HANDBOOK OF MATHEMATICS FOR ENGINEERS AND SCIENTISTS pptx

HANDBOOK OF
MATHEMATICS FOR
ENGINEERS AND
SCIENTISTS
C5025_C000a.indd 1 10/16/06 2:53:21 PM
C5025_C000a.indd 2 10/16/06 2:53:21 PM
HANDBOOK OF
MATHEMATICS FOR
ENGINEERS AND
SCIENTISTS
Andrei D. Polyanin
Alexander V. Manzhirov
C5025_C000a.indd 3 10/16/06 2:53:21 PM
Chapman & Hall/CRC
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CONTENTS
Authors xxv
Preface xxvii
Main Notation xxix
Part I. Definitions, Formulas, Methods, and Theorems 1
1. Arithmetic and Elementary Algebra 3
1.1. RealNumbers 3
1.1.1. IntegerNumbers 3
1.1.2. Real, Rational, and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2. Equalities and Inequalities. Arithmetic Operations. Absolute Value . . . . . . . . . . . . . . . . 5
1.2.1. Equalities and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2. Addition and Multiplication of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3. Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4. Percentage 7
1.2.5. Absolute Value of a Number (Modulus of a Number) . . . . . . . . . . . . . . . . . . . . . . 8
1.3. PowersandLogarithms 8
1.3.1. PowersandRoots 8
1.3.2. Logarithms 9

1.4. BinomialTheoremandRelatedFormulas 10
1.4.1. Factorials. Binomial Coefficients.BinomialTheorem 10
1.4.2. RelatedFormulas 10
1.5. Arithmetic and Geometric Progressions. Finite Sums and Products . . . . . . . . . . . . . . . . . 11
1.5.1. ArithmeticandGeometricProgressions 11
1.5.2. Finite Series and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6. Mean Values and Inequalities of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.1. Arithmetic Mean, Geometric Mean, and Other Mean Values. Inequalities for
MeanValues 13
1.6.2. Inequalities of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7. Some Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7.1. ProofbyContradiction 15
1.7.2. Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.3. Proof by Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.4. Method of Undetermined Coefficients 17
ReferencesforChapter1 18
2. Elementary Functions 19
2.1. Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1. Power Function: y = x
α
19
2.1.2. Exponential Function: y = a
x
21
2.1.3. Logarithmic Function: y =log
a
x 22
2.2. Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1. Trigonometric Circle. Definition of Trigonometric Functions . . . . . . . . . . . . . . . 24
2.2.2. Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.3. Properties of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
v
vi CONTENTS
2.3. Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1. Definitions. Graphs of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . 30
2.3.2. Properties of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4. HyperbolicFunctions 34
2.4.1. Definitions. Graphs of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2. Properties of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5. InverseHyperbolicFunctions 39
2.5.1. Definitions. Graphs of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 39
2.5.2. Properties of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ReferencesforChapter2 42
3. Elementary Geometry 43
3.1. PlaneGeometry 43
3.1.1. Triangles 43
3.1.2. Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.3. Circle 56
3.2. SolidGeometry 59
3.2.1. StraightLines,Planes,andAnglesinSpace 59
3.2.2. Polyhedra 61
3.2.3. SolidsFormedbyRevolutionofLines 65
3.3. Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1. SphericalGeometry 70
3.3.2. SphericalTriangles 71
ReferencesforChapter3 75
4. Analytic Geometry 77
4.1. Points,Segments,andCoordinatesonLineandPlane 77
4.1.1. CoordinatesonLine 77
4.1.2. CoordinatesonPlane 78

