{"product_id":"mathematical-methods-in-science-and-engineering-isbn-9781119425397","title":"Mathematical Methods in Science and Engineering","description":"\u003cp\u003e\u003cb\u003eA Practical, Interdisciplinary Guide to Advanced Mathematical Methods for Scientists and Engineers \u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eMathematical Methods in Science and Engineering, Second Edition\u003c\/i\u003e, 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. \u003c\/p\u003e \u003cp\u003eRecent 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.\u003c\/p\u003e \u003cp\u003eRevised and expanded for increased utility, this new Second Edition:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eIncludes over 60 new sections and subsections more useful to a multidisciplinary audience\u003c\/li\u003e \u003cli\u003eContains new examples, new figures, new problems, and more fluid arguments\u003c\/li\u003e \u003cli\u003ePresents a detailed discussion on the most frequently encountered special functions in science and engineering\u003c\/li\u003e \u003cli\u003eProvides a systematic treatment of special functions in terms of the Sturm-Liouville theory\u003c\/li\u003e \u003cli\u003eApproaches second-order differential equations of physics and engineering from the factorization perspective\u003c\/li\u003e \u003cli\u003eIncludes extensive discussion of coordinate transformations and tensors, complex analysis, fractional calculus, integral transforms, Green's functions, path integrals, and more\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eExtensively 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. \u003c\/p\u003e \u003cp\u003ePreface xix\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Legendre Equation and Polynomials 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Second-Order Differential Equations of Physics 1\u003c\/p\u003e \u003cp\u003e1.2 Legendre Equation 2\u003c\/p\u003e \u003cp\u003e1.2.1 Method of Separation of Variables 4\u003c\/p\u003e \u003cp\u003e1.2.2 Series Solution of the Legendre Equation 4\u003c\/p\u003e \u003cp\u003e1.2.3 Frobenius Method – Review 7\u003c\/p\u003e \u003cp\u003e1.3 Legendre Polynomials 8\u003c\/p\u003e \u003cp\u003e1.3.1 Rodriguez Formula 10\u003c\/p\u003e \u003cp\u003e1.3.2 Generating Function 10\u003c\/p\u003e \u003cp\u003e1.3.3 Recursion Relations 12\u003c\/p\u003e \u003cp\u003e1.3.4 Special Values 12\u003c\/p\u003e \u003cp\u003e1.3.5 Special Integrals 13\u003c\/p\u003e \u003cp\u003e1.3.6 Orthogonality and Completeness 14\u003c\/p\u003e \u003cp\u003e1.3.7 Asymptotic Forms 17\u003c\/p\u003e \u003cp\u003e1.4 Associated Legendre Equation and Polynomials 18\u003c\/p\u003e \u003cp\u003e1.4.1 Associated Legendre Polynomials Pm l (x) 20\u003c\/p\u003e \u003cp\u003e1.4.2 Orthogonality 21\u003c\/p\u003e \u003cp\u003e1.4.3 Recursion Relations 22\u003c\/p\u003e \u003cp\u003e1.4.4 Integral Representations 24\u003c\/p\u003e \u003cp\u003e1.4.5 Associated Legendre Polynomials for m \u0026lt; 0 26\u003c\/p\u003e \u003cp\u003e1.5 Spherical Harmonics 27\u003c\/p\u003e \u003cp\u003e1.5.1 Addition Theorem of Spherical Harmonics 30\u003c\/p\u003e \u003cp\u003e1.5.2 Real Spherical Harmonics 33\u003c\/p\u003e \u003cp\u003eBibliography 33\u003c\/p\u003e \u003cp\u003eProblems 34\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Laguerre Polynomials 39\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Central Force Problems in Quantum Mechanics 39\u003c\/p\u003e \u003cp\u003e2.2 Laguerre Equation and Polynomials 41\u003c\/p\u003e \u003cp\u003e2.2.1 Generating Function 42\u003c\/p\u003e \u003cp\u003e2.2.2 Rodriguez Formula 43\u003c\/p\u003e \u003cp\u003e2.2.3 Orthogonality 44\u003c\/p\u003e \u003cp\u003e2.2.4 Recursion Relations 45\u003c\/p\u003e \u003cp\u003e2.2.5 Special Values 46\u003c\/p\u003e \u003cp\u003e2.3 Associated Laguerre Equation and Polynomials 46\u003c\/p\u003e \u003cp\u003e2.3.1 Generating Function 48\u003c\/p\u003e \u003cp\u003e2.3.2 Rodriguez Formula and Orthogonality 49\u003c\/p\u003e \u003cp\u003e2.3.3 Recursion Relations 49\u003c\/p\u003e \u003cp\u003eBibliography 49\u003c\/p\u003e \u003cp\u003eProblems 50\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Hermite Polynomials 53\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Harmonic Oscillator in Quantum Mechanics 53\u003c\/p\u003e \u003cp\u003e3.2 Hermite Equation and Polynomials 54\u003c\/p\u003e \u003cp\u003e3.2.1 Generating Function 56\u003c\/p\u003e \u003cp\u003e3.2.2 Rodriguez Formula 56\u003c\/p\u003e \u003cp\u003e3.2.3 Recursion Relations and Orthogonality 57\u003c\/p\u003e \u003cp\u003eBibliography 61\u003c\/p\u003e \u003cp\u003eProblems 62\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Gegenbauer and Chebyshev Polynomials 65\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Wave Equation on a Hypersphere 65\u003c\/p\u003e \u003cp\u003e4.2 Gegenbauer Equation and Polynomials 68\u003c\/p\u003e \u003cp\u003e4.2.1 Orthogonality and the Generating Function 68\u003c\/p\u003e \u003cp\u003e4.2.2 Another Representation of the Solution 69\u003c\/p\u003e \u003cp\u003e4.2.3 The Second Solution 70\u003c\/p\u003e \u003cp\u003e4.2.4 Connection with the Gegenbauer Polynomials 71\u003c\/p\u003e \u003cp\u003e4.2.5 Evaluation of the Normalization Constant 72\u003c\/p\u003e \u003cp\u003e4.3 Chebyshev Equation and Polynomials 72\u003c\/p\u003e \u003cp\u003e4.3.1 Chebyshev Polynomials of the First Kind 72\u003c\/p\u003e \u003cp\u003e4.3.2 Chebyshev and Gegenbauer Polynomials 73\u003c\/p\u003e \u003cp\u003e4.3.3 Chebyshev Polynomials of the Second Kind 73\u003c\/p\u003e \u003cp\u003e4.3.4 Orthogonality and Generating Function 74\u003c\/p\u003e \u003cp\u003e4.3.5 Another Definition 75\u003c\/p\u003e \u003cp\u003eBibliography 76\u003c\/p\u003e \u003cp\u003eProblems 76\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Bessel Functions 81\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Bessel’s Equation 83\u003c\/p\u003e \u003cp\u003e5.2 Bessel Functions 83\u003c\/p\u003e \u003cp\u003e5.2.1 Asymptotic Forms 84\u003c\/p\u003e \u003cp\u003e5.3 Modified Bessel Functions 86\u003c\/p\u003e \u003cp\u003e5.4 Spherical Bessel Functions 87\u003c\/p\u003e \u003cp\u003e5.5 Properties of Bessel Functions 88\u003c\/p\u003e \u003cp\u003e5.5.1 Generating Function 88\u003c\/p\u003e \u003cp\u003e5.5.2 Integral Definitions 89\u003c\/p\u003e \u003cp\u003e5.5.3 Recursion Relations of the Bessel Functions 89\u003c\/p\u003e \u003cp\u003e5.5.4 Orthogonality and Roots of Bessel Functions 90\u003c\/p\u003e \u003cp\u003e5.5.5 Boundary Conditions for the Bessel Functions 91\u003c\/p\u003e \u003cp\u003e5.5.6 Wronskian of Pairs of Solutions 94\u003c\/p\u003e \u003cp\u003e5.6 Transformations of Bessel Functions 95\u003c\/p\u003e \u003cp\u003e5.6.1 Critical Length of a Rod 96\u003c\/p\u003e \u003cp\u003eBibliography 98\u003c\/p\u003e \u003cp\u003eProblems 99\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Hypergeometric Functions 103\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Hypergeometric Series 103\u003c\/p\u003e \u003cp\u003e6.2 Hypergeometric Representations of Special Functions 107\u003c\/p\u003e \u003cp\u003e6.3 Confluent Hypergeometric Equation 108\u003c\/p\u003e \u003cp\u003e6.4 Pochhammer Symbol and Hypergeometric Functions 109\u003c\/p\u003e \u003cp\u003e6.5 Reduction of Parameters 113\u003c\/p\u003e \u003cp\u003eBibliography 115\u003c\/p\u003e \u003cp\u003eProblems 115\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Sturm–Liouville Theory 119\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Self-Adjoint Differential Operators 119\u003c\/p\u003e \u003cp\u003e7.2 Sturm–Liouville Systems 120\u003c\/p\u003e \u003cp\u003e7.3 Hermitian Operators 121\u003c\/p\u003e \u003cp\u003e7.4 Properties of Hermitian Operators 122\u003c\/p\u003e \u003cp\u003e7.4.1 Real Eigenvalues 122\u003c\/p\u003e \u003cp\u003e7.4.2 Orthogonality of Eigenfunctions 123\u003c\/p\u003e \u003cp\u003e7.4.3 Completeness and the Expansion Theorem 123\u003c\/p\u003e \u003cp\u003e7.5 Generalized Fourier Series 125\u003c\/p\u003e \u003cp\u003e7.6 Trigonometric Fourier Series 126\u003c\/p\u003e \u003cp\u003e7.7 Hermitian Operators in Quantum Mechanics 127\u003c\/p\u003e \u003cp\u003eBibliography 129\u003c\/p\u003e \u003cp\u003eProblems 130\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Factorization Method 133\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Another Form for the Sturm–Liouville Equation 133\u003c\/p\u003e \u003cp\u003e8.2 Method of Factorization 135\u003c\/p\u003e \u003cp\u003e8.3 Theory of Factorization and the Ladder Operators 136\u003c\/p\u003e \u003cp\u003e8.4 Solutions via the Factorization Method 141\u003c\/p\u003e \u003cp\u003e8.4.1 Case I (m \u0026gt; 0 and 𝜇(m) is an increasing function) 141\u003c\/p\u003e \u003cp\u003e8.4.2 Case II (m \u0026gt; 0 and 𝜇(m) is a decreasing function) 142\u003c\/p\u003e \u003cp\u003e8.5 Technique and the Categories of Factorization 143\u003c\/p\u003e \u003cp\u003e8.5.1 Possible Forms for k(z, m) 143\u003c\/p\u003e \u003cp\u003e8.5.1.1 Positive powers of m 143\u003c\/p\u003e \u003cp\u003e8.5.1.2 Negative powers of m 146\u003c\/p\u003e \u003cp\u003e8.6 Associated Legendre Equation (Type A) 148\u003c\/p\u003e \u003cp\u003e8.6.1 Determining the Eigenvalues, 𝜆l 149\u003c\/p\u003e \u003cp\u003e8.6.2 Construction of the Eigenfunctions 150\u003c\/p\u003e \u003cp\u003e8.6.3 Ladder Operators for m 151\u003c\/p\u003e \u003cp\u003e8.6.4 Interpretation of the L+ and L− Operators 153\u003c\/p\u003e \u003cp\u003e8.6.5 Ladder Operators for l 155\u003c\/p\u003e \u003cp\u003e8.6.6 Complete Set of Ladder Operators 159\u003c\/p\u003e \u003cp\u003e8.7 Schrödinger Equation and Single-Electron Atom (Type F) 160\u003c\/p\u003e \u003cp\u003e8.8 Gegenbauer Functions (Type A) 162\u003c\/p\u003e \u003cp\u003e8.9 Symmetric Top (Type A) 163\u003c\/p\u003e \u003cp\u003e8.10 Bessel Functions (Type C) 164\u003c\/p\u003e \u003cp\u003e8.11 Harmonic Oscillator (Type D) 165\u003c\/p\u003e \u003cp\u003e8.12 Differential Equation for the Rotation Matrix 166\u003c\/p\u003e \u003cp\u003e8.12.1 Step-Up\/Down Operators for m 166\u003c\/p\u003e \u003cp\u003e8.12.2 Step-Up\/Down Operators for m′ 167\u003c\/p\u003e \u003cp\u003e8.12.3 Normalized Functions