A Practical, Interdisciplinary Guide to Advanced Mathematical Methods for Scientists and Engineers
Mathematical Methods in Science and Engineering, Second Edition, provides students and scientists with a detailed mathematical reference for advanced analysis and computational methodologies. Making complex tools accessible, this invaluable resource is designed for both the classroom and the practitioners; the modular format allows flexibility of coverage, while the text itself is formatted to provide essential information without detailed study. Highly practical discussion focuses on the “how-to” aspect of each topic presented, yet provides enough theory to reinforce central processes and mechanisms.
Recent growing interest in interdisciplinary studies has brought scientists together from physics, chemistry, biology, economy, and finance to expand advanced mathematical methods beyond theoretical physics. This book is written with this multi-disciplinary group in mind, emphasizing practical solutions for diverse applications and the development of a new interdisciplinary science.
Revised and expanded for increased utility, this new Second Edition:
- Includes over 60 new sections and subsections more useful to a multidisciplinary audience
- Contains new examples, new figures, new problems, and more fluid arguments
- Presents a detailed discussion on the most frequently encountered special functions in science and engineering
- Provides a systematic treatment of special functions in terms of the Sturm-Liouville theory
- Approaches second-order differential equations of physics and engineering from the factorization perspective
- Includes extensive discussion of coordinate transformations and tensors, complex analysis, fractional calculus, integral transforms, Green's functions, path integrals, and more
Extensively reworked to provide increased utility to a broader audience, this book provides a self-contained three-semester course for curriculum, self-study, or reference. As more scientific disciplines begin to lean more heavily on advanced mathematical analysis, this resource will prove to be an invaluable addition to any bookshelf.
Table of Contents
Preface xix
1 Legendre Equation and Polynomials 1
1.1 Second-Order Differential Equations of Physics 1
1.2 Legendre Equation 2
1.2.1 Method of Separation of Variables 4
1.2.2 Series Solution of the Legendre Equation 4
1.2.3 Frobenius Method – Review 7
1.3 Legendre Polynomials 8
1.3.1 Rodriguez Formula 10
1.3.2 Generating Function 10
1.3.3 Recursion Relations 12
1.3.4 Special Values 12
1.3.5 Special Integrals 13
1.3.6 Orthogonality and Completeness 14
1.3.7 Asymptotic Forms 17
1.4 Associated Legendre Equation and Polynomials 18
1.4.1 Associated Legendre Polynomials Pm l (x) 20
1.4.2 Orthogonality 21
1.4.3 Recursion Relations 22
1.4.4 Integral Representations 24
1.4.5 Associated Legendre Polynomials for m < 0 26
1.5 Spherical Harmonics 27
1.5.1 AdditionTheorem of Spherical Harmonics 30
1.5.2 Real Spherical Harmonics 33
Bibliography 33
Problems 34
2 Laguerre Polynomials 39
2.1 Central Force Problems in Quantum Mechanics 39
2.2 Laguerre Equation and Polynomials 41
2.2.1 Generating Function 42
2.2.2 Rodriguez Formula 43
2.2.3 Orthogonality 44
2.2.4 Recursion Relations 45
2.2.5 Special Values 46
2.3 Associated Laguerre Equation and Polynomials 46