4.1.3. PointsandSegmentsonPlane 81
4.2. CurvesonPlane 84
4.2.1. CurvesandTheirEquations 84
4.2.2. MainProblemsofAnalyticGeometryforCurves 88
4.3. StraightLinesandPointsonPlane 89
4.3.1. EquationsofStraightLinesonPlane 89
4.3.2. MutualArrangementofPointsandStraightLines 93
4.4. Second-Order Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.1. Circle 97
4.4.2. Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.3. Hyperbola 101
4.4.4. Parabola 104
4.4.5. Transformation of Second-Order Curves to Canonical Form . . . . . . . . . . . . . . . . 107
4.5. Coordinates, Vectors, Curves, and Surfaces in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.1. Vectors.CartesianCoordinateSystem 113
4.5.2. CoordinateSystems 114
4.5.3. Vectors. Products of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.5.4. Curves and Surfaces in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
CONTENTS vii
4.6. LineandPlaneinSpace 124
4.6.1. PlaneinSpace 124
4.6.2. LineinSpace 131
4.6.3. MutualArrangementofPoints,Lines,andPlanes 135
4.7. Quadric Surfaces (Quadrics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.7.1. Quadrics (Canonical Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.7.2. Quadrics(GeneralTheory) 148
ReferencesforChapter4 153
5. Algebra 155
5.1. Polynomials and Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.1.1. Polynomials and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.1.2. LinearandQuadraticEquations 157
5.1.3. CubicEquations 158
5.1.4. Fourth-DegreeEquation 159
5.1.5. Algebraic Equations of Arbitrary Degree and Their Properties . . . . . . . . . . . . . . 161
5.2. MatricesandDeterminants 167
5.2.1. Matrices 167
5.2.2. Determinants 175
5.2.3. EquivalentMatrices.Eigenvalues 180
5.3. Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.3.1. Concept of a Linear Space. Its Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . 187
5.3.2. Subspaces of Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.3.3. Coordinate Transformations Corresponding to Basis Transformations in a Linear
Space 191
5.4. Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.4.1. RealEuclideanSpace 192
5.4.2. Complex Euclidean Space (Unitary Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.4.3. Banach Spaces and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.5. SystemsofLinearAlgebraicEquations 197
5.5.1. Consistency Condition for a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.5.2. Finding Solutions of a System of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 198
5.6. LinearOperators 204
5.6.1. NotionofaLinearOperator.ItsProperties 204
5.6.2. LinearOperatorsinMatrixForm 208
5.6.3. EigenvectorsandEigenvaluesofLinearOperators 209
5.7. Bilinear and Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.7.1. Linear and Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.7.2. Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.7.3. QuadraticForms 216
5.7.4. Bilinear and Quadratic Forms in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . 219
5.7.5. Second-Order Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5.8. SomeFactsfromGroupTheory 225
5.8.1. Groups and Their Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.8.2. Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.8.3. GroupRepresentations 230
ReferencesforChapter5 233
viii CONTENTS
6. Limits and Derivatives 235
6.1. BasicConceptsofMathematicalAnalysis 235
6.1.1. NumberSets.FunctionsofRealVariable 235
6.1.2. LimitofaSequence 237
6.1.3. LimitofaFunction.Asymptotes 240
6.1.4. Infinitely Small and InfinitelyLargeFunctions 242
6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind . . . . . . . 243
6.1.6. ConvexandConcaveFunctions 245
6.1.7. Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.1.8. ConvergenceofFunctions 249
6.2. DifferentialCalculusforFunctionsofaSingleVariable 250
6.2.1. Derivative and Differential, Their Geometrical and Physical Meaning . . . . . . . . . 250
6.2.2. TableofDerivativesandDifferentiationRules 252
6.2.3. Theorems about Differentiable Functions. L’Hospital Rule . . . . . . . . . . . . . . . . . 254
6.2.4. Higher-Order Derivatives and Differentials. Taylor’s Formula . . . . . . . . . . . . . . . 255
6.2.5. Extremal Points. Points of Inflection 257
6.2.6. Qualitative Analysis of Functions and Construction of Graphs . . . . . . . . . . . . . . 259
6.2.7. Approximate Solution of Equations (Root-Finding Algorithms for Continuous
Functions) 260
6.3. FunctionsofSeveralVariables.PartialDerivatives 263
6.3.1. PointSets.Functions.LimitsandContinuity 263
6.3.2. DifferentiationofFunctionsofSeveralVariables 264
6.3.3. Directional Derivative. Gradient. GeometricalApplications 267
6.3.4. ExtremalPointsofFunctionsofSeveralVariables 269

