Mathematical Methods in Science and Engineering (E-Book, EPUB)

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ISBN/EAN: 9781119425458
Sprache: Englisch
Umfang: 864 S.
Auflage: 2. Auflage 2018
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A Practical, Interdisciplinary Guide to Advanced Mathematical Methods for Scientists and EngineersMathematical 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.
Selçuk ?. Bayin, PhD,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 ofMathematical Methods in Science and EngineeringandEssentials of Mathematical Methods of Science and Engineering,also published by Wiley.
Preface xix1 Legendre Equation and Polynomials 11.1 Second-Order Differential Equations of Physics 11.2 Legendre Equation 21.2.1 Method of Separation of Variables 41.2.2 Series Solution of the Legendre Equation 41.2.3 Frobenius Method Review 71.3 Legendre Polynomials 81.3.1 Rodriguez Formula 101.3.2 Generating Function 101.3.3 Recursion Relations 121.3.4 Special Values 121.3.5 Special Integrals 131.3.6 Orthogonality and Completeness 141.3.7 Asymptotic Forms 171.4 Associated Legendre Equation and Polynomials 181.4.1 Associated Legendre Polynomials Pm l (x) 201.4.2 Orthogonality 211.4.3 Recursion Relations 221.4.4 Integral Representations 241.4.5 Associated Legendre Polynomials for m0 and ??(m) is an increasing function) 1418.4.2 Case II (m>0 and ??(m) is a decreasing function) 1428.5 Technique and the Categories of Factorization 1438.5.1 Possible Forms for k(z,m) 1438.5.1.1 Positive powers of m 1438.5.1.2 Negative powers of m 1468.6 Associated Legendre Equation (Type A) 1488.6.1 Determining the Eigenvalues, ??l 1498.6.2 Construction of the Eigenfunctions 1508.6.3 Ladder Operators for m 1518.6.4 Interpretation of the L+ and L? Operators 1538.6.5 Ladder Operators for l 1558.6.6 Complete Set of Ladder Operators 1598.7 Schrödinger Equation and Single-Electron Atom (Type F) 1608.8 Gegenbauer Functions (Type A) 1628.9 Symmetric Top (Type A) 1638.10 Bessel Functions (Type C) 1648.11 Harmonic Oscillator (Type D) 1658.12 Differential Equation for the Rotation Matrix 1668.12.1 Step-Up/Down Operators for m 1668.12.2 Step-Up/Down Operators for m? 1678.12.3 Normalized Functions with m = m? = l 1688.12.4 Full Matrix for l = 2 1688.12.5 Step-Up/Down Operators for l 170Bibliography 171Problems 1719 Coordinates and Tensors 1759.1 Cartesian Coordinates 1759.1.1 Algebra of Vectors 1769.1.2 Differentiation of Vectors 1779.2 Orthogonal Transformations 1789.2.1 Rotations About Cartesian Axes 1829.2.2 Formal Properties of the Rotation Matrix 1839.2.3 Euler Angles and Arbitrary Rotations 1839.2.4 Active and Passive Interpretations of Rotations 1859.2.5 Infinitesimal Transformations 1869.2.6 Infinitesimal Transformations Commute 1889.3 Cartesian Tensors 1899.3.1 Operations with Cartesian Tensors 1909.3.2 Tensor Densities or Pseudotensors 1919.4 Cartesian Tensors and theTheory of Elasticity 1929.4.1 Strain Tensor 1929.4.2 Stress Tensor 1939.4.3 Thermodynamics and Deformations 1949.4.4 Connection between Shear and Strain 1969.4.5 Hooks Law 2009.5 Generalized Coordinates and General Tensors 2019.5.1 Contravariant and Covariant Components 2029.5.2 Metric Tensor and the Line Element 2039.5.3 Geometric Interpretation of Components 2069.5.4 Interpretation of the Metric Tensor 2079.6 Operations with General Tensors 2149.6.1 Einstein Summation Convention 2149.6.2 Contraction of Indices 2149.6.3 Multiplication of Tensors 2149.6.4 The Quotient Theorem 2149.6.5 Equality of Tensors 2159.6.6 Tensor Densities 2159.6.7 Differentiation of Tensors 2169.6.8 Some Covariant Derivatives 2199.6.9 Riemann Curvature