Essential MathematicalMethods for Physicists

Hans J. WeberUniversity of Virginia

Charlottesville, VA

George B. ArfkenMiami University

Oxford, Ohio

ELSEV1ERACADEMIC

PRESS

Amsterdam Boston Heidelberg London New York Oxford ParisSan Diego San Francisco Singapore Sydney Tokyo

Preface

VECTOR ANALYSIS

1.1 Elementary ApproachVectors and Vector Space Summary

1.2 Scalar or Dot ProductFree Motion and Other Orbits

1.3 Vector or Cross Product1.4 Triple Scalar Product and Triple Vector Product

Triple Scalar ProductTriple Vector Product

1.5 Gradient, VPartial DerivativesGradient as a Vector OperatorA Geometrical Interpretation

1.6 Divergence, V

A Physical Interpretation1.7 Curl, V x1.8 Successive Applications of V1.9 Vector Integration

Line Integrals

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Contents

Surface Integrals 62

Volume Integrals 65

Integral Definitions of Gradient, Divergence, and Curl 66

1.10 Gauss's Theorem 68

Green's Theorem 70

1.11 Stokes's Theorem 72

1.12 Potential Theory 76

Scalar Potential 76

1.13 Gauss's Law and Poisson's Equation 82

Gauss's Law 82

Poisson's Equation 84

1.14 Dirac Delta Function 86

Additional Reading 95

2 VECTOR ANALYSIS IN CURVED COORDINATESAND TENSORS 96

2.1 Special Coordinate Systems 97

Rectangular Cartesian Coordinates 97

Integrals in Cartesian Coordinates 98

2.2 Circular Cylinder Coordinates 98

Integrals in Cylindrical Coordinates 101

Gradient 107

Divergence 108

Curl '• s 110

2.3 Orthogonal Coordinates 113

2.4 Differential Vector Operators 121

Gradient 121

Divergence 122

Curl 124

2.5 Spherical Polar Coordinates 126

Integrals in Spherical Polar Coordinates 130

2.6 Tensor Analysis 136

Rotation of Coordinate Axes 137

Invariance of the Scalar Product under Rotations 141

Covariance of Cross Product 142

Covariance of Gradient 143

Contents

2.7

2.8

2.9

3

3.1

3.2

3.3

Definition of Tensors of Rank TwoAddition and Subtraction of Tensors

Summation ConventionSymmetry-AntisymmetrySpinorsContraction and Direct Product

Contraction

Direct ProductQuotient RuleDual TensorsLevi-Civita Symbol

Dual TensorsAdditional Reading

DETERMINANTS AND MATRICES

DeterminantsLinear Equations: ExamplesHomogeneous Linear EquationsInhomogeneous Linear Equations

Laplacian Development by MinorsAntisymmetryMatrices

Basic Definitions, Equality, and RankMatrix Multiplication, Inner ProductDirac Bra-ket, Transposition

Multiplication (by a Scalar)AdditionProduct TheoremDirect Product

Diagonal Matrices

TraceMatrix InversionOrthogonal Matrices

Direction CosinesApplications to VectorsOrthogonality Conditions: Two-Dimensional Case

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Contents

Euler Angles 200Symmetry Properties and SimilarityTransformations 202Relation to Tensors 204

3.4 Hermitian Matrices and Unitary Matrices 206Definitions 206Pauli Matrices 208

3.5 Diagonalization of Matrices 211Moment of Inertia Matrix 211Eigenvectors and Eigenvalues 212Hermi tian Matrices 214Anti-Hermitian Matrices 216Normal Modes of Vibration 218Ill-Conditioned Systems 220Functions of Matrices 221Additional Reading 228

4 GROUP THEORY 229

4.1 Introduction to Group Theory 229Definition of Group 230Homomorphism and Isomorphism 234

Matrix Representations: Reducible and Irreducible 2344.2 Generators of Continuous Groups 237

Rotation Groups SO(2) and SO(3) ' 238Rotation of Functions and OrbitalAngular Momentum 239Special Unitary Group SU(2) 240

