Upload
others
View
56
Download
0
Embed Size (px)
Citation preview
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
xix
1
1
9
12
14
20
29
29
31
35
35
40
42
44
45
47
53
58
59
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
144145
145
146
147
149
149
149
151
153
153
154
157
159
159
159
160
161
164
166
174
174
175
178
178
179
180
182
182
184184
193
194195198
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
9
9.1
9.2
9.3
9.4
10
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)
474
476
480
482
483
485
490
490
492
496
496
498
499
501
503
507
510
512
513
515
518
522
523
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
12.2
12.3
12.4
13
13.1
13.2
14
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
595
599
600
600
608
611
611
613
613
613
617
618
622
624
624
627
629
630
637
638
638
638
639
643
645
646
650
650
655662
663
14.1 General Properties 663Completeness 664
Behavior of Discontinuities 666
Contents xv
14.2
14.3
14.4
15
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
15.10
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
671671
674
677
678
683
683
684
685
688
689
689
689
690
693
694
695
696
698
698
699
700
706
712715
718
724
724
726
726729
730
732734734
735
vi Contents
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