Tensor 2209.7 Curvature 2219.7.1 Parallel Transport 2229.7.2 Round Trips via Parallel Transport 2239.7.3 Algebraic Properties of the Curvature Tensor 2259.7.4 Contractions of the Curvature Tensor 2269.7.5 Curvature in n Dimensions 2279.7.6 Geodesics 2299.7.7 Invariance Versus Covariance 2299.8 Spacetime and Four-Tensors 2309.8.1 Minkowski Spacetime 2309.8.2 Lorentz Transformations and Special Relativity 2319.8.3 Time Dilation and Length Contraction 2339.8.4 Addition of Velocities 2339.8.5 Four-Tensors in Minkowski Spacetime 2349.8.6 Four-Velocity 2379.8.7 Four-Momentum and Conservation Laws 2389.8.8 Mass of a Moving Particle 2409.8.9 Wave Four-Vector 2409.8.10 Derivative Operators in Spacetime 2419.8.11 Relative Orientation of Axes in K and K Frames 2419.9 Maxwells Equations in Minkowski Spacetime 2439.9.1 Transformation of Electromagnetic Fields 2469.9.2 Maxwells Equations in Terms of Potentials 2469.9.3 Covariance of Newtons Dynamic Theory 247Bibliography 248Problems 24910 Continuous Groups and Representations 25710.1 Definition of a Group 25810.1.1 Nomenclature 25810.2 Infinitesimal Ring or Lie Algebra 25910.2.1 Properties of rG 26010.3 Lie Algebra of the Rotation Group R(3) 26010.3.1 Another Approach to rR(3) 26210.4 Group Invariants 26410.4.1 Lorentz Transformations 26610.5 Unitary Group in Two Dimensions U(2) 26710.5.1 Special Unitary Group SU(2) 26910.5.2 Lie Algebra of SU(2) 27010.5.3 Another Approach to rSU(2) 27210.6 Lorentz Group and Its Lie Algebra 27410.7 Group Representations 27910.7.1 Schurs Lemma 27910.7.2 Group Character 28010.7.3 Unitary Representation 28010.8 Representations of R(3) 28110.8.1 Spherical Harmonics and Representations of R(3) 28110.8.2 Angular Momentum in Quantum Mechanics 28110.8.3 Rotation of the Physical System 28210.8.4 Rotation Operator in Terms of the Euler Angles 28210.8.5 Rotation Operator in the Original Coordinates 28310.8.6 Eigenvalue Equations for Lz, L±, and L2 28710.8.7 Fourier Expansion in Spherical Harmonics 28710.8.8 Matrix Elements of Lx, Ly, and Lz 28910.8.9 Rotation Matrices of the Spherical Harmonics 29010.8.10 Evaluation of the dlm?m(??) Matrices 29210.8.11 Inverse of the dlm?m(??) Matrices 29210.8.12 Differential Equation for dlm?m(??) 29310.8.13 AdditionTheorem for Spherical Harmonics 29610.8.14 Determination of Il in the AdditionTheorem 29810.8.15 Connection of Dlmm? (??) with Spherical Harmonics 30010.9 Irreducible Representations of SU(2) 30210.10 Relation of SU(2) and R(3) 30310.11 Group Spaces 30610.11.1 Real Vector Space 30610.11.2 Inner Product Space 30710.11.3 Four-Vector Space 30710.11.4 Complex Vector Space 30810.11.5 Function Space and Hilbert Space 30810.11.6 Completeness 30910.12 Hilbert Space and QuantumMechanics 31010.13 Continuous Groups and Symmetries 31110.13.1 Point Groups and Their Generators 31110.13.2 Transformation of Generators and Normal Forms 31210.13.3 The Case of Multiple Parameters 31410.13.4 Action of Generators on Functions 31510.13.5 Extension or Prolongation of Generators 31610.13.6 Symmetries of Differential Equations 318Bibliography 321Problems 32211 Complex Variables and Functions 32711.1 Complex Algebra 32711.2 Complex Functions 32911.3 Complex Derivatives and CauchyRiemann Conditions 33011.3.1 Analytic Functions 33011.3.2 Harmonic Functions 33211.4 Mappings 33411.4.1 Conformal Mappings 34811.4.2 Electrostatics and Conformal Mappings 34911.4.3 Fluid Mechanics and Conformal Mappings 35211.4.4 SchwarzChristoffel Transformations 358Bibliography 368Problems 36812 Complex Integrals and Series 37312.1 Complex Integral Theorems 37312.1.1 CauchyGoursatTheorem 37312.1.2 Cauchy IntegralTheorem 