2.3.1 Generating Function 48
2.3.2 Rodriguez Formula and Orthogonality 49
2.3.3 Recursion Relations 49
Bibliography 49
Problems 50
3 Hermite Polynomials 53
3.1 Harmonic Oscillator in QuantumMechanics 53
3.2 Hermite Equation and Polynomials 54
3.2.1 Generating Function 56
3.2.2 Rodriguez Formula 56
3.2.3 Recursion Relations and Orthogonality 57
Bibliography 61
Problems 62
4 Gegenbauer and Chebyshev Polynomials 65
4.1 Wave Equation on a Hypersphere 65
4.2 Gegenbauer Equation and Polynomials 68
4.2.1 Orthogonality and the Generating Function 68
4.2.2 Another Representation of the Solution 69
4.2.3 The Second Solution 70
4.2.4 Connection with the Gegenbauer Polynomials 71
4.2.5 Evaluation of the Normalization Constant 72
4.3 Chebyshev Equation and Polynomials 72
4.3.1 Chebyshev Polynomials of the First Kind 72
4.3.2 Chebyshev and Gegenbauer Polynomials 73
4.3.3 Chebyshev Polynomials of the Second Kind 73
4.3.4 Orthogonality and Generating Function 74
4.3.5 Another Definition 75
Bibliography 76
Problems 76
5 Bessel Functions 81
5.1 Bessel’s Equation 83
5.2 Bessel Functions 83
5.2.1 Asymptotic Forms 84
5.3 Modified Bessel Functions 86
5.4 Spherical Bessel Functions 87
5.5 Properties of Bessel Functions 88
5.5.1 Generating Function 88
5.5.2 Integral Definitions 89
5.5.3 Recursion Relations of the Bessel Functions 89
5.5.4 Orthogonality and Roots of Bessel Functions 90
5.5.5 Boundary Conditions for the Bessel Functions 91
5.5.6 Wronskian of Pairs of Solutions 94
5.6 Transformations of Bessel Functions 95
5.6.1 Critical Length of a Rod 96
Bibliography 98
Problems 99
6 Hypergeometric Functions 103
6.1 Hypergeometric Series 103
6.2 Hypergeometric Representations of Special Functions 107
6.3 Confluent Hypergeometric Equation 108
6.4 Pochhammer Symbol and Hypergeometric Functions 109
6.5 Reduction of Parameters 113
Bibliography 115
Problems 115
7 Sturm–Liouville Theory 119
7.1 Self-Adjoint Differential Operators 119
7.2 Sturm–Liouville Systems 120
7.3 Hermitian Operators 121
7.4 Properties of Hermitian Operators 122
7.4.1 Real Eigenvalues 122
7.4.2 Orthogonality of Eigenfunctions 123
7.4.3 Completeness and the ExpansionTheorem 123
7.5 Generalized Fourier Series 125
7.6 Trigonometric Fourier Series 126
7.7 Hermitian Operators in Quantum Mechanics 127
Bibliography 129
Problems 130
8 Factorization Method 133
8.1 Another Form for the Sturm–Liouville Equation 133
8.2 Method of Factorization 135
8.3 Theory of Factorization and the Ladder Operators 136
8.4 Solutions via the Factorization Method 141
8.4.1 Case I (m > 0 and 𝜇(m) is an increasing function) 141
8.4.2 Case II (m > 0 and 𝜇(m) is a decreasing function) 142
8.5 Technique and the Categories of Factorization 143
8.5.1 Possible Forms for k(z,m) 143
8.5.1.1 Positive powers of m 143
8.5.1.2 Negative powers of m 146
8.6 Associated Legendre Equation (Type A) 148
8.6.1 Determining the Eigenvalues, 𝜆l 149
8.6.2 Construction of the Eigenfunctions 150
8.6.3 Ladder Operators for m 151
8.6.4 Interpretation of the L+ and L− Operators 153
8.6.5 Ladder Operators for l 155
8.6.6 Complete Set of Ladder Operators 159
8.7 Schrödinger Equation and Single-Electron Atom (Type F) 160
8.8 Gegenbauer Functions (Type A) 162
8.9 Symmetric Top (Type A) 163
8.10 Bessel Functions (Type C) 164
8.11 Harmonic Oscillator (Type D) 165