6.3.5. DifferentialOperatorsoftheFieldTheory 272
ReferencesforChapter6 272
7. Integrals 273
7.1. IndefiniteIntegral 273
7.1.1. Antiderivative. IndefiniteIntegralandItsProperties 273
7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral. Integration
Examples 274
7.1.3. IntegrationofRationalFunctions 276
7.1.4. Integration of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7.1.5. Integration of Exponential and Trigonometric Functions . . . . . . . . . . . . . . . . . . . 281
7.1.6. Integration of Polynomials Multiplied by Elementary Functions . . . . . . . . . . . . . 283
7.2. DefiniteIntegral 286
7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the
DefiniteIntegral 286
7.2.2. Properties of DefiniteIntegralsandUsefulFormulas 287
7.2.3. General Reduction Formulas for the Evaluation of Integrals . . . . . . . . . . . . . . . . 289
7.2.4. General Asymptotic Formulas for the Calculation of Integrals . . . . . . . . . . . . . . . 290
7.2.5. Mean Value Theorems. Properties of Integrals in Terms of Inequalities.
ArithmeticMeanandGeometricMeanofFunctions 295
7.2.6. Geometric and Physical Applications of the DefiniteIntegral 299
7.2.7. Improper Integrals with InfiniteIntegrationLimit 301
7.2.8. General Reduction Formulas for the Calculation of Improper Integrals . . . . . . . . 304
7.2.9. General Asymptotic Formulas for the Calculation of Improper Integrals . . . . . . . 307
7.2.10. Improper Integrals of Unbounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
7.2.11. Cauchy-Type Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
CONTENTS ix
7.2.12. Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7.2.13. SquareIntegrableFunctions 314
7.2.14. Approximate (Numerical) Methods for Computation of Definite Integrals . . . . 315
7.3. Double and Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

7.3.1. Definition and Properties of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 317
7.3.2. Computation of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.3.3. Geometric and Physical Applications of the Double Integral . . . . . . . . . . . . . . . . 323
7.3.4. Definition and Properties of the Triple Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
7.3.5. Computation of the Triple Integral. Some Applications. Iterated Integrals and
AsymptoticFormulas 325
7.4. LineandSurfaceIntegrals 329
7.4.1. LineIntegraloftheFirstKind 329
7.4.2. LineIntegraloftheSecondKind 330
7.4.3. SurfaceIntegraloftheFirstKind 332
7.4.4. SurfaceIntegraloftheSecondKind 333
7.4.5. IntegralFormulasofVectorCalculus 334
ReferencesforChapter7 335
8. Series 337
8.1. Numerical Series and Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.1.1. Convergent Numerical Series and Their Properties. Cauchy’s Criterion . . . . . . . 337
8.1.2. Convergence Criteria for Series with Positive (Nonnegative) Terms . . . . . . . . . . 338
8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional
Convergence 341
8.1.4. Multiplication of Series. Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
8.1.5. Summation Methods. Convergence Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 344
8.1.6. Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
8.2. FunctionalSeries 348
8.2.1. Pointwise and Uniform Convergence of Functional Series . . . . . . . . . . . . . . . . . . 348
8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent
Series 349
8.3. PowerSeries 350
8.3.1. Radius of Convergence of Power Series. Properties of Power Series . . . . . . . . . . 350
8.3.2. TaylorandMaclaurinPowerSeries 352
8.3.3. Operations with Power Series. Summation Formulas for Power Series . . . . . . . . 354

8.4. FourierSeries 357
8.4.1. Representation of 2π-Periodic Functions by Fourier Series. Main Results . . . . . 357
8.4.2. Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions . . . . . . . 359
8.4.3. Criteria of Uniform and Mean-Square Convergence of Fourier Series . . . . . . . . . 361
8.4.4. Summation Formulas for Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
8.5. AsymptoticSeries 363
8.5.1. Asymptotic Series of Poincar
´
e Type. Formulas for the Coefficients 363
8.5.2. OperationswithAsymptoticSeries 364
ReferencesforChapter8 366
9. Differential Geometry 367
9.1. TheoryofCurves 367
9.1.1. PlaneCurves 367
9.1.2. SpaceCurves 379
x CONTENTS
9.2. Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.2.1. Elementary Notions in Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.2.2. CurvatureofCurvesonSurface 392
9.2.3. IntrinsicGeometryofSurface 395
ReferencesforChapter9 397
10. Functions of Complex Variable 399
10.1. BasicNotions 399
10.1.1. ComplexNumbers.FunctionsofComplexVariable 399
10.1.2. FunctionsofComplexVariable 401
10.2. MainApplications 419
10.2.1. ConformalMappings 419
10.2.2. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
ReferencesforChapter10 433
11. Integral Transforms 435