37412.1.3 CauchyTheorem 37612.2 Taylor Series 37812.3 Laurent Series 37912.4 Classification of Singular Points 38512.5 ResidueTheorem 38612.6 Analytic Continuation 38912.7 Complex Techniques in Taking Some Definite Integrals 39212.8 Gamma and Beta Functions 39912.8.1 Gamma Function 39912.8.2 Beta Function 40112.8.3 Useful Relations of the Gamma Functions 40312.8.4 Incomplete Gamma and Beta Functions 40312.8.5 Analytic Continuation of the Gamma Function 40412.9 Cauchy Principal Value Integral 40612.10 Integral Representations of Special Functions 41012.10.1 Legendre Polynomials 41012.10.2 Laguerre Polynomials 41112.10.3 Bessel Functions 413Bibliography 416Problems 41613 Fractional Calculus 42313.1 Unified Expression of Derivatives and Integrals 42513.1.1 Notation and Definitions 42513.1.2 The nth Derivative of a Function 42613.1.3 Successive Integrals 42713.1.4 Unification of Derivative and Integral Operators 42913.2 Differintegrals 42913.2.1 Grünwalds Definition of Differintegrals 42913.2.2 RiemannLiouville Definition of Differintegrals 43113.3 Other Definitions of Differintegrals 43413.3.1 Cauchy Integral Formula 43413.3.2 Riemann Formula 43913.3.3 Differintegrals via Laplace Transforms 44013.4 Properties of Differintegrals 44213.4.1 Linearity 44313.4.2 Homogeneity 44313.4.3 Scale Transformations 44313.4.4 Differintegral of a Series 44313.4.5 Composition of Differintegrals 44413.4.5.1 Composition Rule for General q and Q 44713.4.6 Leibniz Rule 45013.4.7 Right- and Left-Handed Differintegrals 45013.4.8 Dependence on the Lower Limit 45213.5 Differintegrals of Some Functions 45313.5.1 Differintegral of a Constant 45313.5.2 Differintegral of [x ? a] 45413.5.3 Differintegral of [x ? a]p (p>?1) 45513.5.4 Differintegral of [1 ? x]p 45613.5.5 Differintegral of exp(±x) 45613.5.6 Differintegral of ln(x) 45713.5.7 Some Semiderivatives and Semi-Integrals 45913.6 Mathematical Techniques with Differintegrals 45913.6.1 Laplace Transform of Differintegrals 45913.6.2 Extraordinary Differential Equations 46313.6.3 MittagLeffler Functions 46313.6.4 Semidifferential Equations 46413.6.5 Evaluating Definite Integrals by Differintegrals 46613.6.6 Evaluation of Sums of Series by Differintegrals 46813.6.7 Special Functions Expressed as Differintegrals 46913.7 Caputo Derivative 46913.7.1 Caputo and the RiemannLiouville Derivative 47013.7.2 MittagLeffler Function and the Caputo Derivative 47313.7.3 Right- and Left-Handed Caputo Derivatives 47413.7.4 A Useful Relation of the Caputo Derivative 47513.8 Riesz Fractional Integral and Derivative 47713.8.1 Riesz Fractional Integral 47713.8.2 Riesz Fractional Derivative 48013.8.3 Fractional Laplacian 48213.9 Applications of Differintegrals in Science and Engineering 48213.9.1 Fractional Relaxation 48213.9.2 Continuous Time RandomWalk (CTRW) 48313.9.3 Time Fractional Diffusion Equation 48613.9.4 Fractional FokkerPlanck Equations 487Bibliography 489Problems 49014 Infinite Series 49514.1 Convergence of Infinite Series 49514.2 Absolute Convergence 49614.3 Convergence Tests 49614.3.1 Comparison Test 49714.3.2 Ratio Test 49714.3.3 Cauchy Root Test 49714.3.4 Integral Test 49714.3.5 Raabe Test 49914.3.6 CauchyTheorem 49914.3.7 Gauss Test and Legendre Series 50014.3.8 Alternating Series 50314.4 Algebra of Series 50314.4.1 Rearrangement of Series 50414.5 Useful Inequalities About Series 50514.6 Series of Functions 50614.6.1 Uniform Convergence 50614.6.2 Weierstrass M-Test 50714.6.3 Abel Test 50714.6.4 Properties of Uniformly Convergent Series 50814.7 Taylor Series 50814.7.1 Maclaurin Theorem 50914.7.2 BinomialTheorem 50914.7.3 Taylor Series with Multiple Variables 51014.8 Power