Essential Mathematical Methods for Physicists, ISE

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ISBN: 9780120598779 | 0120598779
Cover: Hardcover
Copyright: 8/8/2003

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This new adaptation of Arfken and Weber's bestselling Mathematical Methods for Physicists, Fifth Editionlt;/b>, is the most comprehensive, modern, and accessible text for using mathematics to solve physics problems. Additional explanations and examples make it student-friendly and more adaptable to a course syllabus. KEY FEATURES: This is a more accessible version of Arfken and Weber's blockbuster reference, Mathematical Methods for Physicists, 5th Edition Many more detailed, worked-out examples illustrate how to use and apply mathematical techniques to solve physics problems More frequent and thorough explanations help readers understand, recall, and apply the theory New introductions and review material provide context and extra support for key ideas Many more routine problems reinforce basic concepts and computations

Preface

xix

Vector Analysis

(95)

Elementary Approach

(11)

Vectors and Vector Space Summary

(3)

Scalar or Dot Product

(8)

Free Motion and Other Orbits

(6)

Vector or Cross Product

(9)

Triple Scalar Product and Triple Vector Product

(6)

Triple Scalar Product

(2)

Triple Vector Product

(4)

Gradient

(9)

Partial Derivatives

(5)

Gradient as a Vector Operator

(2)

A Geometrical Interpretation

(2)

Divergence

(3)

A Physical Interpretation

(2)

Curl, x

(6)

Successive Applications of

(5)

Vector Integration

(10)

Line Integrals

(3)

Surface Integrals

(3)

Volume Integrals

(1)

Integral Definitions of Gradient, Divergence, and Curl

(2)

Gauss's Theorem

(4)

Green's Theorem

(2)

Stokes's Theorem

(4)

Potential Theory

(6)

Scalar Potential

(6)

Gauss's Law and Poisson's Equation

(4)

Gauss's Law

(2)

Poisson's Equation

(2)

Dirac Delta Function

(10)

Additional Reading

(1)

Vector Analysis in Curved Coordinates and Tensors

(63)

Special Coordinate Systems

(1)

Rectangular Cartesian Coordinates

(1)

Integrals in Cartesian Coordinates

(1)

Circular Cylinder Coordinates

(15)

Integrals in Cylindrical Coordinates

101

(6)

Gradient

107

(1)

Divergence

108

(2)

Curl

110

(3)

Orthogonal Coordinates

113

(8)

Differential Vector Operators

121

(5)

Gradient

121

(1)

Divergence

122

(2)

Curl

124

(2)

Spherical Polar Coordinates

126

(10)

Integrals in Spherical Polar Coordinates

130

(6)

Tensor Analysis

136

(13)

Rotation of Coordinate Axes

137

(4)

Invariance of the Scalar Product under Rotations

141

(1)

Covariance of Cross Product

142

(1)

Covariance of Gradient

143

(1)

Definition of Tensors of Rank Two

144

(1)

Addition and Subtraction of Tensors

145

(1)

Summation Convention

145

(1)

Symmetry--Antisymmetry

146

(1)

Spinors

147

(2)

Contraction and Direct Product

149

(2)

Contraction

149

(1)

Direct Product

149

(2)

Quotient Rule

151

(2)

Dual Tensors

153

(6)

Levi--Civita Symbol

153

(1)

Dual Tensors

154

(3)

Additional Reading

157

(2)

Determinants and Matrices

159

(70)

Determinants

159

(15)

Linear Equations: Examples

159

(1)

Homogeneous Linear Equations

160

(1)

Inhomogeneous Linear Equations

161

(3)

Laplacian Development by Minors

164

(2)

Antisymmetry

166

(8)

Matrices

174

(19)

Basic Definitions, Equality, and Rank

174

(1)

Matrix Multiplication, Inner Product

175

(3)

Dirac Bra-ket, Transposition

178

(1)

Multiplication (by a Scalar)

178

(1)

Addition

179

(1)

Product Theorem

180

(2)

Direct Product

182

(1)

Diagonal Matrices

182

(2)

Trace

184

(1)

Matrix Inversion

184

(9)

Orthogonal Matrices

193

(13)

Direction Cosines

194

(1)

Applications to Vectors

195

(3)

Orthogonality Conditions: Two-Dimensional Case

198

(2)

Euler Angles

200

(2)

Symmetry Properties and Similarity Transformations

202

(2)

Relation to Tensors

204

(2)

Hermitian Matrices and Unitary Matrices

206

(5)

Definitions

206

(2)

Pauli Matrices

208

(3)

Diagonalization of Matrices

211

(18)

Moment of Inertia Matrix

211

(1)

Eigenvectors and Eigenvalues

212

(2)

Hermitian Matrices

214

(2)