11.1. GeneralFormofIntegralTransforms.SomeFormulas 435
11.1.1. IntegralTransformsandInversionFormulas 435
11.1.2. Residues.JordanLemma 435
11.2. LaplaceTransform 436
11.2.1. LaplaceTransformandtheInverseLaplaceTransform 436
11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some
Functions 437
11.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series
andAsymptoticExpansions 440
11.3. Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.3.1. Mellin Transform and the Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.3.2. Main Properties of the Mellin Transform. Relation Among the Mellin,
Laplace, and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
11.4. VariousFormsoftheFourierTransform 443
11.4.1. FourierTransformandtheInverseFourierTransform 443
11.4.2. FourierCosineandSineTransforms 445
11.5. OtherIntegralTransforms 446
11.5.1. Integral Transforms Whose Kernels Contain Bessel Functions and Modified
BesselFunctions 446
11.5.2. Summary Table of Integral Transforms. Areas of Application of Integral
Transforms 448
ReferencesforChapter11 451
12. Ordinary Differential Equations 453
12.1. First-OrderDifferentialEquations 453
12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 453
12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration . . . . 456
12.1.3. ExactDifferentialEquations.IntegratingFactor 458
12.1.4. Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
12.1.5. AbelEquationsoftheFirstKind 462
12.1.6. Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

12.1.7. EquationsNotSolvedfortheDerivative 465
12.1.8. ContactTransformations 468
12.1.9. Approximate Analytic Methods for Solution of Equations . . . . . . . . . . . . . . . . 469
12.1.10. NumericalIntegrationofDifferentialEquations 471
CONTENTS xi
12.2. Second-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
12.2.1. Formulas for the General Solution. Some Transformations . . . . . . . . . . . . . . . 472
12.2.2. Representation of Solutions as a Series in the Independent Variable . . . . . . . . 475
12.2.3. AsymptoticSolutions 477
12.2.4. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
12.2.5. EigenvalueProblems 482
12.2.6. TheoremsonEstimatesandZerosofSolutions 487
12.3. Second-Order Nonlinear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
12.3.1. Form of the General Solution. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . 488
12.3.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
12.3.3. Methods of Regular Series Expansions with Respect to the Independent
Variable 492
12.3.4. Movable Singularities of Solutions of Ordinary Differential Equations.
Painlev
´
eTranscendents 494
12.3.5. Perturbation Methods of Mechanics and Physics . . . . . . . . . . . . . . . . . . . . . . . 499
12.3.6. Galerkin Method and Its Modifications (Projection Methods) . . . . . . . . . . . . . 508
12.3.7. Iteration and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
12.4. LinearEquationsofArbitraryOrder 514
12.4.1. Linear Equations with Constant Coefficients 514
12.4.2. Linear Equations with Variable Coefficients 518
12.4.3. AsymptoticSolutionsofLinearEquations 522
12.4.4. CollocationMethodandItsConvergence 523
12.5. NonlinearEquationsofArbitraryOrder 524

12.5.1. StructureoftheGeneralSolution.CauchyProblem 524
12.5.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
12.6. LinearSystemsofOrdinaryDifferentialEquations 528
12.6.1. Systems of Linear Constant-CoefficientEquations 528
12.6.2. Systems of Linear Variable-CoefficientEquations 539
12.7. NonlinearSystemsofOrdinaryDifferentialEquations 542
12.7.1. Solutions and First Integrals. Uniqueness and Existence Theorems . . . . . . . . . 542
12.7.2. Integrable Combinations. Autonomous Systems of Equations . . . . . . . . . . . . . 545
12.7.3. Elements of Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
ReferencesforChapter12 550
13. First-Order Partial Differential Equations 553
13.1. Linear and Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
13.1.1. CharacteristicSystem.GeneralSolution 553
13.1.2. CauchyProblem.ExistenceandUniquenessTheorem 556
13.1.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations . . 558
13.1.4. Quasilinear Equations of General Form. Generalized Solution, Jump
Condition, and Stability Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
13.2. NonlinearEquations 570
13.2.1. Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
13.2.2. CauchyProblem.ExistenceandUniquenessTheorem 576
13.2.3. Generalized Viscosity Solutions and Their Applications . . . . . . . . . . . . . . . . . 579
ReferencesforChapter13 584
xii CONTENTS
14. Linear Partial Differential Equations 585
14.1. Classification of Second-Order Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 585
14.1.1. EquationswithTwoIndependentVariables 585
14.1.2. EquationswithManyIndependentVariables 589
14.2. BasicProblemsofMathematicalPhysics 590
14.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems 590
14.2.2. First, Second, Third, and Mixed Boundary Value Problems . . . . . . . . . . . . . . . 593

