Mathematical Methods in Science and Engineering

Selçuk Ş. 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 of Mathematical Methods in Science and Engineering and Essentials 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 m < 0 261.5 Spherical Harmonics 271.5.1 Addition Theorem of Spherical Harmonics 301.5.2 Real Spherical Harmonics 33Bibliography 33Problems 342 Laguerre Polynomials 392.1 Central Force Problems in Quantum Mechanics 392.2 Laguerre Equation and Polynomials 412.2.1 Generating Function 422.2.2 Rodriguez Formula 432.2.3 Orthogonality 442.2.4 Recursion Relations 452.2.5 Special Values 462.3 Associated Laguerre Equation and Polynomials 462.3.1 Generating Function 482.3.2 Rodriguez Formula and Orthogonality 492.3.3 Recursion Relations 49Bibliography 49Problems 503 Hermite Polynomials 533.1 Harmonic Oscillator in Quantum Mechanics 533.2 Hermite Equation and Polynomials 543.2.1 Generating Function 563.2.2 Rodriguez Formula 563.2.3 Recursion Relations and Orthogonality 57Bibliography 61Problems 624 Gegenbauer and Chebyshev Polynomials 654.1 Wave Equation on a Hypersphere 654.2 Gegenbauer Equation and Polynomials 684.2.1 Orthogonality and the Generating Function 684.2.2 Another Representation of the Solution 694.2.3 The Second Solution 704.2.4 Connection with the Gegenbauer Polynomials 714.2.5 Evaluation of the Normalization Constant 724.3 Chebyshev Equation and Polynomials 724.3.1 Chebyshev Polynomials of the First Kind 724.3.2 Chebyshev and Gegenbauer Polynomials 734.3.3 Chebyshev Polynomials of the Second Kind 734.3.4 Orthogonality and Generating Function 744.3.5 Another Definition 75Bibliography 76Problems 765 Bessel Functions 815.1 Bessel’s Equation 835.2 Bessel Functions 835.2.1 Asymptotic Forms 845.3 Modified Bessel Functions 865.4 Spherical Bessel Functions 875.5 Properties of Bessel Functions 885.5.1 Generating Function 885.5.2 Integral Definitions 895.5.3 Recursion Relations of the Bessel Functions 895.5.4 Orthogonality and Roots of Bessel Functions 905.5.5 Boundary Conditions for the Bessel Functions 915.5.6 Wronskian of Pairs of Solutions 945.6 Transformations of Bessel Functions 955.6.1 Critical Length of a Rod 96Bibliography 98Problems 996 Hypergeometric Functions 1036.1 Hypergeometric Series 1036.2 Hypergeometric Representations of Special Functions 1076.3 Confluent Hypergeometric Equation 1086.4 Pochhammer Symbol and Hypergeometric Functions 1096.5 Reduction of Parameters 113Bibliography 115Problems 1157 Sturm–Liouville Theory 1197.1 Self-Adjoint Differential Operators 1197.2 Sturm–Liouville Systems 1207.3 Hermitian Operators 1217.4 Properties of Hermitian Operators 1227.4.1 Real Eigenvalues 1227.4.2 Orthogonality of Eigenfunctions 1237.4.3 Completeness and the Expansion Theorem 1237.5 Generalized Fourier Series 1257.6 Trigonometric Fourier Series 1267.7 Hermitian Operators in Quantum Mechanics 127Bibliography 129Problems 1308 Factorization Method 1338.1 Another Form for the Sturm–Liouville Equation 1338.2 Method of Factorization 1358.3 Theory of Factorization and the Ladder Operators 1368.4 Solutions via the Factorization Method 1418.4.1 Case I (m > 0 and 𝜇(m) is an increasing function) 1418.4.2 Case II (m > 0 and 𝜇(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, 𝜆l 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 the Theory 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 Hook’s 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 Maxwell’s Equations in Minkowski Spacetime 2439.9.1 Transformation of Electromagnetic Fields 2469.9.2 Maxwell’s Equations in Terms of Potentials 2469.9.3 Covariance of Newton’s 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 r G 26010.3 Lie Algebra of the Rotation Group R(3) 26010.3.1 Another Approach to r R(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 r SU(2) 27210.6 Lorentz Group and Its Lie Algebra 27410.7 Group Representations 