Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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James J. Kelly
Graduate Mathematical Physics
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JamesJ.Kelly
Handbook of Time Series Analysis
With MATHEMATICA supplements
WILEY-VCH Verlag GmbH & Co. KGaA
The Author
Prof. James J. Kelly
University of Maryland
Dept. of Physics
jjkelly@umd.edu
For a Solutions Manual, lecturers should
contact the editorial department at
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stating their affiliation and the course in which
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©2006 WILEY-VCH Verlag GmbH & Co. KGaA,
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ISBN-13: 978-3-527-40637-1
ISBN-10: 978-3-527-40637-1
Preface
This textbook is intended to serve a course on mathematical methods of physics that is
often taken by graduate students in their first semester or by undergraduates in their senior
year. I believe the most important topic for first-year graduate students in physics is the
theory of analytic functions. Some students may have had a brief exposure to that subject
as undergraduates, but few are adequately prepared to apply such methods to physics prob-
lems. Therefore, I start with the theory of analytic functions and practically all subsequent
material is based upon it. The primary topics include: theory of analytic functions, integral
transforms, generalized functions, eigenfunction expansions, Green functions, boundary-
value problems, and group theory. This course is designed to prepare students for advanced
treatments of electromagnetic theory and quantum mechanics, but the methods and appli-
cations are more general. Although this is a fairly standard course taught in most major
universities, I was not satisfied with the available textbooks. Some popular but encyclope-
dic books include a broader range of topics, much too broad to cover in one semester at
the depth that I thought necessary for graduate students. Others with a more manageable
length appear to be targeted primarily at undergraduates and relegate to appendices some
of the topics that I believe to be most important. Therefore, I soon found that preparation
of lecture notes for distribution to students was evolving into a textbook-writing project.
I was not able to avoid producing too much material either. I usually chose to skip
most of the chapter on Legendre and Bessel functions, assuming that graduate students
already had some familiarity with them, and instead referred them to a summary of the
properties that are useful for the chapter on boundary-value problems. Other instructors
might choose to omit the chapter on dispersion theory instead because most of it will
probably be covered in the subsequent course on electromagnetism, but I find that subject
more interesting and more fun to discuss than special functions. The chapter on group
theory was prepared at the request of reviewers; although I never reached that topic in one
semester, I hope that it will be useful for those teaching a two-semester course or as a
resource that students will use later on. It may also be useful for one-semester courses at
institutions where the average student already has a sufficiently strong mastery of analytic
functions that the first couple of chapters can be abbreviated or omitted. I believe that
it should be possible to cover most of the remaining material well in a single semester
at any mid-level university. I assume that the calculus of variations will be covered in a
concurrent course on classical mechanics and that the students are already comfortable
with linear algebra, differential equations, and vector calculus. Probability theory, tensor
analysis, and differential geometry are omitted.
A CD containing detailed solutions to all of the problems is available to instructors.
These solutions often employ to perform some of the routine but tedious
manipulations and to prepare figures. Some of these solutions may also be presented as
additional examples of the techniques covered in this course.