14.3. Properties and Exact Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
14.3.1. Homogeneous Linear Equations and Their Particular Solutions . . . . . . . . . . . . 594
14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions . . . . . . . . . 598
14.3.3. General Solutions of Some Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . 600
14.4. Method of Separation of Variables (Fourier Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
14.4.1. Description of the Method of Separation of Variables. General Stage of
Solution 602
14.4.2. Problems for Parabolic Equations: Final Stage of Solution . . . . . . . . . . . . . . . 605
14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution . . . . . . . . . . . . . . 607
14.4.4. Solution of Boundary Value Problems for Elliptic Equations . . . . . . . . . . . . . . 609
14.5. IntegralTransformsMethod 611
14.5.1. Laplace Transform and Its Application in Mathematical Physics . . . . . . . . . . . 611
14.5.2. Fourier Transform and Its Application in Mathematical Physics . . . . . . . . . . . 614
14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution . . 615
14.6.1. CauchyProblemforParabolicEquations 615
14.6.2. CauchyProblemforHyperbolicEquations 617
14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green’s
Function 618
14.7.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 618
14.7.2. Problems for Equation s(x)
∂w
∂t
=
∂
∂x

p(x)
∂w
∂x


–q(x)w + Φ(x, t) 620
14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s
Function.GoursatProblem 623
14.8.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 623
14.8.2. Problems for Equation s(x)
∂
2
w
∂t
2
=
∂
∂x

p(x)
∂w
∂x

–q(x)w + Φ(x, t) 624
14.8.3. Problems for Equation
∂
2
w
∂t
2
+ a(t)
∂w
∂t
= b(t)


∂
∂x

p(x)
∂w
∂x

– q(x)w

+ Φ(x, t) 626
14.8.4. Generalized Cauchy Problem with Initial Conditions Set Along a Curve . . . . . 627
14.8.5. Goursat Problem (a Problem with Initial Data of Characteristics) . . . . . . . . . . 629
14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables . . . . . . . . . 631
14.9.1. Problems and the Green’s Functions for Equation
a(x)
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+ b(x)
∂w
∂x
+ c(x)w =–Φ(x, y) 631

14.9.2. Representation of Solutions to Boundary Value Problems via the Green’s
Functions 633
14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions
viatheGreen’sFunction 634
14.10.1. ProblemsforParabolicEquations 634
14.10.2. Problems for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
14.10.3. Problems for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
14.10.4. Comparison of the Solution Structures for Boundary Value Problems for
EquationsofVariousTypes 638
CONTENTS xiii
14.11. Construction of the Green’s Functions. General Formulas and Relations . . . . . . . . . . 639
14.11.1. Green’s Functions of Boundary Value Problems for Equations of Various
Types in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
14.11.2. Green’s Functions Admitting Incomplete Separation of Variables . . . . . . . . 640
14.11.3. Construction of Green’s Functions via Fundamental Solutions . . . . . . . . . . 642
14.12. Duhamel’s Principles in Nonstationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
14.12.1. Problems for Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 646
14.12.2. Problems for Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . 648
14.13. Transformations Simplifying Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . 649
14.13.1. Transformations That Lead to Homogeneous Boundary Conditions . . . . . . 649
14.13.2. Transformations That Lead to Homogeneous Initial and Boundary
Conditions 650
ReferencesforChapter14 650
15. Nonlinear Partial Differential Equations 653
15.1. Classification of Second-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
15.1.1. Classification of Semilinear Equations in Two Independent Variables . . . . . . . 653
15.1.2. Classification of Nonlinear Equations in Two Independent Variables . . . . . . . . 653
15.2. Transformations of Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . 655
15.2.1. PointTransformations:OverviewandExamples 655
15.2.2. Hodograph Transformations (Special Point Transformations) . . . . . . . . . . . . . 657