with m = m′ = l 168\u003c\/p\u003e \u003cp\u003e8.12.4 Full Matrix for l = 2 168\u003c\/p\u003e \u003cp\u003e8.12.5 Step-Up\/Down Operators for l 170\u003c\/p\u003e \u003cp\u003eBibliography 171\u003c\/p\u003e \u003cp\u003eProblems 171\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Coordinates and Tensors 175\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Cartesian Coordinates 175\u003c\/p\u003e \u003cp\u003e9.1.1 Algebra of Vectors 176\u003c\/p\u003e \u003cp\u003e9.1.2 Differentiation of Vectors 177\u003c\/p\u003e \u003cp\u003e9.2 Orthogonal Transformations 178\u003c\/p\u003e \u003cp\u003e9.2.1 Rotations About Cartesian Axes 182\u003c\/p\u003e \u003cp\u003e9.2.2 Formal Properties of the Rotation Matrix 183\u003c\/p\u003e \u003cp\u003e9.2.3 Euler Angles and Arbitrary Rotations 183\u003c\/p\u003e \u003cp\u003e9.2.4 Active and Passive Interpretations of Rotations 185\u003c\/p\u003e \u003cp\u003e9.2.5 Infinitesimal Transformations 186\u003c\/p\u003e \u003cp\u003e9.2.6 Infinitesimal Transformations Commute 188\u003c\/p\u003e \u003cp\u003e9.3 Cartesian Tensors 189\u003c\/p\u003e \u003cp\u003e9.3.1 Operations with Cartesian Tensors 190\u003c\/p\u003e \u003cp\u003e9.3.2 Tensor Densities or Pseudotensors 191\u003c\/p\u003e \u003cp\u003e9.4 Cartesian Tensors and the Theory of Elasticity 192\u003c\/p\u003e \u003cp\u003e9.4.1 Strain Tensor 192\u003c\/p\u003e \u003cp\u003e9.4.2 Stress Tensor 193\u003c\/p\u003e \u003cp\u003e9.4.3 Thermodynamics and Deformations 194\u003c\/p\u003e \u003cp\u003e9.4.4 Connection between Shear and Strain 196\u003c\/p\u003e \u003cp\u003e9.4.5 Hook’s Law 200\u003c\/p\u003e \u003cp\u003e9.5 Generalized Coordinates and General Tensors 201\u003c\/p\u003e \u003cp\u003e9.5.1 Contravariant and Covariant Components 202\u003c\/p\u003e \u003cp\u003e9.5.2 Metric Tensor and the Line Element 203\u003c\/p\u003e \u003cp\u003e9.5.3 Geometric Interpretation of Components 206\u003c\/p\u003e \u003cp\u003e9.5.4 Interpretation of the Metric Tensor 207\u003c\/p\u003e \u003cp\u003e9.6 Operations with General Tensors 214\u003c\/p\u003e \u003cp\u003e9.6.1 Einstein Summation Convention 214\u003c\/p\u003e \u003cp\u003e9.6.2 Contraction of Indices 214\u003c\/p\u003e \u003cp\u003e9.6.3 Multiplication of Tensors 214\u003c\/p\u003e \u003cp\u003e9.6.4 The Quotient Theorem 214\u003c\/p\u003e \u003cp\u003e9.6.5 Equality of Tensors 215\u003c\/p\u003e \u003cp\u003e9.6.6 Tensor Densities 215\u003c\/p\u003e \u003cp\u003e9.6.7 Differentiation of Tensors 216\u003c\/p\u003e \u003cp\u003e9.6.8 Some Covariant Derivatives 219\u003c\/p\u003e \u003cp\u003e9.6.9 Riemann Curvature Tensor 220\u003c\/p\u003e \u003cp\u003e9.7 Curvature 221\u003c\/p\u003e \u003cp\u003e9.7.1 Parallel Transport 222\u003c\/p\u003e \u003cp\u003e9.7.2 Round Trips via Parallel Transport 223\u003c\/p\u003e \u003cp\u003e9.7.3 Algebraic Properties of the Curvature Tensor 225\u003c\/p\u003e \u003cp\u003e9.7.4 Contractions of the Curvature Tensor 226\u003c\/p\u003e \u003cp\u003e9.7.5 Curvature in n Dimensions 227\u003c\/p\u003e \u003cp\u003e9.7.6 Geodesics 229\u003c\/p\u003e \u003cp\u003e9.7.7 Invariance Versus Covariance 229\u003c\/p\u003e \u003cp\u003e9.8 Spacetime and Four-Tensors 230\u003c\/p\u003e \u003cp\u003e9.8.1 Minkowski Spacetime 230\u003c\/p\u003e \u003cp\u003e9.8.2 Lorentz Transformations and Special Relativity 231\u003c\/p\u003e \u003cp\u003e9.8.3 Time Dilation and Length Contraction 233\u003c\/p\u003e \u003cp\u003e9.8.4 Addition of Velocities 233\u003c\/p\u003e \u003cp\u003e9.8.5 Four-Tensors in Minkowski Spacetime 234\u003c\/p\u003e \u003cp\u003e9.8.6 Four-Velocity 237\u003c\/p\u003e \u003cp\u003e9.8.7 Four-Momentum and Conservation Laws 238\u003c\/p\u003e \u003cp\u003e9.8.8 Mass of a Moving Particle 240\u003c\/p\u003e \u003cp\u003e9.8.9 Wave Four-Vector 240\u003c\/p\u003e \u003cp\u003e9.8.10 Derivative Operators in Spacetime 241\u003c\/p\u003e \u003cp\u003e9.8.11 Relative Orientation of Axes in K and K Frames 241\u003c\/p\u003e \u003cp\u003e9.9 Maxwell’s Equations in Minkowski Spacetime 243\u003c\/p\u003e \u003cp\u003e9.9.1 Transformation of Electromagnetic Fields 246\u003c\/p\u003e \u003cp\u003e9.9.2 Maxwell’s Equations in Terms of Potentials 246\u003c\/p\u003e \u003cp\u003e9.9.3 Covariance of Newton’s Dynamic Theory 247\u003c\/p\u003e \u003cp\u003eBibliography 248\u003c\/p\u003e \u003cp\u003eProblems 249\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Continuous Groups and Representations 257\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Definition of a Group 258\u003c\/p\u003e \u003cp\u003e10.1.1 Nomenclature 258\u003c\/p\u003e \u003cp\u003e10.2 Infinitesimal Ring or Lie Algebra 259\u003c\/p\u003e \u003cp\u003e10.2.1 Properties of r G 260\u003c\/p\u003e \u003cp\u003e10.3 Lie Algebra of the Rotation Group R(3) 260\u003c\/p\u003e \u003cp\u003e10.3.1 Another Approach to r R(3) 262\u003c\/p\u003e \u003cp\u003e10.4 Group Invariants 264\u003c\/p\u003e \u003cp\u003e10.4.1 Lorentz Transformations 266\u003c\/p\u003e \u003cp\u003e10.5 Unitary Group in Two Dimensions U(2) 267\u003c\/p\u003e \u003cp\u003e10.5.1 Special Unitary Group SU(2) 269\u003c\/p\u003e \u003cp\u003e10.5.2 Lie Algebra of SU(2) 270\u003c\/p\u003e \u003cp\u003e10.5.3 Another Approach to r SU(2) 272\u003c\/p\u003e \u003cp\u003e10.6 Lorentz Group and Its Lie Algebra 274\u003c\/p\u003e \u003cp\u003e10.7 Group Representations 279\u003c\/p\u003e \u003cp\u003e10.7.1 Schur’s Lemma 279\u003c\/p\u003e \u003cp\u003e10.7.2 Group Character 280\u003c\/p\u003e \u003cp\u003e10.7.3 Unitary Representation 280\u003c\/p\u003e \u003cp\u003e10.8 Representations of R(3) 281\u003c\/p\u003e \u003cp\u003e10.8.1 Spherical Harmonics and Representations of R(3) 281\u003c\/p\u003e \u003cp\u003e10.8.2 Angular Momentum in Quantum Mechanics 281\u003c\/p\u003e \u003cp\u003e10.8.3 Rotation of the Physical System 282\u003c\/p\u003e \u003cp\u003e10.8.4 Rotation Operator in Terms of the Euler Angles 282\u003c\/p\u003e \u003cp\u003e10.8.5 Rotation Operator in the Original Coordinates 283\u003c\/p\u003e \u003cp\u003e10.8.6 Eigenvalue Equations for Lz, L±, and L2 287\u003c\/p\u003e \u003cp\u003e10.8.7 Fourier Expansion in Spherical Harmonics 287\u003c\/p\u003e \u003cp\u003e10.8.8 Matrix Elements of Lx, Ly, and Lz 289\u003c\/p\u003e \u003cp\u003e10.8.9 Rotation Matrices of the Spherical Harmonics 290\u003c\/p\u003e \u003cp\u003e10.8.10 Evaluation of the dl m′m(𝛽) Matrices 292\u003c\/p\u003e \u003cp\u003e10.8.11 Inverse of the dl m′m(𝛽) Matrices 292\u003c\/p\u003e \u003cp\u003e10.8.12 Differential Equation for dl m′m(𝛽) 293\u003c\/p\u003e \u003cp\u003e10.8.13 Addition Theorem for Spherical Harmonics 296\u003c\/p\u003e \u003cp\u003e10.8.14 Determination of Il in the Addition Theorem 298\u003c\/p\u003e \u003cp\u003e10.8.15 Connection of Dl mm′ (𝛽) with Spherical Harmonics 300\u003c\/p\u003e \u003cp\u003e10.9 Irreducible Representations of SU(2) 302\u003c\/p\u003e \u003cp\u003e10.10 Relation of SU(2) and R(3) 303\u003c\/p\u003e \u003cp\u003e10.11 Group Spaces 306\u003c\/p\u003e \u003cp\u003e10.11.1 Real Vector Space 306\u003c\/p\u003e \u003cp\u003e10.11.2 Inner Product Space 307\u003c\/p\u003e \u003cp\u003e10.11.3 Four-Vector Space 307\u003c\/p\u003e \u003cp\u003e10.11.4 Complex Vector Space 308\u003c\/p\u003e \u003cp\u003e10.11.5 Function Space and Hilbert Space 308\u003c\/p\u003e \u003cp\u003e10.11.6 Completeness 309\u003c\/p\u003e \u003cp\u003e10.12 Hilbert Space and Quantum Mechanics 310\u003c\/p\u003e \u003cp\u003e10.13 Continuous Groups and Symmetries 311\u003c\/p\u003e \u003cp\u003e10.13.1 Point Groups and Their Generators 311\u003c\/p\u003e \u003cp\u003e10.13.2 Transformation of Generators and Normal Forms 312\u003c\/p\u003e \u003cp\u003e10.13.3 The Case of Multiple Parameters 314\u003c\/p\u003e \u003cp\u003e10.13.4 Action of Generators on Functions 315\u003c\/p\u003e \u003cp\u003e10.13.5 Extension or Prolongation of Generators 316\u003c\/p\u003e \u003cp\u003e10.13.6 Symmetries of Differential Equations 318\u003c\/p\u003e \u003cp\u003eBibliography 321\u003c\/p\u003e \u003cp\u003eProblems 322\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Complex Variables and Functions 327\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Complex Algebra 327\u003c\/p\u003e \u003cp\u003e11.2 Complex Functions 329\u003c\/p\u003e \u003cp\u003e11.3 Complex Derivatives and Cauchy–Riemann Conditions 330\u003c\/p\u003e \u003cp\u003e11.3.1 Analytic Functions 330\u003c\/p\u003e \u003cp\u003e11.3.2 Harmonic Functions 332\u003c\/p\u003e \u003cp\u003e11.4 Mappings 334\u003c\/p\u003e \u003cp\u003e11.4.1 Conformal Mappings 348\u003c\/p\u003e \u003cp\u003e11.4.2 Electrostatics and Conformal Mappings 349\u003c\/p\u003e \u003cp\u003e11.4.3 Fluid Mechanics and Conformal Mappings 352\u003c\/p\u003e \u003cp\u003e11.4.4 Schwarz–Christoffel Transformations 358\u003c\/p\u003e \u003cp\u003eBibliography 368\u003c\/p\u003e \u003cp\u003eProblems 368\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Complex Integrals and Series 373\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Complex Integral Theorems 373\u003c\/p\u003e \u003cp\u003e12.1.1 Cauchy–Goursat Theorem 373\u003c\/p\u003e \u003cp\u003e12.1.2 Cauchy Integral Theorem 374\u003c\/p\u003e \u003cp\u003e12.1.3 Cauchy Theorem 376\u003c\/p\u003e \u003cp\u003e12.2 Taylor Series 378\u003c\/p\u003e \u003cp\u003e12.3 Laurent Series 379\u003c\/p\u003e \u003cp\u003e12.4 Classification of Singular Points 385\u003c\/p\u003e \u003cp\u003e12.5 Residue Theorem 386\u003c\/p\u003e \u003cp\u003e12.6 Analytic Continuation 389\u003c\/p\u003e \u003cp\u003e12.7 Complex Techniques in Taking Some Definite Integrals 392\u003c\/p\u003e \u003cp\u003e12.8 Gamma and Beta Functions 399\u003c\/p\u003e \u003cp\u003e12.8.1 Gamma Function 399\u003c\/p\u003e \u003cp\u003e12.8.2 Beta Function 401\u003c\/p\u003e \u003cp\u003e12.8.3 Useful Relations of the Gamma Functions 403\u003c\/p\u003e \u003cp\u003e12.8.4 Incomplete Gamma and Beta Functions 403\u003c\/p\u003e \u003cp\u003e12.8.5 Analytic Continuation of the Gamma Function 404\u003c\/p\u003e \u003cp\u003e12.9 Cauchy Principal Value Integral 406\u003c\/p\u003e \u003cp\u003e12.10 Integral Representations of Special Functions 410\u003c\/p\u003e \u003cp\u003e12.10.1 Legendre Polynomials 410\u003c\/p\u003e \u003cp\u003e12.10.2 Laguerre Polynomials 411\u003c\/p\u003e \u003cp\u003e12.10.3 Bessel Functions 413\u003c\/p\u003e \u003cp\u003eBibliography 416\u003c\/p\u003e \u003cp\u003eProblems 416\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Fractional Calculus 423\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Unified Expression of Derivatives and Integrals 425\u003c\/p\u003e \u003cp\u003e13.1.1 Notation and Definitions 425\u003c\/p\u003e \u003cp\u003e13.1.2 The nth Derivative of a Function 426\u003c\/p\u003e \u003cp\u003e13.1.3 Successive Integrals 427\u003c\/p\u003e \u003cp\u003e13.1.4 Unification of Derivative and Integral Operators 429\u003c\/p\u003e \u003cp\u003e13.2 Differintegrals 429\u003c\/p\u003e \u003cp\u003e13.2.1 Grünwald’s Definition of Differintegrals 429\u003c\/p\u003e \u003cp\u003e13.2.2 Riemann–Liouville Definition of Differintegrals 431\u003c\/p\u003e \u003cp\u003e13.3 Other Definitions of Differintegrals 434\u003c\/p\u003e \u003cp\u003e13.3.1 Cauchy Integral Formula 434\u003c\/p\u003e \u003cp\u003e13.3.2 Riemann Formula 439\u003c\/p\u003e \u003cp\u003e13.3.3 Differintegrals via Laplace Transforms 440\u003c\/p\u003e \u003cp\u003e13.4 Properties of Differintegrals 442\u003c\/p\u003e \u003cp\u003e13.4.1 Linearity 443\u003c\/p\u003e \u003cp\u003e13.4.2 Homogeneity 443\u003c\/p\u003e \u003cp\u003e13.4.3 Scale Transformations 443\u003c\/p\u003e \u003cp\u003e13.4.4 Differintegral of a Series 443\u003c\/p\u003e \u003cp\u003e13.4.5 Composition of Differintegrals 444\u003c\/p\u003e \u003cp\u003e13.4.5.1 Composition Rule for General q and Q 447\u003c\/p\u003e \u003cp\u003e13.4.6 Leibniz Rule 450\u003c\/p\u003e \u003cp\u003e13.4.7 Right- and Left-Handed Differintegrals 450\u003c\/p\u003e \u003cp\u003e13.4.8 Dependence on the Lower Limit 452\u003c\/p\u003e \u003cp\u003e13.5 Differintegrals of Some Functions 453\u003c\/p\u003e \u003cp\u003e13.5.1 Differintegral of a Constant 453\u003c\/p\u003e \u003cp\u003e13.5.2 Differintegral of [x − a] 454\u003c\/p\u003e \u003cp\u003e13.5.3 Differintegral of [x − a] p (p \u0026gt; −1) 455\u003c\/p\u003e \u003cp\u003e13.5.4 Differintegral of [1 − x] p 456\u003c\/p\u003e \u003cp\u003e13.5.5 Differintegral of exp(±x) 456\u003c\/p\u003e \u003cp\u003e13.5.6 Differintegral of ln(x) 457\u003c\/p\u003e \u003cp\u003e13.5.7 Some Semiderivatives and Semi-Integrals 459\u003c\/p\u003e \u003cp\u003e13.6 Mathematical Techniques with Differintegrals 459\u003c\/p\u003e \u003cp\u003e13.6.1 Laplace Transform of Differintegrals 459\u003c\/p\u003e \u003cp\u003e13.6.2 Extraordinary Differential Equations 463\u003c\/p\u003e \u003cp\u003e13.6.3 Mittag–Leffler Functions 463\u003c\/p\u003e \u003cp\u003e13.6.4 Semidifferential Equations 464\u003c\/p\u003e \u003cp\u003e13.6.5 Evaluating Definite Integrals by Differintegrals 466\u003c\/p\u003e \u003cp\u003e13.6.6 Evaluation of Sums of Series by Differintegrals 468\u003c\/p\u003e \u003cp\u003e13.6.7 Special Functions Expressed as Differintegrals 469\u003c\/p\u003e \u003cp\u003e13.7 Caputo Derivative 469\u003c\/p\u003e \u003cp\u003e13.7.1 Caputo and the Riemann–Liouville Derivative 470\u003c\/p\u003e \u003cp\u003e13.7.2 Mittag–Leffler Function and the Caputo Derivative 473\u003c\/p\u003e \u003cp\u003e13.7.3 Right- and Left-Handed Caputo Derivatives 474\u003c\/p\u003e \u003cp\u003e13.7.4 A Useful Relation of the Caputo Derivative 475\u003c\/p\u003e \u003cp\u003e13.8 Riesz Fractional Integral and Derivative 477\u003c\/p\u003e \u003cp\u003e13.8.1 Riesz Fractional Integral 477\u003c\/p\u003e \u003cp\u003e13.8.2 Riesz Fractional Derivative 480\u003c\/p\u003e \u003cp\u003e13.8.3 Fractional Laplacian 482\u003c\/p\u003e \u003cp\u003e13.9 Applications of Differintegrals in Science and Engineering 482\u003c\/p\u003e \u003cp\u003e13.9.1 Fractional Relaxation 482\u003c\/p\u003e \u003cp\u003e13.9.2 Continuous Time Random Walk (CTRW) 483\u003c\/p\u003e \u003cp\u003e13.9.3 Time Fractional Diffusion Equation 486\u003c\/p\u003e \u003cp\u003e13.9.4 Fractional Fokker–Planck Equations 487\u003c\/p\u003e \u003cp\u003eBibliography 489\u003c\/p\u003e \u003cp\u003eProblems 490\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Infinite Series 495\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Convergence of Infinite Series 495\u003c\/p\u003e \u003cp\u003e14.2 Absolute Convergence 496\u003c\/p\u003e \u003cp\u003e14.3 Convergence Tests 496\u003c\/p\u003e \u003cp\u003e14.3.1 Comparison Test 497\u003c\/p\u003e \u003cp\u003e14.3.2 Ratio Test 497\u003c\/p\u003e \u003cp\u003e14.3.3 Cauchy Root Test 497\u003c\/p\u003e \u003cp\u003e14.3.4 Integral Test 497\u003c\/p\u003e \u003cp\u003e14.3.5 Raabe Test 499\u003c\/p\u003e \u003cp\u003e14.3.6 Cauchy Theorem 499\u003c\/p\u003e \u003cp\u003e14.3.7 Gauss Test and Legendre Series 500\u003c\/p\u003e \u003cp\u003e14.3.8 Alternating Series 503\u003c\/p\u003e \u003cp\u003e14.4 Algebra of Series 503\u003c\/p\u003e \u003cp\u003e14.4.1 Rearrangement of Series 504\u003c\/p\u003e \u003cp\u003e14.5 Useful Inequalities About Series 505\u003c\/p\u003e \u003cp\u003e14.6 Series of Functions 506\u003c\/p\u003e \u003cp\u003e14.6.1 Uniform Convergence 506\u003c\/p\u003e \u003cp\u003e14.6.2 Weierstrass M-Test 507\u003c\/p\u003e \u003cp\u003e14.6.3 Abel Test 507\u003c\/p\u003e \u003cp\u003e14.6.4 Properties of Uniformly Convergent Series 508\u003c\/p\u003e \u003cp\u003e14.7 Taylor Series 508\u003c\/p\u003e \u003cp\u003e14.7.1 Maclaurin Theorem 509\u003c\/p\u003e \u003cp\u003e14.7.2 Binomial Theorem 509\u003c\/p\u003e \u003cp\u003e14.7.3 Taylor Series with Multiple Variables 510\u003c\/p\u003e \u003cp\u003e14.8 Power Series 511\u003c\/p\u003e \u003cp\u003e14.8.1 Convergence of Power Series 512\u003c\/p\u003e \u003cp\u003e14.8.2 Continuity 512\u003c\/p\u003e \u003cp\u003e14.8.3 Differentiation and Integration of Power Series 512\u003c\/p\u003e \u003cp\u003e14.8.4 Uniqueness Theorem 513\u003c\/p\u003e \u003cp\u003e14.8.5 Inversion of Power Series 513\u003c\/p\u003e \u003cp\u003e14.9 Summation of Infinite Series 514\u003c\/p\u003e \u003cp\u003e14.9.1 Bernoulli Polynomials and their Properties 514\u003c\/p\u003e \u003cp\u003e14.9.2 Euler–Maclaurin Sum Formula 516\u003c\/p\u003e \u003cp\u003e14.9.3 Using Residue Theorem to Sum Infinite Series 519\u003c\/p\u003e \u003cp\u003e14.9.4 Evaluating Sums of Series by Differintegrals 522\u003c\/p\u003e \u003cp\u003e14.10 Asymptotic Series 523\u003c\/p\u003e \u003cp\u003e14.11 Method of Steepest Descent 525\u003c\/p\u003e \u003cp\u003e14.12 Saddle-Point Integrals 528\u003c\/p\u003e \u003cp\u003e14.13 Padé Approximants 535\u003c\/p\u003e \u003cp\u003e14.14 Divergent Series in Physics 539\u003c\/p\u003e \u003cp\u003e14.14.1 Casimir Effect and Renormalization 540\u003c\/p\u003e \u003cp\u003e14.14.2 Casimir Effect and MEMS 542\u003c\/p\u003e \u003cp\u003e14.15 Infinite Products 542\u003c\/p\u003e \u003cp\u003e14.15.1 Sine, Cosine, and the Gamma Functions 544\u003c\/p\u003e \u003cp\u003eBibliography 546\u003c\/p\u003e \u003cp\u003eProblems 546\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Integral Transforms 553\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Some Commonly Encountered Integral Transforms 553\u003c\/p\u003e \u003cp\u003e15.2 Derivation of the Fourier Integral 555\u003c\/p\u003e \u003cp\u003e15.2.1 Fourier Series 555\u003c\/p\u003e \u003cp\u003e15.2.2 Dirac-Delta Function 557\u003c\/p\u003e \u003cp\u003e15.3 Fourier and Inverse Fourier Transforms 557\u003c\/p\u003e \u003cp\u003e15.3.1 Fourier-Sine and Fourier-Cosine Transforms 558\u003c\/p\u003e \u003cp\u003e15.4 Conventions and Properties of the Fourier Transforms 560\u003c\/p\u003e \u003cp\u003e15.4.1 Shifting 561\u003c\/p\u003e \u003cp\u003e15.4.2 Scaling 561\u003c\/p\u003e \u003cp\u003e15.4.3 Transform of an Integral 561\u003c\/p\u003e \u003cp\u003e15.4.4 Modulation 561\u003c\/p\u003e \u003cp\u003e15.4.5 Fourier Transform of a Derivative 563\u003c\/p\u003e \u003cp\u003e15.4.6 Convolution Theorem 564\u003c\/p\u003e \u003cp\u003e15.4.7 Existence of Fourier Transforms 565\u003c\/p\u003e \u003cp\u003e15.4.8 Fourier Transforms in Three Dimensions 565\u003c\/p\u003e \u003cp\u003e15.4.9 Parseval Theorems 566\u003c\/p\u003e \u003cp\u003e15.5 Discrete Fourier Transform 572\u003c\/p\u003e \u003cp\u003e15.6 Fast Fourier Transform 576\u003c\/p\u003e \u003cp\u003e15.7 Radon Transform 578\u003c\/p\u003e \u003cp\u003e15.8 Laplace Transforms 581\u003c\/p\u003e \u003cp\u003e15.9 Inverse Laplace Transforms 581\u003c\/p\u003e \u003cp\u003e15.9.1 Bromwich Integral 582\u003c\/p\u003e \u003cp\u003e15.9.2 Elementary Laplace Transforms 583\u003c\/p\u003e \u003cp\u003e15.9.3 Theorems About Laplace Transforms 584\u003c\/p\u003e \u003cp\u003e15.9.4 Method of Partial Fractions 591\u003c\/p\u003e \u003cp\u003e15.10 Laplace Transform of a Derivative 593\u003c\/p\u003e \u003cp\u003e15.10.1 Laplace Transforms in n Dimensions 600\u003c\/p\u003e \u003cp\u003e15.11 Relation Between Laplace and Fourier Transforms 601\u003c\/p\u003e \u003cp\u003e15.12 Mellin Transforms 601\u003c\/p\u003e \u003cp\u003eBibliography 602\u003c\/p\u003e \u003cp\u003eProblems 602\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Variational Analysis 607\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Presence of One Dependent and One Independent Variable 608\u003c\/p\u003e \u003cp\u003e16.1.1 Euler Equation 608\u003c\/p\u003e \u003cp\u003e16.1.2 Another Form of the Euler Equation 610\u003c\/p\u003e \u003cp\u003e16.1.3 Applications of the Euler Equation 610\u003c\/p\u003e \u003cp\u003e16.2 Presence of More than One Dependent Variable 617\u003c\/p\u003e \u003cp\u003e16.3 Presence of More than One Independent Variable 617\u003c\/p\u003e \u003cp\u003e16.4 Presence of Multiple Dependent and Independent Variables 619\u003c\/p\u003e \u003cp\u003e16.5 Presence of Higher-Order Derivatives 619\u003c\/p\u003e \u003cp\u003e16.6 Isoperimetric Problems and the Presence of Constraints 622\u003c\/p\u003e \u003cp\u003e16.7 Applications to Classical Mechanics 626\u003c\/p\u003e \u003cp\u003e16.7.1 Hamilton’s Principle 626\u003c\/p\u003e \u003cp\u003e16.8 Eigenvalue Problems and Variational Analysis 628\u003c\/p\u003e \u003cp\u003e16.9 Rayleigh–Ritz Method 632\u003c\/p\u003e \u003cp\u003e16.10 Optimum Control Theory 637\u003c\/p\u003e \u003cp\u003e16.11 