Graduate Mathematical Physics, With MATHEMATICA Supplements

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ISBN: 9783527406371 | 3527406379
Cover: Paperback
Copyright: 11/10/2006

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This up-to-date textbook on mathematical methods of physics is designed for a one-semester graduate or two-semester advanced undergraduate course. The formal methods are supplemented by applications that use MATHEMATICA to perform both symbolic and numerical calculations.The book is written by a physicist lecturer who knows the difficulties involved in applying mathematics to real problems. As many as 40 exercises are included at the end of each chapter. A student CD includes a basic introduction to MATHEMATICA, notebook files for each chapter, and solutions to selected exercises.Free solutions manual available for lecturers at college@wiley-vch.de.

Preface

Note to the Reader

Analytic Functions

(64)

Complex Numbers

(7)

Motivation and Definitions

(3)

Triangle Inequalities

(1)

Polar Representation

(1)

Argument Function

(3)

Take Care with Multivalued Functions

(5)

Functions as Mappings

(4)

Mapping: w = ez

(2)

Mapping: w = Sin[z]

(1)

Elementary Functions and Their Inverses

(4)

Exponential and Logarithm

(1)

Powers

(1)

Trigonometric and Hyperbolic Functions

(1)

Standard Branch Cuts

(1)

Sets, Curves, Regions and Domains

(1)

Limits and Continuity

(1)

Differentiability

(3)

Cauchy--Riemann Equations

(2)

Differentiation Rules

(1)

Properties of Analytic Functions

(2)

Cauchy--Goursat Theorem

(4)

Simply Connected Regions

(1)

Proof

(2)

Example

(1)

Cauchy Integral Formula

(5)

Integration Around Nonanalytic Regions

(1)

Cauchy Integral Formula

(1)

Example: Yukawa Field

(2)

Derivatives of Analytic Functions

(1)

Morera's Theorem

(1)

Complex Sequences and Series

(4)

Convergence Tests

(3)

Uniform Convergence

(1)

Derivatives and Taylor Series for Analytic Functions

(4)

Taylor Series

(3)

Cauchy Inequality

(1)

Liouville's Theorem

(1)

Fundamental Theorem of Algebra

(1)

Zeros of Analytic Functions

(1)

Laurent Series

(6)

Derivation

(2)

Example

(1)

Classification of Singularities

(1)

Poles and Residues

(2)

Meromorphic Functions

(14)

Pole Expansion

(2)

Example: Tan[z]

(1)

Product Expansion

(1)

Example: Sin[z]

(1)

Problems for Chapter 1

(10)

Integration

(30)

Introduction

(1)

Good Tricks

(1)

Parametric Differentiation

(1)

Convergence Factors

(1)

Contour Integration

(7)

Residue Theorem

(1)

Definite Integrals of the Form ∫2π0f[sin Ø, cos Ø] dØ

(2)

Definite Integrals of the Form ∫∞-∞f[x]dx

(1)

Fourier Integrals

(2)

Custom Contours

(1)

Isolated Singularities on the Contour

(4)

Removable Singularity

(2)

Cauchy Principal Value

(2)

Integration Around a Branch Point

(2)

Reduction to Tabulated Integrals

(3)

Example: ∫∞-∞e-x4 dx

(1)

Example: The Beta Function

(1)

Example: ∫∞0 ωn eβω-1 dω

(1)

Integral Representations for Analytic Functions

(4)

Using Mathematica to Evaluate Integrals

(9)

Symbolic Integration

(2)

Numerical Integration

(1)

Further Information

(1)

Problems for Chapter 2

(6)

Asymptotic Series

(16)

Introduction

(1)

Method of Steepest Descent

(5)

Example: Gamma Function

(2)

Partial Integration

101

(3)

Example: Complementary Error Function

102

(2)

Expansion of an Integrand

104

(7)

Example: Modified Bessel Function

105

(3)

Problems for Chapter 3

108

(3)

Generalized Functions

111

(14)

