Handbook of mathematics for engineers and scienteists part 3 ppsx

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