Series 51114.8.1 Convergence of Power Series 51214.8.2 Continuity 51214.8.3 Differentiation and Integration of Power Series 51214.8.4 Uniqueness Theorem 51314.8.5 Inversion of Power Series 51314.9 Summation of Infinite Series 51414.9.1 Bernoulli Polynomials and their Properties 51414.9.2 EulerMaclaurin Sum Formula 51614.9.3 Using ResidueTheorem to Sum Infinite Series 51914.9.4 Evaluating Sums of Series by Differintegrals 52214.10 Asymptotic Series 52314.11 Method of Steepest Descent 52514.12 Saddle-Point Integrals 52814.13 Padé Approximants 53514.14 Divergent Series in Physics 53914.14.1 Casimir Effect and Renormalization 54014.14.2 Casimir Effect and MEMS 54214.15 Infinite Products 54214.15.1 Sine, Cosine, and the Gamma Functions 544Bibliography 546Problems 54615 Integral Transforms 55315.1 Some Commonly Encountered Integral Transforms 55315.2 Derivation of the Fourier Integral 55515.2.1 Fourier Series 55515.2.2 Dirac-Delta Function 55715.3 Fourier and Inverse Fourier Transforms 55715.3.1 Fourier-Sine and Fourier-Cosine Transforms 55815.4 Conventions and Properties of the Fourier Transforms 56015.4.1 Shifting 56115.4.2 Scaling 56115.4.3 Transform of an Integral 56115.4.4 Modulation 56115.4.5 Fourier Transform of a Derivative 56315.4.6 Convolution Theorem 56415.4.7 Existence of Fourier Transforms 56515.4.8 Fourier Transforms inThree Dimensions 56515.4.9 ParsevalTheorems 56615.5 Discrete Fourier Transform 57215.6 Fast Fourier Transform 57615.7 Radon Transform 57815.8 Laplace Transforms 58115.9 Inverse Laplace Transforms 58115.9.1 Bromwich Integral 58215.9.2 Elementary Laplace Transforms 58315.9.3 Theorems About Laplace Transforms 58415.9.4 Method of Partial Fractions 59115.10 Laplace Transform of a Derivative 59315.10.1 Laplace Transforms in n Dimensions 60015.11 Relation Between Laplace and Fourier Transforms 60115.12 Mellin Transforms 601Bibliography 602Problems 60216 Variational Analysis 60716.1 Presence of One Dependent and One Independent Variable 60816.1.1 Euler Equation 60816.1.2 Another Form of the Euler Equation 61016.1.3 Applications of the Euler Equation 61016.2 Presence of More than One Dependent Variable 61716.3 Presence of More than One Independent Variable 61716.4 Presence of Multiple Dependent and Independent Variables 61916.5 Presence of Higher-Order Derivatives 61916.6 Isoperimetric Problems and the Presence of Constraints 62216.7 Applications to Classical Mechanics 62616.7.1 Hamiltons Principle 62616.8 Eigenvalue Problems and Variational Analysis 62816.9 RayleighRitzMethod 63216.10 Optimum Control Theory 63716.11 BasicTheory: Dynamics versus Controlled Dynamics 63816.11.1 Connection with Variational Analysis 64116.11.2 Controllability of a System 642Bibliography 646Problems 64717 Integral Equations 65317.1 Classification of Integral Equations 65417.2 Integral and Differential Equations 65417.2.1 Converting Differential Equations into Integral Equations 65617.2.2 Converting Integral Equations into Differential Equations 65817.3 Solution of Integral Equations 65917.3.1 Method of Successive Iterations: Neumann Series 65917.3.2 Error Calculation in Neumann Series 66017.3.3 Solution for the Case of Separable Kernels 66117.3.4 Solution by Integral Transforms 66317.3.4.1 Fourier Transform Method 66317.3.4.2 Laplace Transform Method 66417.4 HilbertSchmidt Theory 66517.4.1 Eigenvalues for Hermitian Operators 66517.4.2 Orthogonality of Eigenfunctions 66617.4.3 Completeness of the Eigenfunction Set 66617.5 Neumann Series and the SturmLiouville Problem 66817.6 Eigenvalue Problem for the Non-Hermitian Kernels 672Bibliography 672Problems 67218 Greens Functions 67518.1 Time-Independent