Anti-Hermitian Matrices

216

(2)

Normal Modes of Vibration

218

(2)

Ill-Conditioned Systems

220

(1)

Functions of Matrices

221

(7)

Additional Reading

228

(1)

Group Theory

229

(28)

Introduction to Group Theory

229

(8)

Definition of Group

230

(4)

Homomorphism and Isomorphism

234

(1)

Matrix Representations: Reducible and Irreducible

234

(3)

Generators of Continuous Groups

237

(6)

Rotation Groups SO(2) and SO(3)

238

(1)

Rotation of Functions and Orbital Angular Momentum

239

(1)

Special Unitary Group SU(2)

240

(3)

Orbital Angular Momentum

243

(5)

Ladder Operator Approach

244

(4)

Homogeneous Lorentz Group

248

(9)

Vector Analysis in Minkowski Space-Time

251

(4)

Additional Reading

255

(2)

Infinite Series

257

(61)

Fundamental Concepts

257

(5)

Addition and Subtraction of Series

260

(2)

Convergence Tests

262

(7)

Comparison Test

262

(1)

Cauchy Root Test

263

(1)

d'Alembert or Cauchy Ratio Test

263

(1)

Cauchy or Maclaurin Integral Test

264

(5)

Alternating Series

269

(5)

Leibniz Criterion

270

(1)

Absolute and Conditional Convergence

271

(3)

Algebra of Series

274

(2)

Multiplication of Series

275

(1)

Series of Functions

276

(5)

Uniform Convergence

276

(2)

Weierstrass M (Majorant) Test

278

(1)

Abel's Test

279

(2)

Taylor's Expansion

281

(10)

Maclaurin Theorem

283

(1)

Binomial Theorem

284

(2)

Taylor Expansion---More Than One Variable

286

(5)

Power Series

291

(5)

Convergence

291

(1)

Uniform and Absolute Convergence

291

(1)

Continuity

292

(1)

Differentiation and Integration

292

(1)

Uniqueness Theorem

292

(1)

Inversion of Power Series

293

(3)

Elliptic Integrals

296

(6)

Definitions

297

(1)

Series Expansion

298

(2)

Limiting Values

300

(2)

Bernoulli Numbers and the Euler-Maclaurin Formula

302

(12)

Bernoulli Functions

305

(1)

Euler-Maclaurin Integration Formula

306

(1)

Improvement of Convergence

307

(2)

Improvement of Convergence by Rational Approximations

309

(5)

Asymptotic Series

314

(4)

Error Function

314

(3)

Additional Reading

317

(1)

Functions of a Complex Variable I

318

(54)

Complex Algebra

319

(12)

Complex Conjugation

321

(4)

Functions of a Complex Variable

325

(6)

Cauchy-Riemann Conditions

331

(6)

Analytic Functions

335

(2)

Cauchy's Integral Theorem

337

(7)

Contour Integrals

337

(2)

Stokes's Theorem Proof of Cauchy's Integral Theorem

339

(2)

Multiply Connected Regions

341

(3)

Cauchy's Integral Formula

344

(6)

Derivatives

346

(1)

Morera's Theorem

346

(4)

Laurent Expansion

350

(10)

Taylor Expansion

350

(1)

Schwarz Reflection Principle

351

(1)

Analytic Continuation

352

(2)

Laurent Series

354

(6)

Mapping

360

(8)

Translation

360

(1)

Rotation

361

(1)

Inversion

361

(2)

Branch Points and Multivalent Functions

363

(5)

Conformal Mapping

368

(4)

Additional Reading

370

(2)

Functions of a Complex Variable II

372

(38)

Singularities

372

(6)

Poles

373

(1)

Branch Points

374

(4)

Calculus of Residues

378

(22)

Residue Theorem

378

(1)

Evaluation of Definite Integrals

379

(5)

Cauchy Principal Value

384

(6)

Pole Expansion of Meromorphic Functions

390

(2)

Product Expansion of Entire Functions

392

(8)

Method of Steepest Descents

400

(10)

Analytic Landscape

400

(2)

Saddle Point Method

402

(7)

Additional Reading

409

(1)

Differential Equations

410

(72)

Introduction

410

(1)

First-Order ODEs

411

(13)

Separable Variables

411

(2)

Exact Differential Equations

413

(1)

Linear First-Order ODEs

414

(4)

ODEs of Special Type

418

(6)

Second-Order ODEs

424

(15)

Inhomogeneous Linear ODEs and Particular Solutions

430

(1)

Inhomogeneous Euler ODE

430

(1)

Inhomogeneous ODE with Constant Coefficients

431

(3)

Linear Independence of Solutions

434

(5)

Singular Points

439

(2)

Series Solutions---Frobenius's Method

441

(13)

Expansion about xo

445

(1)