15.2.3. Contact Transformations. Legendre and Euler Transformations . . . . . . . . . . . . 660
15.2.4. B
¨
acklund Transformations. Differential Substitutions . . . . . . . . . . . . . . . . . . . 663
15.2.5. Differential Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions.
SimilarityMethod 667
15.3.1. PreliminaryRemarks 667
15.3.2. Traveling-Wave Solutions. Invariance of Equations Under Translations . . . . . 667
15.3.3. Self-Similar Solutions. Invariance of Equations Under Scaling
Transformations 669
15.3.4. Equations Invariant Under Combinations of Translation and Scaling
Transformations,andTheirSolutions 674
15.3.5. GeneralizedSelf-SimilarSolutions 677
15.4. ExactSolutionswithSimpleSeparationofVariables 678
15.4.1. Multiplicative and Additive Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . 678
15.4.2. Simple Separation of Variables in Nonlinear Partial Differential Equations . . . 678
15.4.3. Complex Separation of Variables in Nonlinear Partial Differential Equations . 679
15.5. MethodofGeneralizedSeparationofVariables 681
15.5.1. Structure of Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 681
15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System
ofCoordinateFunctions 683
15.5.3. Solution of Functional Differential Equations by Differentiation . . . . . . . . . . . 684
15.5.4. Solution of Functional-Differential Equations by Splitting . . . . . . . . . . . . . . . . 688
15.5.5. Titov–GalaktionovMethod 693
15.6. MethodofFunctionalSeparationofVariables 697
15.6.1. Structure of Functional Separable Solutions. Solution by Reduction to
Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
15.6.2. Special Functional Separable Solutions. Generalized Traveling-Wave
Solutions 697

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
CONTENTS xxi
T5. Ordinary Differential Equations 1207
T5.1. First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207
T5.2. Second-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212
T5.2.1. Equations Involving Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213
T5.2.2. Equations Involving Exponential and Other Functions . . . . . . . . . . . . . . . . . . . 1220
T5.2.3. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222
T5.3. Second-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223
T5.3.1. Equations of the Form y

xx
= f (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223

T5.3.2. Equations of the Form f (x, y)y

xx
= g(x, y, y

x
) . . . . . . . . . . . . . . . . . . . . . . . . . 1225
References for Chapter T5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228
T6. Systems of Ordinary Differential Equations 1229
T6.1. Linear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229
T6.1.1. Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229
T6.1.2. Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232
T6.2. Linear Systems of Three and More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237
T6.3. Nonlinear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239
T6.3.1. Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239
T6.3.2. Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1240
T6.4. Nonlinear Systems of Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244
References for Chapter T6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246
T7. First-Order Partial Differential Equations 1247
T7.1. Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247
T7.1.1. Equations of the Form f (x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= 0 . . . . . . . . . . . . . . . . . . . . . . 1247
T7.1.2. Equations of the Form f (x, y)
∂w
∂x

+ g(x, y)
∂w
∂y
= h(x, y) . . . . . . . . . . . . . . . . . 1248
T7.1.3. Equations of the Form f (x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h(x, y)w + r(x, y) . . . . . . . . 1250
T7.2. Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252
T7.2.1. Equations of the Form f (x, y)
∂w
∂x
+ g(x, y)
∂w
∂y
= h(x, y, w) . . . . . . . . . . . . . . . 1252
T7.2.2. Equations of the Form
∂w
∂x
+ f (x, y, w)
∂w
∂y
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 1254
T7.2.3. Equations of the Form
∂w
∂x
+ f (x, y, w)

∂w
∂y
= g(x, y, w) . . . . . . . . . . . . . . . . . . 1256
T7.3. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258
T7.3.1. Equations Quadratic in One Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258
T7.3.2. Equations Quadratic in Two Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259
T7.3.3. Equations with Arbitrary Nonlinearities in Derivatives . . . . . . . . . . . . . . . . . . . 1261
References for Chapter T7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265
T8. Linear Equations and Problems of Mathematical Physics 1267
T8.1. Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267
T8.1.1. Heat Equation
∂w
∂t
= a
∂
2
w
∂x
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267
T8.1.2. Nonhomogeneous Heat Equation
∂w
∂t
= a
∂
2
w
∂x
2
+ Φ(x, t) . . . . . . . . . . . . . . . . . . 1268