8.12 Differential Equation for the Rotation Matrix 166
8.12.1 Step-Up/Down Operators for m 166
8.12.2 Step-Up/Down Operators for m′ 167
8.12.3 Normalized Functions with m = m′ = l 168
8.12.4 Full Matrix for l = 2 168
8.12.5 Step-Up/Down Operators for l 170
Bibliography 171
Problems 171
9 Coordinates and Tensors 175
9.1 Cartesian Coordinates 175
9.1.1 Algebra of Vectors 176
9.1.2 Differentiation of Vectors 177
9.2 Orthogonal Transformations 178
9.2.1 Rotations About Cartesian Axes 182
9.2.2 Formal Properties of the Rotation Matrix 183
9.2.3 Euler Angles and Arbitrary Rotations 183
9.2.4 Active and Passive Interpretations of Rotations 185
9.2.5 Infinitesimal Transformations 186
9.2.6 Infinitesimal Transformations Commute 188
9.3 Cartesian Tensors 189
9.3.1 Operations with Cartesian Tensors 190
9.3.2 Tensor Densities or Pseudotensors 191
9.4 Cartesian Tensors and theTheory of Elasticity 192
9.4.1 Strain Tensor 192
9.4.2 Stress Tensor 193
9.4.3 Thermodynamics and Deformations 194
9.4.4 Connection between Shear and Strain 196
9.4.5 Hook’s Law 200
9.5 Generalized Coordinates and General Tensors 201
9.5.1 Contravariant and Covariant Components 202
9.5.2 Metric Tensor and the Line Element 203
9.5.3 Geometric Interpretation of Components 206
9.5.4 Interpretation of the Metric Tensor 207
9.6 Operations with General Tensors 214
9.6.1 Einstein Summation Convention 214
9.6.2 Contraction of Indices 214
9.6.3 Multiplication of Tensors 214
9.6.4 The Quotient Theorem 214
9.6.5 Equality of Tensors 215
9.6.6 Tensor Densities 215
9.6.7 Differentiation of Tensors 216
9.6.8 Some Covariant Derivatives 219
9.6.9 Riemann Curvature Tensor 220
9.7 Curvature 221
9.7.1 Parallel Transport 222
9.7.2 Round Trips via Parallel Transport 223
9.7.3 Algebraic Properties of the Curvature Tensor 225
9.7.4 Contractions of the Curvature Tensor 226
9.7.5 Curvature in n Dimensions 227
9.7.6 Geodesics 229
9.7.7 Invariance Versus Covariance 229
9.8 Spacetime and Four-Tensors 230
9.8.1 Minkowski Spacetime 230
9.8.2 Lorentz Transformations and Special Relativity 231
9.8.3 Time Dilation and Length Contraction 233
9.8.4 Addition of Velocities 233
9.8.5 Four-Tensors in Minkowski Spacetime 234
9.8.6 Four-Velocity 237
9.8.7 Four-Momentum and Conservation Laws 238
9.8.8 Mass of a Moving Particle 240
9.8.9 Wave Four-Vector 240
9.8.10 Derivative Operators in Spacetime 241
9.8.11 Relative Orientation of Axes in K and K Frames 241
9.9 Maxwell’s Equations in Minkowski Spacetime 243
9.9.1 Transformation of Electromagnetic Fields 246
9.9.2 Maxwell’s Equations in Terms of Potentials 246
9.9.3 Covariance of Newton’s Dynamic Theory 247
Bibliography 248
Problems 249
10 Continuous Groups and Representations 257
10.1 Definition of a Group 258
10.1.1 Nomenclature 258
10.2 Infinitesimal Ring or Lie Algebra 259
10.2.1 Properties of rG 260
10.3 Lie Algebra of the Rotation Group R(3) 260
10.3.1 Another Approach to rR(3) 262
10.4 Group Invariants 264
10.4.1 Lorentz Transformations 266
10.5 Unitary Group in Two Dimensions U(2) 267
10.5.1 Special Unitary Group SU(2) 269
10.5.2 Lie Algebra of SU(2) 270
10.5.3 Another Approach to rSU(2) 272
10.6 Lorentz Group and Its Lie Algebra 274
10.7 Group Representations 279
10.7.1 Schur’s Lemma 279
10.7.2 Group Character 280
10.7.3 Unitary Representation 280
10.8 Representations of R(3) 281
