xiv CONTENTS 15.6.3. DifferentiationMethod 700 15.6.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and TheirApplications 704 15.7. Direct Method of Symmetry Reductions of Nonlinear Equations . . . . . . . . . . . . . . . . . . 708 15.7.1. Clarkson–Kruskal Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 15.7.2. Some ModificationsandGeneralizations 712 15.8. Classical Method of Studying Symmetries of Differential Equations . . . . . . . . . . . . . . . 716 15.8.1. One-Parameter Transformations and Their Local Properties . . . . . . . . . . . . . . 716 15.8.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition . . . . 719 15.8.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions 724 15.8.4. Some Generalizations. Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . 730 15.9. NonclassicalMethodofSymmetryReductions 732 15.9.1. Description of the Method. Invariant Surface Condition . . . . . . . . . . . . . . . . . 732 15.9.2. Examples: The Newell–Whitehead Equation and a Nonlinear Wave Equation 733 15.10. DifferentialConstraintsMethod 737 15.10.1. DescriptionoftheMethod 737 15.10.2. First-OrderDifferentialConstraints 739 15.10.3. Second- and Higher-Order Differential Constraints . . . . . . . . . . . . . . . . . . . 744 15.10.4. Connection Between the Differential Constraints Method and Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 15.11. Painlev ´ e Test for Nonlinear Equations of Mathematical Physics . . . . . . . . . . . . . . . . . 748 15.11.1. Solutions of Partial Differential Equations with a Movable Pole. Method Description 748 15.11.2. Examples of Performing the Painlev ´ e Test and Truncated Expansions for Studying Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750 15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painlev ´ e Test,UsingTruncatedExpansions 753 15.12. Methods of the Inverse Scattering Problem (Soliton Theory) . . . . . . . . . . . . . . . . . . . . 755 15.12.1. MethodBasedonUsingLaxPairs 755 15.12.2. Method Based on a Compatibility Condition for Systems of Linear Equations 757 15.12.3. Solution of the Cauchy Problem by the Inverse Scattering Problem Method 760 15.13. ConservationLawsandIntegralsofMotion 766 15.13.1. Basic DefinitionsandExamples 766 15.13.2. Equations Admitting Variational Formulation. Noetherian Symmetries . . . 767 15.14. NonlinearSystemsofPartialDifferentialEquations 770 15.14.1. OverdeterminedSystemsofTwoEquations 770 15.14.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems 772 15.14.3. Systems of First-Order Equations Describing Convective Mass Transfer withVolumeReaction 775 15.14.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of ConservationLawsofGasDynamicType 780 15.14.5. Systems of Second-Order Equations of Reaction-Diffusion Type . . . . . . . . 796 ReferencesforChapter15 798 CONTENTS xv 16. Integral Equations 801 16.1. Linear Integral Equations of the First Kind with Variable Integration Limit . . . . . . . . . 801 16.1.1. VolterraEquationsoftheFirstKind 801 16.1.2. Equations with Degenerate Kernel: K(x, t)=g 1 (x)h 1 (t)+···+ g n (x)h n (t) . . 802 16.1.3. Equations with Difference Kernel: K(x, t)=K(x – t) 804 16.1.4. Reduction of Volterra Equations of the First Kind to Volterra Equations of the SecondKind 807 16.1.5. MethodofQuadratures 808 16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit . . . . . . . 810 16.2.1. VolterraEquationsoftheSecondKind 810 16.2.2. Equations with Degenerate Kernel: K(x, t)=g 1 (x)h 1 (t)+···+ g n (x)h n (t) . . 811 16.2.3. Equations with Difference Kernel: K(x, t)=K(x – t) 813 16.2.4. Construction of Solutions of Integral Equations with Special Right-Hand Side 815 16.2.5. MethodofModelSolutions 818 16.2.6. SuccessiveApproximationMethod 822 16.2.7. MethodofQuadratures 823 16.3. Linear Integral Equations of the First Kind with Constant Limits of Integration . . . . . . 824 16.3.1. Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . 824 16.3.2. MethodofIntegralTransforms 825 16.3.3. Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration . . . . 829 16.4.1. Fredholm Integral Equations of the Second Kind. Resolvent . . . . . . . . . . . . . . 829 16.4.2. Fredholm Equations of the Second Kind with Degenerate Kernel . . . . . . . . . . 830 16.4.3. Solution as a Power Series in the Parameter. Method of Successive Approximations 832 16.4.4. Fredholm Theorems and the Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . 834 16.4.5. Fredholm Integral Equations of the Second Kind with Symmetric Kernel . . . . 835 16.4.6. Methods of Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 16.4.7. Method of Approximating a Kernel by a Degenerate One . . . . . . . . . . . . . . . . 844 16.4.8. CollocationMethod 847 16.4.9. MethodofLeastSquares 849 16.4.10. Bubnov–Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 16.4.11. QuadratureMethod 852 16.4.12. Systems of Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . 854 16.5. NonlinearIntegralEquations 856 16.5.1. NonlinearVolterraandUrysohnIntegralEquations 856 16.5.2. NonlinearVolterraIntegralEquations 856 16.5.3. EquationswithConstantIntegrationLimits 863 ReferencesforChapter16 871 17. Difference Equations and Other Functional Equations 873 17.1. Difference Equations of Integer Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 17.1.1. First-Order Linear Difference Equations of Integer Argument . . . . . . . . . . . . . 