T8.1.3. Equation of the Form
∂w
∂t
= a
∂
2
w
∂x
2
+ b
∂w
∂x
+ cw + Φ(x, t) . . . . . . . . . . . . . . . . . 1270
T8.1.4. Heat Equation with Axial Symmetry
∂w
∂t
= a

∂
2
w
∂r
2
+
1
r
∂w
∂r

. . . . . . . . . . . . . . . 1270

T8.1.5. Equation of the Form
∂w
∂t
= a

∂
2
w
∂r
2
+
1
r
∂w
∂r

+ Φ(r, t) . . . . . . . . . . . . . . . . . . . 1271
T8.1.6. Heat Equation with Central Symmetry
∂w
∂t
= a

∂
2
w
∂r
2
+
2
r

∂w
∂r

. . . . . . . . . . . . . 1272
T8.1.7. Equation of the Form
∂w
∂t
= a

∂
2
w
∂r
2
+
2
r
∂w
∂r

+ Φ(r, t) . . . . . . . . . . . . . . . . . . . 1273
T8.1.8. Equation of the Form
∂w
∂t
=
∂
2
w
∂x
2

+
1–2β
x
∂w
∂x
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1274
xxii CONTENTS
T8.1.9. Equations of the Diffusion (Thermal) Boundary Layer . . . . . . . . . . . . . . . . . . . 1276
T8.1.10. Schr
¨
odinger Equation i
∂w
∂t
=–

2
2m
∂
2
w
∂x
2
+ U (x)w . . . . . . . . . . . . . . . . . . . . . 1276
T8.2. Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278
T8.2.1. Wave Equation
∂
2
w
∂t
2

= a
2
∂
2
w
∂x
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278
T8.2.2. Equation of the Form
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
+ Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . 1279
T8.2.3. Klein–Gordon Equation
∂
2
w
∂t
2
= a
2

∂
2
w
∂x
2
– bw . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280
T8.2.4. Equation of the Form
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
– bw + Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . 1281
T8.2.5. Equation of the Form
∂
2
w
∂t
2
= a
2

∂

2
w
∂r
2
+
1
r
∂w
∂r

+ Φ(r, t) . . . . . . . . . . . . . . . . . . 1282
T8.2.6. Equation of the Form
∂
2
w
∂t
2
= a
2

∂
2
w
∂r
2
+
2
r
∂w
∂r


+ Φ(r, t) . . . . . . . . . . . . . . . . . . 1283
T8.2.7. Equations of the Form
∂
2
w
∂t
2
+ k
∂w
∂t
= a
2
∂
2
w
∂x
2
+ b
∂w
∂x
+ cw + Φ(x, t) . . . . . . . . . 1284
T8.3. Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284
T8.3.1. Laplace Equation Δw = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284
T8.3.2. Poisson Equation Δw + Φ(x) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287
T8.3.3. Helmholtz Equation Δw + λw =–Φ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289
T8.4. Fourth-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294
T8.4.1. Equation of the Form
∂
2

w
∂t
2
+ a
2
∂
4
w
∂x
4
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294
T8.4.2. Equation of the Form
∂
2
w
∂t
2
+ a
2
∂
4
w
∂x
4
= Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . 1295
T8.4.3. Biharmonic Equation ΔΔw = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297
T8.4.4. Nonhomogeneous Biharmonic Equation ΔΔw = Φ(x, y) . . . . . . . . . . . . . . . . 1298
References for Chapter T8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299
T9. Nonlinear Mathematical Physics Equations 1301
T9.1. Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301