27910.7.1 Schur’s 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 dl m′m(𝛽) Matrices 29210.8.11 Inverse of the dl m′m(𝛽) Matrices 29210.8.12 Differential Equation for dl m′m(𝛽) 29310.8.13 Addition Theorem for Spherical Harmonics 29610.8.14 Determination of Il in the Addition Theorem 29810.8.15 Connection of Dl mm′ (𝛽) 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 Quantum Mechanics 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 Cauchy–Riemann 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 Schwarz–Christoffel Transformations 358Bibliography 368Problems 36812 Complex Integrals and Series 37312.1 Complex Integral Theorems 37312.1.1 Cauchy–Goursat Theorem 37312.1.2 Cauchy Integral Theorem 37412.1.3 Cauchy Theorem 37612.2 Taylor Series 37812.3 Laurent Series 37912.4 Classification of Singular Points 38512.5 Residue Theorem 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 Grünwald’s Definition of Differintegrals 42913.2.2 Riemann–Liouville 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 Mittag–Leffler 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 Riemann–Liouville Derivative 47013.7.2 Mittag–Leffler 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 Random Walk (CTRW) 48313.9.3 Time Fractional Diffusion Equation 48613.9.4 Fractional Fokker–Planck 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 Cauchy Theorem 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 Binomial Theorem 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 Euler–Maclaurin Sum Formula 51614.9.3 Using Residue Theorem 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 in Three Dimensions 56515.4.9 Parseval Theorems 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 Hamilton’s Principle 62616.8 Eigenvalue Problems and Variational Analysis 62816.9 Rayleigh–Ritz Method 63216.10 Optimum Control Theory 63716.11 Basic Theory: 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 Hilbert–Schmidt 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 Sturm–Liouville Problem 66817.6 Eigenvalue Problem for the Non-Hermitian Kernels 672Bibliography 672Problems 67218 Green’s Functions 67518.1 Time-Independent Green’s Functions in One Dimension 67518.1.1 Abel’s Formula 67718.1.2 Constructing the Green’s Function 67718.1.3 Differential Equation for the Green’s Function 67918.1.4 Single-Point Boundary Conditions 67918.1.5 Green’s Function for the Operator d2∕dx2 68018.1.6 Inhomogeneous Boundary Conditions 68218.1.7 Green’s Functions and Eigenvalue Problems 68418.1.8 Green’s 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 Green’s function 69718.2 Time-Independent Green’s Functions in Three Dimensions 70118.2.1 Helmholtz Equation in Three Dimensions 70118.2.2 Green’s Functions in Three Dimensions 70218.2.3 Green’s Function for the Laplace Operator 70418.2.4 Green’s 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 Perturbation Theory 72118.3.1 Nondegenerate Perturbation Theory 72118.3.2 Slightly Anharmonic Oscillator in One Dimension 72618.3.3 Degenerate Perturbation Theory 72818.4 First-Order Time-Dependent Green’s 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 Green’s Functions 73818.5.1 Propagators for the Scalar Wave Equation 74118.5.2 Advanced and Retarded Green’s Functions 74318.5.3 Scalar Wave Equation 745Bibliography 747Problems 74819 Green’s 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 Feynman–Kac 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 Fox’s 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 Riemann–Liouville Definition of Differintegral 79819.6.2 Caputo Fractional Derivative 79819.6.3 Fractional Relaxation 79919.6.4 Time Fractional Diffusion via R–L 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 Mittag–Leffler Function and the Caputo Derivative 81419.8.4 Euler Equation for the Mittag–Leffler Function 814Bibliography 817Problems 818Further Reading 825Index 827