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A CONCISE HANDBOOK OF MATHEMATICS, PHYSICS, AND ENGINEERING SCIENCES K10319_FM.indd 1 9/20/10 4:27:31 PM A CONCISE HANDBOOK OF MATHEMATICS, PHYSICS, AND ENGINEERING SCIENCES Edited by Andrei D. Polyanin Alexei I. Chernoutsan Authors A.I. Chernoutsan, A.V. Egorov, A.V. Manzhirov, A.D. Polyanin, V.D. Polyanin, V.A. Popov, B.V. Putyatin, Yu.V. Repina, V.M. Safrai, A.I. Zhurov K10319_FM.indd 2 9/20/10 4:27:32 PM A CONCISE HANDBOOK OF MATHEMATICS, PHYSICS, AND ENGINEERING SCIENCES Edited by Andrei D. Polyanin Alexei I. Chernoutsan Authors A.I. Chernoutsan, A.V. Egorov, A.V. Manzhirov, A.D. Polyanin, V.D. Polyanin, V.A. Popov, B.V. Putyatin, Yu.V. Repina, V.M. Safrai, A.I. Zhurov K10319_FM.indd 3 9/20/10 4:27:32 PM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4398-0640-1 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pagev CONTENTS Preface .................................................................. xxv Editors .................................................................. xxvii PartI. Mathematics 1 M1. ArithmeticandElementaryAlgebra .................................. 3 M1.1. RealNumbers ...................................................... 3 M1.1.1. IntegerNumbers ............................................ 3 M1.1.2. Real,Rational, andIrrational Numbers ......................... 4 M1.2. EqualitiesandInequalities. ArithmeticOperations. AbsoluteValue ........ 5 M1.2.1. EqualitiesandInequalities .................................... 5 M1.2.2. AdditionandMultiplication ofNumbers ....................... 5 M1.2.3. RatiosandProportions ....................................... 6 M1.2.4. Percentage ................................................. 6 M1.2.5. AbsoluteValueofaNumber(ModulusofaNumber) ............ 7 M1.3. PowersandLogarithms .............................................. 7 M1.3.1. PowersandRoots ........................................... 7 M1.3.2. Logarithms ................................................. 8 M1.4. BinomialTheoremandRelatedFormulas ............................... 9 M1.4.1. Factorials. BinomialCoefficients. BinomialTheorem ............ 9 M1.4.2. RelatedFormulas ........................................... 9 M1.5. Progressions ........................................................ 10 M1.5.1. ArithmeticProgression ....................................... 10 M1.5.2. GeometricProgression ....................................... 10 M1.6. MeanValuesandSomeInequalities .................................... 11 M1.6.1. ArithmeticMean,GeometricMean,andOtherMeanValues ...... 11 M1.6.2. Inequalities forMeanValues .................................. 11 M1.6.3. SomeInequalities ofGeneralForm ............................ 11 M1.7. SomeMathematicalMethods ......................................... 12 M1.7.1. ProofbyContradiction ....................................... 12 M1.7.2. Mathematical Induction ...................................... 12 M1.7.3. ProofbyCounterexample .................................... 13 Bibliography forChapterM1 ............................................... 13 M2. ElementaryFunctions ............................................... 15 M2.1. Power,Exponential, andLogarithmicFunctions ......................... 15 M2.1.1. PowerFunction: y = xα ...................................... 15 M2.1.2. Exponential Function: y =ax ................................. 17 M2.1.3. LogarithmicFunction: y = log x ............................. 18 a M2.2. Trigonometric Functions ............................................. 19 M2.2.1. Trigonometric Circle. DefinitionofTrigonometric Functions ...... 19 M2.2.2. GraphsofTrigonometricFunctions ............................ 21 M2.2.3. PropertiesofTrigonometric Functions ......................... 22 v K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pagev K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pagevi vi CONTENTS M2.3. InverseTrigonometric Functions ...................................... 26 M2.3.1. Definitions. GraphsofInverse TrigonometricFunctions .......... 26 M2.3.2. PropertiesofInverseTrigonometric Functions ................... 27 M2.4. Hyperbolic Functions ................................................ 29 M2.4.1. Definitions. GraphsofHyperbolic Functions .................... 29 M2.4.2. PropertiesofHyperbolic Functions ............................ 