
Hankel Functions     297
Neumann Functions     299
Modified Bessel Functions     303
Spherical Bessel Functions     305
Fourier-Bessel Transform     308
Example: Fourier-Bessel Expansion of Nuclear Charge Density     311
Summary     312
Legendre Functions     313
Associated Legendre Functions     314
Spherical Harmonics     315
Cylindrical Bessel Functions     315
Spherical Bessel Functions     316
Fourier-Bessel Expansions      318
Problems for Chapter 8     318
Boundary-Value Problems     327
Introduction     327
Laplace's Equation in Box with Specified Potential on one Side     328
Green Function for Grounded Box     329
Green's Theorem for Electrostatics     332
Separable Coordinate Systems     335
Spherical Polar Coordinates     336
Cylindrical Coordinates     338
Spherical Expansion of Dirichlet Green Function for Poisson's Equation     339
Example: Multipole Expansion for Localized Charge Distribution     342
Example: Point Charge Near Grounded Conducting Sphere     342
Example: Specified Potential on Surface of Empty Sphere     344
Example: Charged Ring at Center of Grounded Conducting Sphere     346
Magnetic Field of Current Loop     346
Inhomogeneous Wave Equation     349
Spatial Representation of Time-Independent Green Function     349
Partial-Wave Expansion     352
Momentum Representation of Time-Independent Green Function     354
Retarded Green Function     356
Lippmann-Schwinger Equation     358
Problems for Chapter 9     360
Group Theory     369
Introduction      369
Finite Groups     370
Definitions     370
Equivalence Classes     373
Subgroups     374
Homomorphism     375
Direct Products     376
Representations     376
Definitions     376
Example: Vibrating triangle     380
Orthogonality Theorem     383
Character     388
Example: Character table for symmetries of a square     393
Example: Vibrational eigenvalues of square     397
Direct-Product Representations     400
Eigenfunctions     403
Wigner-Eckart Theorem     404
Continuous Groups     405
Definitions     405
Transformation of Functions     408
Generators     409
Example: Linear coordinate transformations in one dimension     413
Example: SO(2)     414
Example: SU(2)     416
Example: SO(3)     417
Total angular momentum     418
Transformation of Operators     419
Invariant Functions     420
Lie Algebra     422
Definitions     422
Example: SU(2)      423
Orthogonality Relations for Lie Groups     425
Quantum Mechanical Representations of the Rotation Group     428
Generators and Commutation Relations     428
Euler Parametrization     430
Homomorphism Between SU(2) and SO(3)     431
Irreducible Representations of SU(2)     433
Orthogonality Relations for Rotation Matrices     437
Coupling of Angular Momenta     438
Spherical Tensors     442
Unitary Symmetries in Nuclear and Particle Physics     445
Problems for Chapter 10     447
Bibliography     459
Index     461