Symmetry of ODE and Solutions

445

(1)

Limitations of Series Approach---Bessel's Equation

446

(2)

Regular and Irregular Singularities

448

(2)

Fuchs's Theorem

450

(1)

Summary

450

(4)

A Second Solution

454

(10)

Series Form of the Second Solution

456

(8)

Numerical Solutions

464

(6)

First-Order Differential Equations

464

(1)

Taylor Series Solution

464

(2)

Runge--Kutta Method

466

(1)

Predictor--Corrector Methods

467

(1)

Second-Order ODEs

468

(2)

Introduction to Partial Differential Equations

470

(1)

Separation of Variables

470

(12)

Cartesian Coordinates

471

(3)

Circular Cylindrical Coordinates

474

(2)

Spherical Polar Coordinates

476

(4)

Additional Reading

480

(2)

Sturm--Liouville Theory---Orthogonal Functions

482

(41)

Self-Adjoint ODEs

483

(13)

Eigenfunctions and Eigenvalues

485

(5)

Boundary Conditions

490

(1)

Hermitian Operators

490

(2)

Hermitian Operators in Quantum Mechanics

492

(4)

Hermitian Operators

496

(7)

Real Eigenvalues

496

(2)

Orthogonal Eigenfunctions

498

(1)

Expansion in Orthogonal Eigenfunctions

499

(2)

Degeneracy

501

(2)

Gram-Schmidt Orthogonalization

503

(7)

Orthogonal Polynomials

507

(3)

Completeness of Eigenfunctions

510

(13)

Bessel's Inequality

512

(1)

Schwarz Inequality

513

(2)

Summary of Vector Spaces---Completeness

515

(3)

Expansion (Fourier) Coefficients

518

(4)

Additional Reading

522

(1)

The Gamma Function (Factorial Function)

523

(29)

Definitions and Simple Properties

523

(12)

Infinite Limit (Euler)

523

(1)

Definite Integral (Euler)

524

(2)

Infinite Product (Weierstrass)

526

(2)

Factorial Notation

528

(2)

Double Factorial Notation

530

(1)

Integral Representation

531

(4)

Digamma and Polygamma Functions

535

(5)

Digamma Function

535

(1)

Polygamma Function

536

(1)

Maclaurin Expansion, Computation

537

(1)

Series Summation

537

(3)

Stirling's Series

540

(5)

Derivation from Euler--Maclaurin Integration Formula

540

(1)

Stirling's Series

541

(1)

Numerical Computation

542

(3)

The Incomplete Gamma Functions and Related Functions

545

(7)

Exponential Integral

546

(2)

Error Integrals

548

(2)

Additional Reading

550

(2)

Legendre Polynomials and Spherical Harmonics

552

(37)

Introduction

552

(11)

Physical Basis: Electrostatics

552

(1)

Generating Function

553

(3)

Power Series

556

(2)

Linear Electric Multipoles

558

(1)

Vector Expansion

559

(4)

Recurrence Relations and Special Properties

563

(5)

Recurrence Relations

563

(1)

Differential Equations

564

(2)

Upper and Lower Bounds for Pn (cos θ)

566

(2)

Orthogonality

568

(11)

Expansion of Functions, Legendre Series

569

(10)

Alternate Definitions of Legendre Polynomials

579

(2)

Rodrigues's Formula

579

(2)

Associated Legendre Functions

581

(8)

Spherical Harmonics

584

(4)

Additional Reading

588

(1)

Bessel Functions

589

(49)

Bessel Functions of the First Kind, Jv(x)

589

(22)

Generating Function for Integral Order

590

(3)

Applications of Recurrence Relations

593

(1)

Bessel's Differential Equation

594

(1)

Integral Representations

595

(4)

Orthogonality

599

(1)

Normalization

600

(1)

Bessel Series

600

(8)

Bessel Functions of Nonintegral Order

608

(3)

Neumann Functions, Bessel Functions of the Second Kind

611

(6)

Definition and Series Form

611

(2)

Other Forms

613

(1)

Recurrence Relations

613

(1)

Wronskian Formulas

613

(4)

Asymptotic Expansions

617

(7)

Expansion of an Integral Representation

618

(4)

Numerical Evaluation

622

(2)

Spherical Bessel Functions

624

(14)

Definitions

624

(3)

Limiting Values

627

(2)

Recurrence Relations

629

(1)

Numerical Computation

630

(7)

Additional Reading

637

(1)

Hermite and Laguerre Polynomials

638

(25)

Hermite Polynomials

638

(12)

Quantum Mechanical Simple Harmonic Oscillator

638

(1)

Raising and Lowering Operators

639

(4)

Recurrence Relations and Generating Function

643

(2)

Alternate Representations

645

(1)

