**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.

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 (m) is an increasing function) 141

8.4.2 Case II (m > 0 and (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, l 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( ) Matrices 292

10.8.11 Inverse of the dlm m( ) Matrices 292

10.8.12 Differential Equation for dlm m( ) 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 ( ) 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

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