10.8.1 Spherical Harmonics and Representations of R(3) 281
10.8.2 Angular Momentum in Quantum Mechanics 281
10.8.3 Rotation of the Physical System 282
10.8.4 Rotation Operator in Terms of the Euler Angles 282
10.8.5 Rotation Operator in the Original Coordinates 283
10.8.6 Eigenvalue Equations for Lz, L±, and L2 287
10.8.7 Fourier Expansion in Spherical Harmonics 287
10.8.8 Matrix Elements of Lx, Ly, and Lz 289
10.8.9 Rotation Matrices of the Spherical Harmonics 290
10.8.10 Evaluation of the dlm′m(𝛽) Matrices 292
10.8.11 Inverse of the dlm′m(𝛽) Matrices 292
10.8.12 Differential Equation for dlm′m(𝛽) 293
10.8.13 AdditionTheorem for Spherical Harmonics 296
10.8.14 Determination of Il in the AdditionTheorem 298
10.8.15 Connection of Dlmm′ (𝛽) with Spherical Harmonics 300
10.9 Irreducible Representations of SU(2) 302
10.10 Relation of SU(2) and R(3) 303
10.11 Group Spaces 306
10.11.1 Real Vector Space 306
10.11.2 Inner Product Space 307
10.11.3 Four-Vector Space 307
10.11.4 Complex Vector Space 308
10.11.5 Function Space and Hilbert Space 308
10.11.6 Completeness 309
10.12 Hilbert Space and QuantumMechanics 310
10.13 Continuous Groups and Symmetries 311
10.13.1 Point Groups and Their Generators 311
10.13.2 Transformation of Generators and Normal Forms 312
10.13.3 The Case of Multiple Parameters 314
10.13.4 Action of Generators on Functions 315
10.13.5 Extension or Prolongation of Generators 316
10.13.6 Symmetries of Differential Equations 318
Bibliography 321
Problems 322
11 Complex Variables and Functions 327
11.1 Complex Algebra 327
11.2 Complex Functions 329
11.3 Complex Derivatives and Cauchy–Riemann Conditions 330
11.3.1 Analytic Functions 330
11.3.2 Harmonic Functions 332
11.4 Mappings 334
11.4.1 Conformal Mappings 348
11.4.2 Electrostatics and Conformal Mappings 349
11.4.3 Fluid Mechanics and Conformal Mappings 352
11.4.4 Schwarz–Christoffel Transformations 358
Bibliography 368
Problems 368
12 Complex Integrals and Series 373
12.1 Complex Integral Theorems 373
12.1.1 Cauchy–GoursatTheorem 373
12.1.2 Cauchy IntegralTheorem 374
12.1.3 CauchyTheorem 376
12.2 Taylor Series 378
12.3 Laurent Series 379
12.4 Classification of Singular Points 385
12.5 ResidueTheorem 386
12.6 Analytic Continuation 389
12.7 Complex Techniques in Taking Some Definite Integrals 392
12.8 Gamma and Beta Functions 399
12.8.1 Gamma Function 399
12.8.2 Beta Function 401
12.8.3 Useful Relations of the Gamma Functions 403
12.8.4 Incomplete Gamma and Beta Functions 403
12.8.5 Analytic Continuation of the Gamma Function 404
12.9 Cauchy Principal Value Integral 406
12.10 Integral Representations of Special Functions 410
12.10.1 Legendre Polynomials 410
12.10.2 Laguerre Polynomials 411
12.10.3 Bessel Functions 413
Bibliography 416
Problems 416
13 Fractional Calculus 423
13.1 Unified Expression of Derivatives and Integrals 425
13.1.1 Notation and Definitions 425
13.1.2 The nth Derivative of a Function 426
13.1.3 Successive Integrals 427
13.1.4 Unification of Derivative and Integral Operators 429
13.2 Differintegrals 429
13.2.1 Grünwald’s Definition of Differintegrals 429
13.2.2 Riemann–Liouville Definition of Differintegrals 431
13.3 Other Definitions of Differintegrals 434
13.3.1 Cauchy Integral Formula 434
13.3.2 Riemann Formula 439
13.3.3 Differintegrals via Laplace Transforms 440