Orthogonality

646

(4)

Laguerre Functions

650

(13)

Differential Equation---Laguerre Polynomials

650

(5)

Associated Laguerre Polynomials

655

(7)

Additional Reading

662

(1)

Fourier Series

663

(26)

General Properties

663

(8)

Completeness

664

(2)

Behavior of Discontinuities

666

(5)

Advantages and Uses of Fourier Series

671

(6)

Periodic Functions

671

(3)

Change of Interval

674

(3)

Complex Fourier Series

677

(6)

Abel's Theorem

678

(5)

Properties of Fourier Series

683

(6)

Convergence

683

(1)

Integration

684

(1)

Differentiation

685

(3)

Additional Reading

688

(1)

Integral Transforms

689

(67)

Introduction and Definitions

689

(1)

Linearity

689

(1)

Fourier Transform

690

(4)

Laplace Transform

693

(1)

Development of the Inverse Fourier Transform

694

(4)

Inverse Fourier Transform---Exponential Form

695

(1)

Dirac Delta Function Derivation

696

(2)

Fourier Transforms---Inversion Theorem

698

(8)

Exponential Transform

698

(1)

Cosine Transform

699

(1)

Sine Transform

700

(6)

Fourier Transform of Derivatives

706

(6)

Convolution Theorem

712

(6)

Parseval's Relation

715

(3)

Momentum Representation

718

(6)

Laplace Transforms

724

(6)

Definition

724

(2)

Inverse Transform

726

(1)

Partial Fraction Expansion

726

(3)

Numerical Inversion

729

(1)

Laplace Transform of Derivatives

730

(4)

Dirac Delta Function

732

(2)

Other Properties

734

(8)

Substitution

734

(1)

RLC Analog

735

(1)

Translation

736

(1)

Derivative of a Transform

737

(1)

Integration of Transforms

738

(1)

Limits of Integration---Unit Step Function

738

(4)

Convolution or Faltungs Theorem

742

(4)

Inverse Laplace Transform

746

(10)

Bromwich Integral

746

(6)

Summary: Inversion of Laplace Transform

752

(2)

Additional Reading

754

(2)

Partial Differential Equations

756

(26)

Examples of Partial Differential Equations and Boundary Conditions

756

(4)

Boundary Conditions

758

(2)

Heat Flow or Diffusion PDE

760

(9)

Alternate Solutions

763

(6)

Inhomogeneous PDE---Green's Function

769

(13)

Additional Reading

780

(2)

Probability

782

(44)

Definitions, Simple Properties

782

(7)

Counting of Permutations and Combinations

787

(2)

Random Variables

789

(13)

Binomial Distribution

802

(2)

Poisson Distribution

804

(3)

Gauss's Normal Distribution

807

(5)

Statistics

812

(14)

Error Propagation

812

(3)

Fitting Curves to Data

815

(2)

The x2 Distribution

817

(4)

The Student t Distribution

821

(4)

Additional Reading

825

(1)

Calculus of Variations

826

(41)

Uses of the Calculus of Variations

826

(1)

A Dependent and an Independent Variable

827

(10)

Concept of Variation

827

(10)

Several Dependent Variables

837

(8)

Transversality Condition

839

(2)

Hamilton's Principle

841

(4)

Several Independent Variables

845

(2)

Several Dependent and Independent Variables

847

(1)

Relation to Physics

848

(1)

Lagrangian Multipliers: Variation with Constraints

848

(13)

Variation with Constraints

850

(3)

Lagrangian Equations

853

(8)

Rayleigh-Ritz Variational Technique

861

(6)

Ground-State Eigenfunction

862

(4)

Additional Reading

866

(1)

Nonlinear Methods and Chaos

867

(38)

Introduction

867

(2)

The Logistic Map

869

(5)

Sensitivity to Initial Conditions and Parameters

874

(4)

Lyapunov Exponents

874

(1)

Fractals

875

(3)

Nonlinear Differential Equations

878

(27)

Bernoulli and Riccati Equations

879

(2)

Fixed and Movable Singularities, Special Solutions

881

(1)

Autonomous Differential Equations

882

(3)

Local and Global Behavior in Higher Dimensions

885

(11)

Dissipation in Dynamical Systems

896

(2)

Bifurcations in Dynamical Systems

898

(2)

Chaos in Dynamical Systems

900

(1)

Routes to Chaos in Dynamical Systems

901

(2)

Additional Reading

903

(2)

APPENDIX 1 REAL ZEROS OF A FUNCTION

905

(6)

Bisection Method

905

(2)

Three Warnings

907

(1)

Additional Reading

908

(1)

General References

908

(3)

Index

911

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Essential Mathematical Methods for Physicists, ISE

Summary

Table of Contents

Supplemental Materials