4.3 Orbital Angular Momentum 243Ladder Operator Approach 244

4.4 Homogeneous Lorentz Group 248Vector Analysis in Minkowski Space-Time 251Additional Reading 255

5 INFINITE SERIES 257

5.1 Fundamental Concepts 257Addition and Subtraction of Series 260

Contents ix

5.2 Convergence Tests 262

Comparison Test 262

Cauchy Root Test 263

d'Alembert or Cauchy Ratio Test 263

Cauchy or Maclaurin Integral Test 264

5.3 Alternating Series 269

Leibniz Criterion 270

Absolute and Conditional Convergence 271

5.4 Algebra of Series 274

Multiplication of Series 275

5.5 Series of Functions 276

Uniform Convergence 276

Weierstrass M (Majorant) Test 278

Abel's Test 279

5.6 Taylor's Expansion 281

Maclaurin Theorem 283

Binomial Theorem 284

Taylor Expansion—More Than One Variable 286

5.7 Power Series 291

Convergence 291

Uniform and Absolute Convergence 291

Continuity 292

Differentiation and Integration 292

Uniqueness Theorem 292

Inversion of Power Series 293

5.8 Elliptic Integrals 296

Definitions 297

Series Expansion 298

Limiting Values 300

5.9 Bernoulli Numbers and the Euler-Maclaurin Formula 302

Bernoulli Functions 305

Euler-Maclaunn Integration Formula 306

Improvement of Convergence 307

Improvement of Convergence by Rational Approximations 309

5.10 Asymptotic Series 314

Error Function 314

Additional Reading 317

Contents

6 FUNCTIONS OF A COMPLEX VARIABLE I 318

6.1 Complex Algebra 319

Complex Conjugation 321Functions of a Complex Variable ' 325

6.2 Cauchy-Riemann Conditions 331Analytic Functions 335

6.3 Cauchy's Integral Theorem 337Contour Integrals 337Stokes's Theorem Proof of Cauchy's Integral Theorem 339Multiply Connected Regions 341

6.4 Cauchy's Integral Formula 344Derivatives 346Morera's Theorem 346

6.5 Laurent Expansion 350Taylor Expansion 350Schwarz Reflection Principle 351Analytic Continuation 352Laurent Series 354

6.6 Mapping 360Translation 360Rotation 361Inversion 361Branch Points and Multivalent Functions 363

6.7 Conformal Mapping ; 368Additional Reading 370

7 FUNCTIONS OF A COMPLEX VARIABLE II 372

7.1 Singularities 372Poles 373

Branch Points 374

7.2 Calculus of Residues 378Residue Theorem 378Evaluation of Definite Integrals 379

Cauchy Principal Value 384Pole Expansion of Merornorphic Functions 390Product Expansion of Entire Functions 392

Contents

7.3 Method of Steepest Descents 400

Analytic Landscape 400

Saddle Point Method 402

Additional Reading 409

8 DIFFERENTIAL EQUATIONS 410

8.1 Introduction 410

8.2 First-Order ODEs 411

Separable Variables 411

Exact Differential Equations 413

Linear First-Order ODEs 414

ODEs of Special Type 418

8.3 Second-Order ODEs 424

Inhomogeneous Linear ODEs and Particular Solutions 430

Inhomogeneous Euler ODE 430

Inhomogeneous ODE with Constant Coefficients 431

Linear Independence of Solutions 434

8.4 Singular Points 439

8.5 Series Solutions—Frobenius's Method 441

Expansion about XQ 445

Symmetry of ODE and Solutions 445

Limitations of Series Approach—Bessel's Equation 446

Regular and Irregular Singularities 448

Fuchs's Theorem • 4 5 0

Summary 450

8.6 A Second Solution 454

Series Form of the Second Solution 456

8.7 Numerical Solutions 464

First-Order Differential Equations 464

Taylor Series Solution 464

Runge-Kutta Method 466

Predictor-Corrector Methods 467

Second-Order ODEs 468

8.8 Introduction to Partial Differential Equations 470

8.9 Separation of Variables 470

Cartesian Coordinates 471

Contents

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Circular Cylindrical Coordinates

Spherical Polar Coordinates

Additional Reading

STURM-LIOUVILLE THEORY—ORTHOGONALFUNCTIONS

Self-Adjoint ODEs

Eigenfunctions and EigenvaluesBoundary Conditions •Hermitian OperatorsHermitian Operators in Quantum MechanicsHermitian OperatorsReal Eigenvalues

Orthogonal EigenfunctionsExpansion in Orthogonal EigenfunctionsDegeneracyGram-Schmidt OrthogonalizationOrthogonal Polynomials

Completeness of EigenfunctionsBessel's Inequality

Schwarz InequalitySummary of Vector Spaces—CompletenessExpansion (Fourier) Coefficients

Additional Reading

THE GAMMA FUNCTION (FACTORIAL FUNCTION)

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10.1 Definitions and Simple Properties 523Infinite Limit (Euler) 523Definite Integral (Euler) 524Infinite Product (Weierstrass) 526

Factorial Notation 528Double Factorial Notation 530Integral Representation 531

10.2 Digamma and Polygamma Functions 535Digamma Function 535Polygamma Function 536

Contents

Maclaurin Expansion, Computation 537Series Summation 537

10.3 Stirling's Series 540Derivation from Euler-Maclaurin Integration Formula 540Stirling's Series 541Numerical Computation 542

10.4 The Incomplete Gamma Functions and Related Functions 545Exponential Integral 546Error Integrals 548Additional Reading 550

11 LEGENDRE POLYNOMIALS AND SPHERICALHARMONICS 552

11.1 Introduction 552Physical Basis: Electrostatics 552

Generating Function 553Power Series 556Linear Electric Multipoles 558Vector Expansion 559