13.4 Properties of Differintegrals 442
13.4.1 Linearity 443
13.4.2 Homogeneity 443
13.4.3 Scale Transformations 443
13.4.4 Differintegral of a Series 443
13.4.5 Composition of Differintegrals 444
13.4.5.1 Composition Rule for General q and Q 447
13.4.6 Leibniz Rule 450
13.4.7 Right- and Left-Handed Differintegrals 450
13.4.8 Dependence on the Lower Limit 452
13.5 Differintegrals of Some Functions 453
13.5.1 Differintegral of a Constant 453
13.5.2 Differintegral of [x − a] 454
13.5.3 Differintegral of [x − a]p (p > −1) 455
13.5.4 Differintegral of [1 − x]p 456
13.5.5 Differintegral of exp(±x) 456
13.5.6 Differintegral of ln(x) 457
13.5.7 Some Semiderivatives and Semi-Integrals 459
13.6 Mathematical Techniques with Differintegrals 459
13.6.1 Laplace Transform of Differintegrals 459
13.6.2 Extraordinary Differential Equations 463
13.6.3 Mittag–Leffler Functions 463
13.6.4 Semidifferential Equations 464
13.6.5 Evaluating Definite Integrals by Differintegrals 466
13.6.6 Evaluation of Sums of Series by Differintegrals 468
13.6.7 Special Functions Expressed as Differintegrals 469
13.7 Caputo Derivative 469
13.7.1 Caputo and the Riemann–Liouville Derivative 470
13.7.2 Mittag–Leffler Function and the Caputo Derivative 473
13.7.3 Right- and Left-Handed Caputo Derivatives 474
13.7.4 A Useful Relation of the Caputo Derivative 475
13.8 Riesz Fractional Integral and Derivative 477
13.8.1 Riesz Fractional Integral 477
13.8.2 Riesz Fractional Derivative 480
13.8.3 Fractional Laplacian 482
13.9 Applications of Differintegrals in Science and Engineering 482
13.9.1 Fractional Relaxation 482
13.9.2 Continuous Time RandomWalk (CTRW) 483
13.9.3 Time Fractional Diffusion Equation 486
13.9.4 Fractional Fokker–Planck Equations 487
Bibliography 489
Problems 490
14 Infinite Series 495
14.1 Convergence of Infinite Series 495
14.2 Absolute Convergence 496
14.3 Convergence Tests 496
14.3.1 Comparison Test 497
14.3.2 Ratio Test 497
14.3.3 Cauchy Root Test 497
14.3.4 Integral Test 497
14.3.5 Raabe Test 499
14.3.6 CauchyTheorem 499
14.3.7 Gauss Test and Legendre Series 500
14.3.8 Alternating Series 503
14.4 Algebra of Series 503
14.4.1 Rearrangement of Series 504
14.5 Useful Inequalities About Series 505
14.6 Series of Functions 506
14.6.1 Uniform Convergence 506
14.6.2 Weierstrass M-Test 507
14.6.3 Abel Test 507
14.6.4 Properties of Uniformly Convergent Series 508
14.7 Taylor Series 508
14.7.1 Maclaurin Theorem 509
14.7.2 BinomialTheorem 509
14.7.3 Taylor Series with Multiple Variables 510
14.8 Power Series 511
14.8.1 Convergence of Power Series 512
14.8.2 Continuity 512
14.8.3 Differentiation and Integration of Power Series 512
14.8.4 Uniqueness Theorem 513
14.8.5 Inversion of Power Series 513
14.9 Summation of Infinite Series 514
14.9.1 Bernoulli Polynomials and their Properties 514
14.9.2 Euler–Maclaurin Sum Formula 516
14.9.3 Using ResidueTheorem to Sum Infinite Series 519
14.9.4 Evaluating Sums of Series by Differintegrals 522
14.10 Asymptotic Series 523
14.11 Method of Steepest Descent 525
14.12 Saddle-Point Integrals 528
14.13 Padé Approximants 535
14.14 Divergent Series in Physics 539
14.14.1 Casimir Effect and Renormalization 540
14.14.2 Casimir Effect and MEMS 542
14.15 Infinite Products 542
14.15.1 Sine, Cosine, and the Gamma Functions 544
Bibliography 546
Problems 546
15 Integral Transforms 553