Basic Theory: Dynamics versus Controlled Dynamics 638\u003c\/p\u003e \u003cp\u003e16.11.1 Connection with Variational Analysis 641\u003c\/p\u003e \u003cp\u003e16.11.2 Controllability of a System 642\u003c\/p\u003e \u003cp\u003eBibliography 646\u003c\/p\u003e \u003cp\u003eProblems 647\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Integral Equations 653\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Classification of Integral Equations 654\u003c\/p\u003e \u003cp\u003e17.2 Integral and Differential Equations 654\u003c\/p\u003e \u003cp\u003e17.2.1 Converting Differential Equations into Integral Equations 656\u003c\/p\u003e \u003cp\u003e17.2.2 Converting Integral Equations into Differential Equations 658\u003c\/p\u003e \u003cp\u003e17.3 Solution of Integral Equations 659\u003c\/p\u003e \u003cp\u003e17.3.1 Method of Successive Iterations: Neumann Series 659\u003c\/p\u003e \u003cp\u003e17.3.2 Error Calculation in Neumann Series 660\u003c\/p\u003e \u003cp\u003e17.3.3 Solution for the Case of Separable Kernels 661\u003c\/p\u003e \u003cp\u003e17.3.4 Solution by Integral Transforms 663\u003c\/p\u003e \u003cp\u003e17.3.4.1 Fourier Transform Method 663\u003c\/p\u003e \u003cp\u003e17.3.4.2 Laplace Transform Method 664\u003c\/p\u003e \u003cp\u003e17.4 Hilbert–Schmidt Theory 665\u003c\/p\u003e \u003cp\u003e17.4.1 Eigenvalues for Hermitian Operators 665\u003c\/p\u003e \u003cp\u003e17.4.2 Orthogonality of Eigenfunctions 666\u003c\/p\u003e \u003cp\u003e17.4.3 Completeness of the Eigenfunction Set 666\u003c\/p\u003e \u003cp\u003e17.5 Neumann Series and the Sturm–Liouville Problem 668\u003c\/p\u003e \u003cp\u003e17.6 Eigenvalue Problem for the Non-Hermitian Kernels 672\u003c\/p\u003e \u003cp\u003eBibliography 672\u003c\/p\u003e \u003cp\u003eProblems 672\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Green’s Functions 675\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Time-Independent Green’s Functions in One Dimension 675\u003c\/p\u003e \u003cp\u003e18.1.1 Abel’s Formula 677\u003c\/p\u003e \u003cp\u003e18.1.2 Constructing the Green’s Function 677\u003c\/p\u003e \u003cp\u003e18.1.3 Differential Equation for the Green’s Function 679\u003c\/p\u003e \u003cp\u003e18.1.4 Single-Point Boundary Conditions 679\u003c\/p\u003e \u003cp\u003e18.1.5 Green’s Function for the Operator d2∕dx2 680\u003c\/p\u003e \u003cp\u003e18.1.6 Inhomogeneous Boundary Conditions 682\u003c\/p\u003e \u003cp\u003e18.1.7 Green’s Functions and Eigenvalue Problems 684\u003c\/p\u003e \u003cp\u003e18.1.8 Green’s Functions and the Dirac-Delta Function 686\u003c\/p\u003e \u003cp\u003e18.1.9 Helmholtz Equation with Discrete Spectrum 687\u003c\/p\u003e \u003cp\u003e18.1.10 Helmholtz Equation in the Continuum Limit 688\u003c\/p\u003e \u003cp\u003e18.1.11 Another Approach for the Green’s function 697\u003c\/p\u003e \u003cp\u003e18.2 Time-Independent Green’s Functions in Three Dimensions 701\u003c\/p\u003e \u003cp\u003e18.2.1 Helmholtz Equation in Three Dimensions 701\u003c\/p\u003e \u003cp\u003e18.2.2 Green’s Functions in Three Dimensions 702\u003c\/p\u003e \u003cp\u003e18.2.3 Green’s Function for the Laplace Operator 704\u003c\/p\u003e \u003cp\u003e18.2.4 Green’s Functions for the Helmholtz Equation 705\u003c\/p\u003e \u003cp\u003e18.2.5 General Boundary Conditions and Electrostatics 710\u003c\/p\u003e \u003cp\u003e18.2.6 Helmholtz Equation in Spherical Coordinates 712\u003c\/p\u003e \u003cp\u003e18.2.7 Diffraction from a Circular Aperture 716\u003c\/p\u003e \u003cp\u003e18.3 Time-Independent Perturbation Theory 721\u003c\/p\u003e \u003cp\u003e18.3.1 Nondegenerate Perturbation Theory 721\u003c\/p\u003e \u003cp\u003e18.3.2 Slightly Anharmonic Oscillator in One Dimension 726\u003c\/p\u003e \u003cp\u003e18.3.3 Degenerate Perturbation Theory 728\u003c\/p\u003e \u003cp\u003e18.4 First-Order Time-Dependent Green’s Functions 729\u003c\/p\u003e \u003cp\u003e18.4.1 Propagators 732\u003c\/p\u003e \u003cp\u003e18.4.2 Compounding Propagators 732\u003c\/p\u003e \u003cp\u003e18.4.3 Diffusion Equation with Discrete Spectrum 733\u003c\/p\u003e \u003cp\u003e18.4.4 Diffusion Equation in the Continuum Limit 734\u003c\/p\u003e \u003cp\u003e18.4.5 Presence of Sources or Interactions 736\u003c\/p\u003e \u003cp\u003e18.4.6 Schrödinger Equation for Free Particles 737\u003c\/p\u003e \u003cp\u003e18.4.7 Schrödinger Equation with Interactions 738\u003c\/p\u003e \u003cp\u003e18.5 Second-Order Time-Dependent Green’s Functions 738\u003c\/p\u003e \u003cp\u003e18.5.1 Propagators for the Scalar Wave Equation 741\u003c\/p\u003e \u003cp\u003e18.5.2 Advanced and Retarded Green’s Functions 743\u003c\/p\u003e \u003cp\u003e18.5.3 Scalar Wave Equation 745\u003c\/p\u003e \u003cp\u003eBibliography 747\u003c\/p\u003e \u003cp\u003eProblems 748\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Green’s Functions and Path Integrals 755\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 Brownian Motion and the Diffusion Problem 755\u003c\/p\u003e \u003cp\u003e19.1.1 Wiener Path Integral and Brownian Motion 757\u003c\/p\u003e \u003cp\u003e19.1.2 Perturbative Solution of the Bloch Equation 760\u003c\/p\u003e \u003cp\u003e19.1.3 Derivation of the Feynman–Kac Formula 763\u003c\/p\u003e \u003cp\u003e19.1.4 Interpretation of V(x) in the Bloch Equation 765\u003c\/p\u003e \u003cp\u003e19.2 Methods of Calculating Path Integrals 767\u003c\/p\u003e \u003cp\u003e19.2.1 Method of Time Slices 769\u003c\/p\u003e \u003cp\u003e19.2.2 Path Integrals with the ESKC Relation 770\u003c\/p\u003e \u003cp\u003e19.2.3 Path Integrals by the Method of Finite Elements 771\u003c\/p\u003e \u003cp\u003e19.2.4 Path Integrals by the “Semiclassical” Method 772\u003c\/p\u003e \u003cp\u003e19.3 Path Integral Formulation of Quantum Mechanics 776\u003c\/p\u003e \u003cp\u003e19.3.1 Schrödinger Equation For a Free Particle 776\u003c\/p\u003e \u003cp\u003e19.3.2 Schrödinger Equation with a Potential 778\u003c\/p\u003e \u003cp\u003e19.3.3 Feynman Phase Space Path Integral 780\u003c\/p\u003e \u003cp\u003e19.3.4 The Case of Quadratic Dependence on Momentum 781\u003c\/p\u003e \u003cp\u003e19.4 Path Integrals Over Lévy Paths and Anomalous Diffusion 783\u003c\/p\u003e \u003cp\u003e19.5 Fox’s H-Functions 788\u003c\/p\u003e \u003cp\u003e19.5.1 Properties of the H-Functions 789\u003c\/p\u003e \u003cp\u003e19.5.2 Useful Relations of the H-Functions 791\u003c\/p\u003e \u003cp\u003e19.5.3 Examples of H-Functions 792\u003c\/p\u003e \u003cp\u003e19.5.4 Computable Form of the H-Function 796\u003c\/p\u003e \u003cp\u003e19.6 Applications of H-Functions 797\u003c\/p\u003e \u003cp\u003e19.6.1 Riemann–Liouville Definition of Differintegral 798\u003c\/p\u003e \u003cp\u003e19.6.2 Caputo Fractional Derivative 798\u003c\/p\u003e \u003cp\u003e19.6.3 Fractional Relaxation 799\u003c\/p\u003e \u003cp\u003e19.6.4 Time Fractional Diffusion via R–L Derivative 800\u003c\/p\u003e \u003cp\u003e19.6.5 Time Fractional Diffusion via Caputo Derivative 801\u003c\/p\u003e \u003cp\u003e19.6.6 Derivation of the Lévy Distribution 803\u003c\/p\u003e \u003cp\u003e19.6.7 Lévy Distributions in Nature 806\u003c\/p\u003e \u003cp\u003e19.6.8 Time and Space Fractional Schrödinger Equation 806\u003c\/p\u003e \u003cp\u003e19.6.8.1 Free Particle Solution 808\u003c\/p\u003e \u003cp\u003e19.7 Space Fractional Schrödinger Equation 809\u003c\/p\u003e \u003cp\u003e19.7.1 Feynman Path Integrals Over Lévy Paths 810\u003c\/p\u003e \u003cp\u003e19.8 Time Fractional Schrödinger Equation 812\u003c\/p\u003e \u003cp\u003e19.8.1 Separable Solutions 812\u003c\/p\u003e \u003cp\u003e19.8.2 Time Dependence 813\u003c\/p\u003e \u003cp\u003e19.8.3 Mittag–Leffler Function and the Caputo Derivative 814\u003c\/p\u003e \u003cp\u003e19.8.4 Euler Equation for the Mittag–Leffler Function 814\u003c\/p\u003e \u003cp\u003eBibliography 817\u003c\/p\u003e \u003cp\u003eProblems 818\u003c\/p\u003e \u003cp\u003eFurther Reading 825\u003c\/p\u003e \u003cp\u003eIndex 827\u003c\/p\u003e   \u003cp\u003e \u003cstrong\u003eSelçuk \u0026amp;Scedil;. Bayin, PhD,\u003c\/strong\u003e is Professor of Physics at the Institute of Applied Mathematics in the Middle East Technical University in Ankara, Turkey, and a member of the Turkish Physical Society and the American Physical Society. He is the author of \u003cem\u003eMathematical Methods in Science and Engineering\u003c\/em\u003e and \u003cem\u003eEssentials of Mathematical Methods of Science and Engineering,\u003c\/em\u003e also published by Wiley.    \u003c\/p\u003e\u003cp\u003e\u003cb\u003eA Practical, Interdisciplinary Guide to Advanced Mathematical Methods for Scientists and Engineers \u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eMathematical Methods in Science and Engineering, Second Edition,\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003eRecent 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.\u003c\/p\u003e \u003cp\u003eRevised and expanded for increased utility, this new Second Edition:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eIncludes over 60 new sections and subsections more useful to a multidisciplinary audience\u003c\/li\u003e \u003cli\u003eContains new examples, new figures, new problems, and more fluid arguments\u003c\/li\u003e \u003cli\u003ePresents a detailed discussion on the most frequently encountered special functions in science and engineering\u003c\/li\u003e \u003cli\u003eProvides a systematic treatment of special functions in terms of the Sturm-Liouville theory\u003c\/li\u003e \u003cli\u003eApproaches second-order differential equations of physics and engineering from the factorization perspective\u003c\/li\u003e \u003cli\u003eIncludes extensive discussion of coordinate transformations and tensors, complex analysis, fractional calculus, integral transforms, Green's functions, path integrals, and more\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eExtensively 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.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989587017957,"sku":"NP9781119425397","price":169.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119425397.jpg?v=1761784707","url":"https:\/\/k12savings.com\/es\/products\/mathematical-methods-in-science-and-engineering-isbn-9781119425397","provider":"K12savings","version":"1.0","type":"link"}