11.2 Recurrence Relations and Special Properties 563Recurrence Relations 563Differential Equations 564Upper and Lower Bounds for Pn (cos 6) 566

11.3 Orthogonality 568Expansion of Functions, Legendre Series 569

11.4 Alternate Definitions of Legendre Polynomials 579Rodrigues's Formula 579

11.5 Associated Legendre Functions 581Spherical Harmonics 584Additional Reading 588

12 BESSEL FUNCTIONS 589

12.1 Bessel Functions of the First Kind, «/„(*) 589

Generating Function for Integral Order 590Applications of Recurrence Relations 593Bessel's Differential Equation 594

Contents

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Integral RepresentationsOrthogonality

NormalizationBessel SeriesBessel Functions of Nonintegral OrderNeumann Functions, Bessel Functions of theSecond KindDefinition and Series Form

Other FormsRecurrence RelationsWronskian FormulasAsymptotic ExpansionsExpansion of an Integral RepresentationNumerical EvaluationSpherical Bessel FunctionsDefinitionsLimiting ValuesRecurrence RelationsNumerical Computation

Additional Reading

HERMITE AND LAGUERRE POLYNOMIALS

Hermite PolynomialsQuantum Mechanical Simple Harmonic OscillatorRaising and Lowering Operators

Recurrence Relations and Generating FunctionAlternate RepresentationsOrthogonalityLaguerre FunctionsDifferential Equation—Laguerre Polynomials

Associated Laguerre PolynomialsAdditional Reading

FOURIER SERIES

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14.1 General Properties 663Completeness 664

Behavior of Discontinuities 666

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Advantages and Uses of Fourier SeriesPeriodic Functions

Change of Interval

Complex Fourier SeriesAbel's Theorem

Properties of Fourier SeriesConvergence

Integration

DifferentiationAdditional Reading

INTEGRAL TRANSFORMS

Introduction and DefinitionsLinearityFourier TransformLaplace TransformDevelopment of the Inverse Fourier TransformInverse Fourier Transform—Exponential Form

Dirac Delta Function DerivationFourier Transforms—Inversion TheoremExponential TransformCosine Transform

Sine Transform

Fourier Transform of Derivatives >Convolution TheoremParseval's RelationMomentum RepresentationLaplace TransformsDefinition

Inverse Transform

Partial Fraction ExpansionNumerical InversionLaplace Transform of Derivatives

Dirac Delta Function

Other PropertiesSubstitutionRLC Analog

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Translation 736Derivative of a Transform 737Integration of Transforms 738Limits of Integration—Unit Step Function - 738

15.11 Convolution or Faltungs Theorem 74215.12 Inverse Laplace Transform 746

Bromwich Integral 746Summary: Inversion of Laplace Transform 752Additional Reading 754

16 PARTIAL DIFFERENTIAL EQUATIONS 756

16.1 Examples of Partial Differential Equationsand Boundary Conditions 756Boundary Conditions 758

16.2 Heat Flow or Diffusion PDE 760Alternate Solutions 763

16.3 Inhomogeneous PDE—Green's Function 769Additional Reading 780

17 PROBABILITY 782

17.1 Definitions, Simple Properties 782Counting of Permutations and Combinations 787

17.2 Random Variables 78917.3 Binomial Distribution 80217.4 Poisson Distribution \ 804

17.5 Gauss's Normal Distribution 80717.6 Statistics 812

Error Propagation 812Fitting Curves to Data 815The x2 Distribution 817The Student t Distribution 821Additional Reading 825

18 CALCULUS OF VARIATIONS 826

Uses of the Calculus of Variations 826

18.1 A Dependent and an Independent Variable 827Concept of Variation 827

Contents xvil

18.2 Several Dependent Variables 837

Transversality Condition 839

Hamilton's Principle 841

18.3 Several Independent Variables 845

18.4 Several Dependent and Independent Variables 847

Relation to Physics 848

18.5 Lagrangian Multipliers: Variation with Constraints 848

Variation with Constraints 850

Lagrangian Equations 853

18.6 Rayleigh-Ritz Variational Technique 861

Ground-State Eigenfunction 862

Additional Reading 866

19 NONLINEAR METHODS AND CHAOS 867

19.1 Introduction 867

19.2 The Logistic Map 869

19.3 Sensitivity to Initial Conditions and Parameters 874

Lyapunov Exponents 874

Fractals 875

19.4 Nonlinear Differential Equations 878

Bernoulli and Riccati Equations 879

Fixed and Movable Singularities, Special Solutions 881

Autonomous Differential Equations 882

Local and Global Behavior in Higher Dimensions 885

Dissipation in Dynamical Systems \ 896

Bifurcations in Dynamical Systems 898

Chaos in Dynamical Systems 900

Routes to Chaos in Dynamical Systems 901

Additional Reading 903

APPENDIX 1 REAL ZEROS OF A FUNCTION 905

Bisection Method 905

Three Warnings 907

Additional Reading 908

General References 908

INDEX 911