Greens Functions in One Dimension 67518.1.1 Abels Formula 67718.1.2 Constructing the Greens Function 67718.1.3 Differential Equation for the Greens Function 67918.1.4 Single-Point Boundary Conditions 67918.1.5 Greens Function for the Operator d2?Mdx2 68018.1.6 Inhomogeneous Boundary Conditions 68218.1.7 Greens Functions and Eigenvalue Problems 68418.1.8 Greens Functions and the Dirac-Delta Function 68618.1.9 Helmholtz Equation with Discrete Spectrum 68718.1.10 Helmholtz Equation in the Continuum Limit 68818.1.11 Another Approach for the Greens function 69718.2 Time-Independent Greens Functions inThree Dimensions 70118.2.1 Helmholtz Equation in Three Dimensions 70118.2.2 Greens Functions inThree Dimensions 70218.2.3 Greens Function for the Laplace Operator 70418.2.4 Greens Functions for the Helmholtz Equation 70518.2.5 General Boundary Conditions and Electrostatics 71018.2.6 Helmholtz Equation in Spherical Coordinates 71218.2.7 Diffraction from a Circular Aperture 71618.3 Time-Independent PerturbationTheory 72118.3.1 Nondegenerate PerturbationTheory 72118.3.2 Slightly Anharmonic Oscillator in One Dimension 72618.3.3 Degenerate PerturbationTheory 72818.4 First-Order Time-Dependent Greens Functions 72918.4.1 Propagators 73218.4.2 Compounding Propagators 73218.4.3 Diffusion Equation with Discrete Spectrum 73318.4.4 Diffusion Equation in the Continuum Limit 73418.4.5 Presence of Sources or Interactions 73618.4.6 Schrödinger Equation for Free Particles 73718.4.7 Schrödinger Equation with Interactions 73818.5 Second-Order Time-Dependent Greens Functions 73818.5.1 Propagators for the ScalarWave Equation 74118.5.2 Advanced and Retarded Greens Functions 74318.5.3 ScalarWave Equation 745Bibliography 747Problems 74819 Greens Functions and Path Integrals 75519.1 Brownian Motion and the Diffusion Problem 75519.1.1 Wiener Path Integral and Brownian Motion 75719.1.2 Perturbative Solution of the Bloch Equation 76019.1.3 Derivation of the FeynmanKac Formula 76319.1.4 Interpretation of V(x) in the Bloch Equation 76519.2 Methods of Calculating Path Integrals 76719.2.1 Method of Time Slices 76919.2.2 Path Integrals with the ESKC Relation 77019.2.3 Path Integrals by the Method of Finite Elements 77119.2.4 Path Integrals by the Semiclassical Method 77219.3 Path Integral Formulation of Quantum Mechanics 77619.3.1 Schrödinger Equation For a Free Particle 77619.3.2 Schrödinger Equation with a Potential 77819.3.3 Feynman Phase Space Path Integral 78019.3.4 The Case of Quadratic Dependence on Momentum 78119.4 Path Integrals Over Lévy Paths and Anomalous Diffusion 78319.5 Foxs H-Functions 78819.5.1 Properties of the H-Functions 78919.5.2 Useful Relations of the H-Functions 79119.5.3 Examples of H-Functions 79219.5.4 Computable Form of the H-Function 79619.6 Applications of H-Functions 79719.6.1 RiemannLiouville Definition of Differintegral 79819.6.2 Caputo Fractional Derivative 79819.6.3 Fractional Relaxation 79919.6.4 Time Fractional Diffusion via RL Derivative 80019.6.5 Time Fractional Diffusion via Caputo Derivative 80119.6.6 Derivation of the Lévy Distribution 80319.6.7 Lévy Distributions in Nature 80619.6.8 Time and Space Fractional Schrödinger Equation 80619.6.8.1 Free Particle Solution 80819.7 Space Fractional Schrödinger Equation 80919.7.1 Feynman Path Integrals Over Lévy Paths 81019.8 Time Fractional Schrödinger Equation 81219.8.1 Separable Solutions 81219.8.2 Time Dependence 81319.8.3 MittagLeffler Function and the Caputo Derivative 81419.8.4 Euler Equation for the MittagLeffler Function 814Bibliography 817Problems 818Further Reading 825Index 827

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