Graduate Mathematical Physics. James J. Kelly
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40637-9
Contents
Preface V
Note to the Reader XV
1 Analytic Functions 1
1.1 ComplexNumbers ............................. 1
1.1.1 MotivationandDefinitions..................... 1
1.1.2 Triangle Inequalities . . . ..................... 4
1.1.3 PolarRepresentation ........................ 4
1.1.4 Argument Function ......................... 5
1.2 Take Care with Multivalued Functions . . . ................ 8
1.3 Functions as Mappings . . ......................... 13
1.3.1 Mapping: w
z
.......................... 14
1.3.2 Mapping: w Sinz ........................ 16
1.4 Elementary Functions and Their Inverses . . ................ 17
1.4.1 Exponential and Logarithm ..................... 17
1.4.2 Powers ............................... 18
1.4.3 Trigonometric and Hyperbolic Functions . . ........... 19
1.4.4 Standard Branch Cuts . . ..................... 20
1.5 Sets,Curves,RegionsandDomains .................... 21
1.6 LimitsandContinuity............................ 22
1.7 Differentiability . .............................. 23
1.7.1 Cauchy–Riemann Equations .................... 23
1.7.2 DifferentiationRules........................ 25
1.8 Properties of Analytic Functions . ..................... 26
1.9 Cauchy–Goursat Theorem ......................... 28
1.9.1 Simply Connected Regions ..................... 28
1.9.2 Proof ................................ 29
1.9.3 Example .............................. 31
1.10 Cauchy Integral Formula . ......................... 32
1.10.1 Integration Around Nonanalytic Regions . . ........... 32
1.10.2 Cauchy Integral Formula . ..................... 34
1.10.3 Example:YukawaField ...................... 34
1.10.4 Derivatives of Analytic Functions . ................ 36
1.10.5 Morera’s Theorem ......................... 37
1.11 Complex Sequences and Series . . ..................... 37
1.11.1 Convergence Tests ......................... 37
1.11.2 Uniform Convergence . . ..................... 40
Graduate Mathematical Physics. James J. Kelly
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40637-9
VIII Contents
1.12 Derivatives and Taylor Series for Analytic Functions . . . ........ 41
1.12.1 TaylorSeries ............................ 41
1.12.2 Cauchy Inequality . . . ...................... 44
1.12.3 Liouville’s Theorem . . ...................... 44
1.12.4 Fundamental Theorem of Algebra ................. 44
1.12.5 Zeros of Analytic Functions . . . ................. 45
1.13LaurentSeries................................ 46
1.13.1 Derivation.............................. 46
1.13.2 Example .............................. 47
1.13.3 Classification of Singularities . . ................. 49
1.13.4 Poles and Residues . . . ...................... 50
1.14 Meromorphic Functions ........................... 51
1.14.1 Pole Expansion ........................... 51
1.14.2 Example: Tanz .......................... 53
1.14.3 Product Expansion . . . ...................... 54
1.14.4 Example: Sinz ........................... 54
2 Integration 65
2.1 Introduction . ................................ 65
2.2 GoodTricks................................. 65
2.2.1 Parametric Differentiation ..................... 65
2.2.2 Convergence Factors . . ...................... 66
2.3 ContourIntegration............................. 66
2.3.1 Residue Theorem .......................... 66
2.3.2 Definite Integrals of the Form
2Π
0
f sin Θ, cos Θ Θ ........ 67
2.3.3 Definite Integrals of the Form
f xx ............. 69
2.3.4 FourierIntegrals .......................... 70
2.3.5 CustomContours.......................... 72
2.4 Isolated Singularities on the Contour . . . ................. 73
2.4.1 Removable Singularity . ...................... 73
2.4.2 Cauchy Principal Value . ...................... 75
2.5 Integration Around a Branch Point ..................... 77
2.6 Reduction to Tabulated Integrals ...................... 79
2.6.1 Example:
x
4
x ........................ 80
2.6.2 Example: The Beta Function . . . ................. 81
2.6.3 Example:
0
Ω
n
ΒΩ
1
Ω ....................... 81
2.7 Integral Representations for Analytic Functions . . ............ 82
2.8 Using toEvaluateIntegrals................. 86
2.8.1 SymbolicIntegration........................ 86
2.8.2 NumericalIntegration ....................... 88
2.8.3 FurtherInformation......................... 89
3 Asymptotic Series 95
3.1 Introduction . ................................ 95
Contents IX
3.2 Method of Steepest Descent . . . ..................... 96
3.2.1 Example: Gamma Function .................... 99
3.3 PartialIntegration.............................. 101