873 17.1.2. First-Order Nonlinear Difference Equations of Integer Argument . . . . . . . . . . 874 17.1.3. Second-Order Linear Difference Equations with Constant Coefficients 877 17.1.4. Second-Order Linear Difference Equations with Variable Coefficients 879 17.1.5. Linear Difference Equations of Arbitrary Order with Constant Coefficients . . 881 17.1.6. Linear Difference Equations of Arbitrary Order with Variable Coefficients . . . 882 17.1.7. NonlinearDifferenceEquationsofArbitraryOrder 884 xvi CONTENTS 17.2. Linear Difference Equations with a Single Continuous Variable . . . . . . . . . . . . . . . . . . 885 17.2.1. First-OrderLinearDifferenceEquations 885 17.2.2. Second-Order Linear Difference Equations with Integer Differences . . . . . . . . 894 17.2.3. Linear mth-Order Difference Equations with Integer Differences . . . . . . . . . . 898 17.2.4. Linear mth-Order Difference Equations with ArbitraryDifferences 904 17.3. LinearFunctionalEquations 907 17.3.1. Iterations of Functions and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 907 17.3.2. LinearHomogeneousFunctionalEquations 910 17.3.3. Linear Nonhomogeneous Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . 912 17.3.4. Linear Functional Equations Reducible to Linear Difference Equations with Constant Coefficients 916 17.4. Nonlinear Difference and Functional Equations with a Single Variable . . . . . . . . . . . . . 918 17.4.1. Nonlinear Difference Equations with a Single Variable . . . . . . . . . . . . . . . . . . 918 17.4.2. Reciprocal(Cyclic)FunctionalEquations 919 17.4.3. Nonlinear Functional Equations Reducible to Difference Equations . . . . . . . . 921 17.4.4. Power Series Solution of Nonlinear Functional Equations . . . . . . . . . . . . . . . . 922 17.5. FunctionalEquationswithSeveralVariables 922 17.5.1. MethodofDifferentiationinaParameter 922 17.5.2. Method of Differentiation in Independent Variables . . . . . . . . . . . . . . . . . . . . . 925 17.5.3. Method of Substituting Particular Values of Independent Arguments . . . . . . . 926 17.5.4. MethodofArgumentEliminationbyTestFunctions 928 17.5.5. Bilinear Functional Equations and Nonlinear Functional Equations Reducible to Bilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930 ReferencesforChapter17 935 18. Special Functions and Their Properties 937 18.1. Some Coefficients,Symbols,andNumbers 937 18.1.1. Binomial Coefficients 937 18.1.2. PochhammerSymbol 938 18.1.3. BernoulliNumbers 938 18.1.4. EulerNumbers 939 18.2. Error Functions. Exponential and Logarithmic Integrals . . . . . . . . . . . . . . . . . . . . . . . . 939 18.2.1. ErrorFunctionandComplementaryErrorFunction 939 18.2.2. Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940 18.2.3. LogarithmicIntegral 941 18.3. SineIntegralandCosineIntegral.FresnelIntegrals 941 18.3.1. SineIntegral 941 18.3.2. CosineIntegral 942 18.3.3. FresnelIntegrals 942 18.4. GammaFunction,PsiFunction,andBetaFunction 943 18.4.1. GammaFunction 943 18.4.2. PsiFunction(DigammaFunction) 944 18.4.3. BetaFunction 945 18.5. IncompleteGammaandBetaFunctions 946 18.5.1. IncompleteGammaFunction 946 18.5.2. IncompleteBetaFunction 947 CONTENTS xvii 18.6. BesselFunctions(CylindricalFunctions) 947 18.6.1. DefinitionsandBasicFormulas 947 18.6.2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 949 18.6.3. Zeros and Orthogonality Properties of Bessel Functions . . . . . . . . . . . . . . . . . 951 18.6.4. Hankel Functions (Bessel Functions of the Third Kind) . . . . . . . . . . . . . . . . . . 952 18.7. ModifiedBesselFunctions 953 18.7.1. Definitions.BasicFormulas 953 18.7.2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 954 18.8. AiryFunctions 955 18.8.1. Definition and Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 18.8.2. PowerSeriesandAsymptoticExpansions 956 18.9. Degenerate Hypergeometric Functions (Kummer Functions) . . . . . . . . . . . . . . . . . . . . . 956 18.9.1. DefinitionsandBasicFormulas 956 18.9.2. Integral Representations and Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 959 18.9.3. Whittaker Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 18.10. HypergeometricFunctions 960 18.10.1. Various Representations of the Hypergeometric Function . . . . . . . . . . . . . . 960 18.10.2. BasicProperties 960 18.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions . . . 962 18.11.1. Legendre Polynomials and Legendre Functions . . . . . . . . . . . . . . . . . . . . . . 962 18.11.2. Associated Legendre Functions with Integer Indices and Real Argument . . 964 18.11.3. AssociatedLegendreFunctions.GeneralCase 965 18.12. ParabolicCylinderFunctions 967 18.12.1. Definitions.BasicFormulas 967 18.12.2. Integral Representations, Asymptotic Expansions, and Linear Relations . . . 968 18.13. Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 18.13.1. Complete Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 18.13.2. Incomplete Elliptic Integrals (Elliptic Integrals) . . . . . . . . . . . . . . . . . . . . . . 970 18.14. Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972 18.14.1. Jacobi Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972 18.14.2. Weierstrass Elliptic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976 18.15. JacobiThetaFunctions 978 18.15.1. Series Representation of the Jacobi Theta Functions. Simplest Properties . . 978 18.15.2. Various Relations and Formulas. Connection with Jacobi Elliptic Functions 978 18.16. Mathieu Functions and ModifiedMathieuFunctions 980 18.16.1. MathieuFunctions 980 18.16.2. ModifiedMathieuFunctions 982 18.17. Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982 18.17.1. Laguerre Polynomials and Generalized Laguerre Polynomials . . . . . . . . . . . 982 18.17.2. Chebyshev Polynomials and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 18.17.3. Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 18.17.4. Jacobi Polynomials and Gegenbauer Polynomials . . . . . . . . . . . . . . . . . . . . 986 18.18. Nonorthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 18.18.1. Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 18.18.2. Euler Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989 ReferencesforChapter18 990 xviii CONTENTS 19. Calculus of Variations and Optimization 991 19.1. CalculusofVariationsandOptimalControl 991 19.1.1. Some DefinitionsandFormulas 991 19.1.2. SimplestProblemofCalculusofVariations 993 19.1.3. Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002 19.1.4. Problems with Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006 19.1.5. Lagrange Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008 19.1.6. Pontryagin Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 19.2. Mathematical Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 19.2.1. Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 19.2.2. Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027 References for Chapter 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028 20. Probability Theory 1031 20.1. Simplest Probabilistic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 20.1.1. Probabilities of Random Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 20.1.2. Conditional Probability and Simplest Formulas . . . . . . . . . . . . . . . . . . . . . . . . 1035 20.1.3. Sequences of Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 20.2. Random Variables and Their Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 20.2.1. One-Dimensional Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 20.2.2. Characteristics of One-Dimensional Random Variables . . . . . . . . . . . . . . . . . . 1042 20.2.3. Main Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047 20.2.4. Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 20.2.5. Multivariate Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 20.3. Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 20.3.1. Convergence of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 20.3.2. Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 20.4. Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 20.4.1. Theory of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 20.4.2. Models of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 References for Chapter 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 21. Mathematical Statistics 1081 21.1. Introduction to Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 21.1.1. Basic Notions and Problems of Mathematical Statistics . . . . . . . . . . . . . . . . . . 1081 21.1.2. Simplest Statistical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 21.1.3. Numerical Characteristics of Statistical Distribution . . . . . . . . . . . . . . . . . . . . 1087 21.2. Statistical Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 21.2.1. Estimators and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 21.2.2. Estimation Methods for Unknown Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 1091 21.2.3. Interval Estimators (Confidence Intervals) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 21.3. Statistical Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 21.3.1. Statistical Hypothesis. Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 21.3.2. Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 21.3.3. Problems Related to Normal Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 References for Chapter 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109 CONTENTS xix Part II. Mathematical Tables 1111 T1. Finite Sums and Infinite Series 1113 T1.1. Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 T1.1.1. Numerical Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 T1.1.2. Functional Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116 T1.2. Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 T1.2.1. Numerical Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 T1.2.2. Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120 References for Chapter T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127 T2. Integrals 1129 T2.1. Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129 T2.1.1. Integrals Involving Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129 T2.1.2. Integrals Involving Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134 T2.1.3. Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137 T2.1.4. Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137 T2.1.5. Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140 T2.1.6. Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142 T2.1.7. Integrals Involving Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 1147 T2.2. Tables of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 T2.2.1. Integrals Involving Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 T2.2.2. Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 T2.2.3. Integrals Involving Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152 T2.2.4. Integrals Involving Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152 T2.2.5. Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153 References for Chapter T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 T3. Integral Transforms 1157 T3.1. Tables of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 T3.1.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 T3.1.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159 T3.1.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159 T3.1.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160 T3.1.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161 T3.1.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161 T3.1.7. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163 T3.2. Tables of Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164 T3.2.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164 T3.2.2. Expressions with Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166 T3.2.3. Expressions with Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1170 T3.2.4. Expressions with Arbitrary Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172 T3.2.5. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172 T3.2.6. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174 T3.2.7. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174 T3.2.8. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 T3.2.9. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 xx CONTENTS T3.3. Tables of Fourier Cosine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 T3.3.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 T3.3.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 T3.3.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178 T3.3.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179 T3.3.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179 T3.3.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1180 T3.3.7. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181 T3.4. Tables of Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 T3.4.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 T3.4.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 T3.4.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183 T3.4.4. Expressions with Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184 T3.4.5. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184 T3.4.6. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185 T3.4.7. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186 T3.5. Tables of Mellin Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187 T3.5.1. General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187 T3.5.2. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188 T3.5.3. Expressions with Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188 T3.5.4. Expressions with Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189 T3.5.5. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189 T3.5.6. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1190 T3.6. Tables of Inverse Mellin Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1190 T3.6.1. Expressions with Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1190 T3.6.2. Expressions with Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 1191 T3.6.3. Expressions with Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192 T3.6.4. Expressions with Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193 References for Chapter T3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194 T4. Orthogonal Curvilinear Systems of Coordinate 1195 T4.1. Arbitrary Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195 T4.1.1. General Nonorthogonal Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . 1195 T4.1.2. General Orthogonal Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 1196 T4.2. Special Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198 T4.2.1. Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198 T4.2.2. Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199 T4.2.3. Coordinates of a Prolate Ellipsoid of Revolution . . . . . . . . . . . . . . . . . . . . . . . 1200 T4.2.4. Coordinates of an Oblate Ellipsoid of Revolution . . . . . . . . . . . . . . . . . . . . . . . 1201 T4.2.5. Coordinates of an Elliptic Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 T4.2.6. Conical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 T4.2.7. Parabolic Cylinder Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 T4.2.8. Parabolic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 T4.2.9. Bicylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204 T4.2.10. Bipolar Coordinates (in Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204 T4.2.11. Toroidal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 References for Chapter T4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 . 930 ReferencesforChapter17 935 18. Special Functions and Their Properties 937 18.1. Some Coefficients,Symbols,andNumbers 937 18.1.1. Binomial Coefficients 937 18.1.2. PochhammerSymbol 938 18.1 .3. . Equation and a Nonlinear Wave Equation 733 15.10. DifferentialConstraintsMethod 737 15.10.1. DescriptionoftheMethod 737 15.10.2. First-OrderDifferentialConstraints 739 15.10 .3. Second- and Higher-Order. Condition for Systems of Linear Equations 757 15.12 .3. Solution of the Cauchy Problem by the Inverse Scattering Problem Method 760 15. 13. ConservationLawsandIntegralsofMotion 766 15. 13. 1. Basic