15.1 Some Commonly Encountered Integral Transforms 553
15.2 Derivation of the Fourier Integral 555
15.2.1 Fourier Series 555
15.2.2 Dirac-Delta Function 557
15.3 Fourier and Inverse Fourier Transforms 557
15.3.1 Fourier-Sine and Fourier-Cosine Transforms 558
15.4 Conventions and Properties of the Fourier Transforms 560
15.4.1 Shifting 561
15.4.2 Scaling 561
15.4.3 Transform of an Integral 561
15.4.4 Modulation 561
15.4.5 Fourier Transform of a Derivative 563
15.4.6 Convolution Theorem 564
15.4.7 Existence of Fourier Transforms 565
15.4.8 Fourier Transforms inThree Dimensions 565
15.4.9 ParsevalTheorems 566
15.5 Discrete Fourier Transform 572
15.6 Fast Fourier Transform 576
15.7 Radon Transform 578
15.8 Laplace Transforms 581
15.9 Inverse Laplace Transforms 581
15.9.1 Bromwich Integral 582
15.9.2 Elementary Laplace Transforms 583
15.9.3 Theorems About Laplace Transforms 584
15.9.4 Method of Partial Fractions 591
15.10 Laplace Transform of a Derivative 593
15.10.1 Laplace Transforms in n Dimensions 600
15.11 Relation Between Laplace and Fourier Transforms 601
15.12 Mellin Transforms 601
Bibliography 602
Problems 602
16 Variational Analysis 607
16.1 Presence of One Dependent and One Independent Variable 608
16.1.1 Euler Equation 608
16.1.2 Another Form of the Euler Equation 610
16.1.3 Applications of the Euler Equation 610
16.2 Presence of More than One Dependent Variable 617
16.3 Presence of More than One Independent Variable 617
16.4 Presence of Multiple Dependent and Independent Variables 619
16.5 Presence of Higher-Order Derivatives 619
16.6 Isoperimetric Problems and the Presence of Constraints 622
16.7 Applications to Classical Mechanics 626
16.7.1 Hamilton’s Principle 626
16.8 Eigenvalue Problems and Variational Analysis 628
16.9 Rayleigh–RitzMethod 632
16.10 Optimum Control Theory 637
16.11 BasicTheory: Dynamics versus Controlled Dynamics 638
16.11.1 Connection with Variational Analysis 641
16.11.2 Controllability of a System 642
Bibliography 646
Problems 647
17 Integral Equations 653
17.1 Classification of Integral Equations 654
17.2 Integral and Differential Equations 654
17.2.1 Converting Differential Equations into Integral Equations 656
17.2.2 Converting Integral Equations into Differential Equations 658
17.3 Solution of Integral Equations 659
17.3.1 Method of Successive Iterations: Neumann Series 659
17.3.2 Error Calculation in Neumann Series 660
17.3.3 Solution for the Case of Separable Kernels 661
17.3.4 Solution by Integral Transforms 663
17.3.4.1 Fourier Transform Method 663
17.3.4.2 Laplace Transform Method 664
17.4 Hilbert–Schmidt Theory 665
17.4.1 Eigenvalues for Hermitian Operators 665
17.4.2 Orthogonality of Eigenfunctions 666
17.4.3 Completeness of the Eigenfunction Set 666
17.5 Neumann Series and the Sturm–Liouville Problem 668
17.6 Eigenvalue Problem for the Non-Hermitian Kernels 672
Bibliography 672
Problems 672
18 Green’s Functions 675
18.1 Time-Independent Green’s Functions in One Dimension 675
18.1.1 Abel’s Formula 677
18.1.2 Constructing the Green’s Function 677
18.1.3 Differential Equation for the Green’s Function 679
18.1.4 Single-Point Boundary Conditions 679
18.1.5 Green’s Function for the Operator d2¨Mdx2 680
18.1.6 Inhomogeneous Boundary Conditions 682
18.1.7 Green’s Functions and Eigenvalue Problems 684