Motivation

111

(2)

Properties of the Dirac Delta Function

113

(2)

Other Useful Generalized Functions

115

(3)

Heaviside Step Function

115

(1)

Derivatives of the Dirac Delta Function

116

(2)

Green Functions

118

(2)

Multidimensional Delta Functions

120

(5)

Problems for Chapter 4

122

(3)

Integral Transforms

125

(66)

Introduction

125

(1)

Fourier Transform

126

(13)

Motivation

126

(2)

Definition and Inversion

128

(2)

Basic Properties

130

(1)

Parseval's Theorem

131

(1)

Convolution Theorem

132

(1)

Correlation Theorem

133

(1)

Useful Fourier Transforms

134

(4)

Fourier Transform of Derivatives

138

(1)

Summary

139

(1)

Green Functions via Fourier Transform

139

(8)

Example: Green Function for One-Dimensional Diffusion

139

(2)

Example: Three-Dimensional Green Function for Diffusion Equations

141

(2)

Example: Green Function for Damped Oscillator

143

(4)

Operator Method

147

(1)

Cosine or Sine Transforms for Even or Odd Functions

147

(1)

Discrete Fourier Transform

148

(17)

Sampling

149

(4)

Convolution

153

(3)

Temporal Correlation

156

(4)

Power Spectrum Estimation

160

(5)

Laplace Transform

165

(8)

Definition and Inversion

165

(2)

Laplace Transforms for Elementary Functions

167

(3)

Laplace Transform of Derivatives

170

(1)

Convolution Theorem

171

(2)

Summary

173

(1)

Green Functions via Laplace Transform

173

(18)

Example: Series RC Circuit

174

(1)

Example: Damped Oscillator

175

(1)

Example: Diffusion with Constant Boundary Value

176

(5)

Problems for Chapter 5

181

(10)

Analytic Continuation and Dispersion Relations

191

(32)

Analytic Continuation

191

(5)

Motivation

191

(1)

Uniqueness

192

(2)

Reflection Principle

194

(1)

Permanence of Algebraic Form

195

(1)

Example: Gamma Function

195

(1)

Dispersion Relations

196

(11)

Causality

196

(4)

Oscillator Model

200

(3)

Kramers-Kronig Relations

203

(2)

Sum Rules

205

(2)

Hilbert Transform

207

(1)

Spreading of a Wave Packet

208

(4)

Solitons

212

(11)

Problems for Chapter 6

216

(7)

Sturm--Liouville Theory

223

(46)

Introduction: The General String Equation

223

(3)

Hilbert Spaces

226

(6)

Schwartz Inequality

229

(1)

Gram--Schmidt Orthogonalization

230

(2)

Properties of Sturm--Liouville Systems

232

(10)

Self--Adjointness

232

(1)

Reality of Eigenvalues and Orthogonality of Eigenfunctions

233

(2)

Discreteness of Eigenvalues

235

(1)

Completeness of Eigenfunctions

235

(2)

Parseval's Theorem

237

(1)

Reality of Eigenfunctions

238

(1)

Interleaving of Zeros

238

(2)

Comparison Theorems

240

(2)

Green Functions

242

(11)

Interface Matching

242

(4)

Eigenfunction Expansion of Green Function

246

(6)

Example: Vibrating String

252

(1)

Perturbation Theory

253

(3)

Example: Bead at Center of a String

255

(1)

Variational Methods

256

(13)

Example: Vibrating String

259

(1)

Problems for Chapter 7

260

(9)

Legendre and Bessel Functions

269

(58)

Introduction

269

(1)

Legendre Functions

270

(21)

Generating Function for Legendre Polynomials

270

(4)

Series Representation and Rodrigues' Formula

274

(1)

Schlafli's Integral Representation

275

(1)

Legendre Expansion

275

(2)

Associated Legendre Functions

277

(4)

Spherical Harmonics

281

(1)

Multipole Expansion

282

(1)

Addition Theorem

283

(3)