3.3.1 Example: Complementary Error Function . . ........... 102
3.4 Expansion of an Integrand ......................... 104
3.4.1 Example: Modified Bessel Function ................ 105
4 Generalized Functions 111
4.1 Motivation.................................. 111
4.2 Properties of the Dirac Delta Function . . . ................ 113
4.3 Other Useful Generalized Functions .................... 115
4.3.1 Heaviside Step Function . ..................... 115
4.3.2 Derivatives of the Dirac Delta Function . . . ........... 116
4.4 Green Functions . .............................. 118
4.5 Multidimensional Delta Functions ..................... 120
5 Integral Transforms 125
5.1 Introduction . . . .............................. 125
5.2 FourierTransform.............................. 126
5.2.1 Motivation ............................. 126
5.2.2 Definition and Inversion . ..................... 128
5.2.3 Basic Properties . ......................... 130
5.2.4 Parseval’s Theorem ......................... 131
5.2.5 Convolution Theorem . . ..................... 132
5.2.6 Correlation Theorem . . . ..................... 133
5.2.7 UsefulFourierTransforms..................... 134
5.2.8 FourierTransformofDerivatives.................. 138
5.2.9 Summary.............................. 139
5.3 Green Functions via Fourier Transform . . ................ 139
5.3.1 Example: Green Function for One-Dimensional Diffusion . . . . 139
5.3.2 Example: Three-Dimensional Green Function for Diffusion
Equations.............................. 141
5.3.3 Example: Green Function for Damped Oscillator . . . ...... 143
5.3.4 OperatorMethod.......................... 147
5.4 Cosine or Sine Transforms for Even or Odd Functions ........... 147
5.5 DiscreteFourierTransform......................... 148
5.5.1 Sampling .............................. 149
5.5.2 Convolution............................. 153
5.5.3 TemporalCorrelation........................ 156
5.5.4 Power Spectrum Estimation .................... 160
5.6 LaplaceTransform ............................. 165
5.6.1 Definition and Inversion . ..................... 165
5.6.2 Laplace Transforms for Elementary Functions ........... 167
5.6.3 LaplaceTransformofDerivatives ................. 170
X Contents
5.6.4 Convolution Theorem . ...................... 171
5.6.5 Summary.............................. 173
5.7 Green Functions via Laplace Transform . ................. 173
5.7.1 Example:SeriesRCCircuit .................... 174
5.7.2 Example: Damped Oscillator . . . ................. 175
5.7.3 Example: Diffusion with Constant Boundary Value ........ 176
6 Analytic Continuation and Dispersion Relations 191
6.1 AnalyticContinuation............................ 191
6.1.1 Motivation ............................. 191
6.1.2 Uniqueness . . ........................... 192
6.1.3 Reflection Principle . . . ...................... 194
6.1.4 Permanence of Algebraic Form . ................. 195
6.1.5 Example: Gamma Function . . . ................. 195
6.2 DispersionRelations ............................ 196
6.2.1 Causality . . . ........................... 196
6.2.2 Oscillator Model .......................... 200
6.2.3 Kramers–KronigRelations..................... 203
6.2.4 SumRules ............................. 206
6.3 HilbertTransform.............................. 207
6.4 Spreading of a Wave Packet . . . ...................... 209
6.5 Solitons ................................... 213
7 Sturm–Liouville Theory 223
7.1 Introduction: The General String Equation ................. 223
7.2 Hilbert Spaces ................................ 226
7.2.1 Schwartz Inequality . . ...................... 229
7.2.2 Gram–Schmidt Orthogonalization ................. 230
7.3 Properties of Sturm–Liouville Systems . . ................. 232
7.3.1 Self-Adjointness .......................... 232
7.3.2 Reality of Eigenvalues and Orthogonality of Eigenfunctions . . . 233
7.3.3 Discreteness of Eigenvalues . . . ................. 235
7.3.4 Completeness of Eigenfunctions . ................. 235
7.3.5 Parseval’s Theorem . . . ...................... 237
7.3.6 Reality of Eigenfunctions ..................... 238
7.3.7 InterleavingofZeros........................ 238
7.3.8 Comparison Theorems . ...................... 240
7.4 Green Functions ............................... 242
7.4.1 InterfaceMatching......................... 242
7.4.2 Eigenfunction Expansion of Green Function ............ 246
7.4.3 Example:VibratingString ..................... 252
7.5 Perturbation Theory . . ........................... 253
7.5.1 Example:BeadatCenterofaString................ 255
7.6 Variational Methods . . ........................... 256