18.1.8 Green’s Functions and the Dirac-Delta Function 686
18.1.9 Helmholtz Equation with Discrete Spectrum 687
18.1.10 Helmholtz Equation in the Continuum Limit 688
18.1.11 Another Approach for the Green’s function 697
18.2 Time-Independent Green’s Functions inThree Dimensions 701
18.2.1 Helmholtz Equation in Three Dimensions 701
18.2.2 Green’s Functions inThree Dimensions 702
18.2.3 Green’s Function for the Laplace Operator 704
18.2.4 Green’s Functions for the Helmholtz Equation 705
18.2.5 General Boundary Conditions and Electrostatics 710
18.2.6 Helmholtz Equation in Spherical Coordinates 712
18.2.7 Diffraction from a Circular Aperture 716
18.3 Time-Independent PerturbationTheory 721
18.3.1 Nondegenerate PerturbationTheory 721
18.3.2 Slightly Anharmonic Oscillator in One Dimension 726
18.3.3 Degenerate PerturbationTheory 728
18.4 First-Order Time-Dependent Green’s Functions 729
18.4.1 Propagators 732
18.4.2 Compounding Propagators 732
18.4.3 Diffusion Equation with Discrete Spectrum 733
18.4.4 Diffusion Equation in the Continuum Limit 734
18.4.5 Presence of Sources or Interactions 736
18.4.6 Schrödinger Equation for Free Particles 737
18.4.7 Schrödinger Equation with Interactions 738
18.5 Second-Order Time-Dependent Green’s Functions 738
18.5.1 Propagators for the ScalarWave Equation 741
18.5.2 Advanced and Retarded Green’s Functions 743
18.5.3 ScalarWave Equation 745
Bibliography 747
Problems 748
19 Green’s Functions and Path Integrals 755
19.1 Brownian Motion and the Diffusion Problem 755
19.1.1 Wiener Path Integral and Brownian Motion 757
19.1.2 Perturbative Solution of the Bloch Equation 760
19.1.3 Derivation of the Feynman–Kac Formula 763
19.1.4 Interpretation of V(x) in the Bloch Equation 765
19.2 Methods of Calculating Path Integrals 767
19.2.1 Method of Time Slices 769
19.2.2 Path Integrals with the ESKC Relation 770
19.2.3 Path Integrals by the Method of Finite Elements 771
19.2.4 Path Integrals by the “Semiclassical” Method 772
19.3 Path Integral Formulation of Quantum Mechanics 776
19.3.1 Schrödinger Equation For a Free Particle 776
19.3.2 Schrödinger Equation with a Potential 778
19.3.3 Feynman Phase Space Path Integral 780
19.3.4 The Case of Quadratic Dependence on Momentum 781
19.4 Path Integrals Over Lévy Paths and Anomalous Diffusion 783
19.5 Fox’s H-Functions 788
19.5.1 Properties of the H-Functions 789
19.5.2 Useful Relations of the H-Functions 791
19.5.3 Examples of H-Functions 792
19.5.4 Computable Form of the H-Function 796
19.6 Applications of H-Functions 797
19.6.1 Riemann–Liouville Definition of Differintegral 798
19.6.2 Caputo Fractional Derivative 798
19.6.3 Fractional Relaxation 799
19.6.4 Time Fractional Diffusion via R–L Derivative 800
19.6.5 Time Fractional Diffusion via Caputo Derivative 801
19.6.6 Derivation of the Lévy Distribution 803
19.6.7 Lévy Distributions in Nature 806
19.6.8 Time and Space Fractional Schrödinger Equation 806
19.6.8.1 Free Particle Solution 808
19.7 Space Fractional Schrödinger Equation 809
19.7.1 Feynman Path Integrals Over Lévy Paths 810
19.8 Time Fractional Schrödinger Equation 812
19.8.1 Separable Solutions 812
19.8.2 Time Dependence 813
19.8.3 Mittag–Leffler Function and the Caputo Derivative 814
19.8.4 Euler Equation for the Mittag–Leffler Function 814
Bibliography 817
Problems 818
Further Reading 825
Index 827