Legendre Functions of the Second Kind

286

(1)

Relationship to Hypergeometric Functions

287

(2)

Analytic Structure of Legendre Functions

289

(2)

Bessel Functions

291

(17)

Cylindrical

291

(6)

Hankel Functions

297

(2)

Neumann Functions

299

(4)

Modified Bessel Functions

303

(2)

Spherical Bessel Functions

305

(3)

Fourier-Bessel Transform

308

(4)

Example: Fourier--Bessel Expansion of Nuclear Charge Density

311

(1)

Summary

312

(15)

Legendre Functions

313

(1)

Associated Legendre Functions

314

(1)

Spherical Harmonics

315

(1)

Cylindrical Bessel Functions

315

(1)

Spherical Bessel Functions

316

(2)

Fourier--Bessel Expansions

318

(1)

Problems for Chapter 8

318

(9)

Boundary-Value Problems

327

(42)

Introduction

327

(5)

Laplace's Equation in Box with Specified Potential on one Side

328

(1)

Green Function for Grounded Box

329

(3)

Green's Theorem for Electrostatics

332

(3)

Separable Coordinate Systems

335

(4)

Spherical Polar Coordinates

336

(2)

Cylindrical Coordinates

338

(1)

Spherical Expansion of Dirichlet Green Function for Poisson's Equation

339

(7)

Example: Multipole Expansion for Localized Charge Distribution

342

(1)

Example: Point Charge Near Grounded Conducting Sphere

342

(2)

Example: Specified Potential on Surface of Empty Sphere

344

(2)

Example: Charged Ring at Center of Grounded Conducting Sphere

346

(1)

Magnetic Field of Current Loop

346

(3)

Inhomogeneous Wave Equation

349

(20)

Spatial Representation of Time-Independent Green Function

349

(3)

Partial-Wave Expansion

352

(2)

Momentum Representation of Time-Independent Green Function

354

(2)

Retarded Green Function

356

(2)

Lippmann-Schwinger Equation

358

(2)

Problems for Chapter 9

360

(9)

Group Theory

369

(90)

Introduction

369

(1)

Finite Groups

370

(6)

Definitions

370

(3)

Equivalence Classes

373

(1)

Subgroups

374

(1)

Homomorphism

375

(1)

Direct Products

376

(1)

Representations

376

(29)

Definitions

376

(4)

Example: Vibrating triangle

380

(3)

Orthogonality Theorem

383

(5)

Character

388

(5)

Example: Character table for symmetries of a square

393

(4)

Example: Vibrational eigenvalues of square

397

(3)

Direct-Product Representations

400

(3)

Eigenfunctions

403

(1)

Wigner-Eckart Theorem

404

(1)

Continuous Groups

405

(17)

Definitions

405

(3)

Transformation of Functions

408

(1)

Generators

409

(4)

Example: Linear coordinate transformations in one dimension

413

(1)

Example: SO(2)

414

(2)

Example: SU(2)

416

(1)

Example: SO(3)

417

(1)

Total angular momentum

418

(1)

Transformation of Operators

419

(1)

Invariant Functions

420

(2)

Lie Algebra

422

(3)

Definitions

422

(1)

Example: SU(2)

423

(2)

Orthogonality Relations for Lie Groups

425

(3)

Quantum Mechanical Representations of the Rotation Group

428

(17)

Generators and Commutation Relations

428

(2)

Euler Parametrization

430

(1)

Homomorphism Between SU(2) and SO(3)

431

(2)

Irreducible Representations of SU(2)

433

(4)

Orthogonality Relations for Rotation Matrices

437

(1)

Coupling of Angular Momenta

438

(4)

Spherical Tensors

442

(3)

Unitary Symmetries in Nuclear and Particle Physics

445

(14)

Problems for Chapter 10

447

(12)

Bibliography

459

(2)

Index

461

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Graduate Mathematical Physics, With MATHEMATICA Supplements

Summary

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Table of Contents

Supplemental Materials