T9.1.1. Nonlinear Heat Equations of the Form
∂w
∂t
=
∂
2
w
∂x
2
+ f (w) . . . . . . . . . . . . . . . . 1301
T9.1.2. Equations of the Form
∂w
∂t
=
∂
∂x

f(w)
∂w
∂x

+ g(w) . . . . . . . . . . . . . . . . . . . . . . 1303
T9.1.3. Burgers Equation and Nonlinear Heat Equation in Radial Symmetric Cases . . 1307
T9.1.4. Nonlinear Schr
¨
odinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309
T9.2. Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312
T9.2.1. Nonlinear Wave Equations of the Form
∂
2

w
∂t
2
= a
∂
2
w
∂x
2
+ f(w) . . . . . . . . . . . . . . 1312
T9.2.2. Other Nonlinear Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316
T9.3. Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318
T9.3.1. Nonlinear Heat Equations of the Form
∂
2
w
∂x
2
+
∂
2
w
∂y
2
= f(w) . . . . . . . . . . . . . . . . 1318
T9.3.2. Equations of the Form
∂
∂x

f(x)

∂w
∂x

+
∂
∂y

g(y)
∂w
∂y

= f(w) . . . . . . . . . . . . . . 1321
T9.3.3. Equations of the Form
∂
∂x

f(w)
∂w
∂x

+
∂
∂y

g(w)
∂w
∂y

= h(w) . . . . . . . . . . . . . . 1322
T9.4. Other Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324

T9.4.1. Equations of Transonic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324
T9.4.2. Monge–Amp
`
ere Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326
T9.5. Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327
T9.5.1. Third-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327
T9.5.2. Fourth-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332
References for Chapter T9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335
T10. Systems of Partial Differential Equations 1337
T10.1. Nonlinear Systems of Two First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337
T10.2. Linear Systems of Two Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1341
CONTENTS xxiii
T10.3. Nonlinear Systems of Two Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1343
T10.3.1. Systems of the Form
∂u
∂t
= a
∂
2
u
∂x
2
+ F (u, w),
∂w
∂t
= b
∂
2
w
∂x

2
+ G(u, w) . . . . . . 1343
T10.3.2. Systems of the Form
∂u
∂t
=
a
x
n
∂
∂x

x
n
∂u
∂x

+ F (u, w),
∂w
∂t
=
b
x
n
∂
∂x

x
n
∂w

∂x

+ G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357
T10.3.3. Systems of the Form Δu = F (u, w), Δw = G(u, w) . . . . . . . . . . . . . . . . . . 1364
T10.3.4. Systems of the Form
∂
2
u
∂t
2
=
a
x
n
∂
∂x

x
n
∂u
∂x

+ F (u, w),
∂
2
w
∂t
2
=
b

x
n
∂
∂x

x
n
∂w
∂x

+ G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368
T10.3.5. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373
T10.4. Systems of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374
T10.4.1. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374
T10.4.2. Nonlinear Systems of Two Equations Involving the First Derivatives in t . . 1374
T10.4.3. Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1378
T10.4.4. Nonlinear Systems of Many Equations Involving the First Derivatives in t . 1381
References for Chapter T10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382
T11. Integral Equations 1385
T11.1. Linear Equations of the First Kind with Variable Limit of Integration . . . . . . . . . . . . . 1385
T11.2. Linear Equations of the Second Kind with Variable Limit of Integration . . . . . . . . . . . 1391
T11.3. Linear Equations of the First Kind with Constant Limits of Integration . . . . . . . . . . . . 1396
T11.4. Linear Equations of the Second Kind with Constant Limits of Integration . . . . . . . . . 1401
References for Chapter T11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406
T12. Functional Equations 1409
T12.1. Linear Functional Equations in One Independent Variable . . . . . . . . . . . . . . . . . . . . . . 1409
T12.1.1. Linear Difference and Functional Equations Involving Unknown Function
with Two Different Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409
T12.1.2. Other Linear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421
T12.2. Nonlinear Functional Equations in One Independent Variable . . . . . . . . . . . . . . . . . . . 1428

T12.2.1. Functional Equations with Quadratic Nonlinearity . . . . . . . . . . . . . . . . . . . . 1428
T12.2.2. Functional Equations with Power Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 1433
T12.2.3. Nonlinear Functional Equation of General Form . . . . . . . . . . . . . . . . . . . . . 1434
T12.3. Functional Equations in Several Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . 1438
T12.3.1. Linear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438
T12.3.2. Nonlinear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443
References for Chapter T12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1450
Supplement. Some Useful Electronic Mathematical Resources 1451
Index 1453