30 M2.5. InverseHyperbolic Functions ......................................... 33 M2.5.1. Definitions. GraphsofInverse HyperbolicFunctions ............. 33 M2.5.2. PropertiesofInverseHyperbolic Functions ..................... 34 Bibliography forChapterM2 ............................................... 35 M3. ElementaryGeometry ............................................... 37 M3.1. PlaneGeometry ..................................................... 37 M3.1.1. Triangles ................................................... 37 M3.1.2. Polygons ................................................... 43 M3.1.3. Circle ...................................................... 46 M3.2. SolidGeometry ..................................................... 49 M3.2.1. StraightLines,Planes,andAnglesinSpace ..................... 49 M3.2.2. Polyhedra .................................................. 51 M3.2.3. SolidsFormedbyRevolution ofLines ......................... 55 Bibliography forChapterM3 ............................................... 60 M4. AnalyticGeometry .................................................. 61 M4.1. Points,Segments,andCoordinate Plane ................................ 61 M4.1.1. CartesianandPolarCoordinates onPlane ....................... 61 M4.1.2. DistanceBetweenPoints. DivisionofSegmentinGivenRatio. Area ofaPolygon ................................................ 62 M4.2. StraightLinesandPointsonPlane ..................................... 64 M4.2.1. EquationsofStraightLinesonPlane ........................... 64 M4.2.2. MutualArrangement ofPointsandStraightLines ................ 66 M4.3. QuadraticCurves .................................................... 68 M4.3.1. Circle ...................................................... 68 M4.3.2. Ellipse ..................................................... 69 M4.3.3. Hyperbola .................................................. 71 M4.3.4. Parabola ................................................... 74 M4.3.5. Transformation ofQuadraticCurvestoCanonicalForm .......... 75 M4.4. Coordinates, Vectors,Curves,andSurfacesinSpace ..................... 78 M4.4.1. VectorsandTheirProperties .................................. 78 M4.4.2. Coordinate Systems ......................................... 79 M4.4.3. Scalar,Cross,andScalarTripleProductsofVectors .............. 82 M4.5. LineandPlaneinSpace .............................................. 84 M4.5.1. PlaneinSpace .............................................. 84 M4.5.2. LineinSpace ............................................... 89 M4.5.3. MutualArrangement ofPoints,Lines,andPlanes ................ 91 M4.6. QuadricSurfaces(Quadrics) .......................................... 96 M4.6.1. QuadricsandTheirCanonicalEquations ....................... 96 M4.6.2. Quadrics(GeneralTheory) ................................... 99 Bibliography forChapterM4 ............................................... 102 K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pagevi K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pagevii CONTENTS vii M5. Algebra ............................................................. 103 M5.1. PolynomialsandAlgebraicEquations .................................. 103 M5.1.1. PolynomialsandTheirProperties .............................. 103 M5.1.2. LinearandQuadraticEquations ............................... 105 M5.1.3. CubicEquations ............................................ 105 M5.1.4. Fourth DegreeEquation ...................................... 107 M5.1.5. AlgebraicEquations ofArbitraryDegreeandTheirProperties ..... 108 M5.2. DeterminantsandMatrices ........................................... 112 M5.2.1. Determinants ............................................... 112 M5.2.2. Matrices. TypesofMatrices. Operations withMatrices ........... 114 M5.2.3. InverseMatrix. Functions ofMatrices .......................... 119 M5.2.4. Eigenvalues and Characteristic Equation of a Matrix. The Cayley–Hamilton Theorem ................................... 120 M5.3. SystemsofLinearAlgebraicEquations ................................. 123 M5.3.1. Consistency ConditionforaLinearSystem ..................... 123 M5.3.2. FindingSolutionsofaSystemofLinearEquations .............. 124 M5.4. QuadraticForms .................................................... 127 M5.4.1. QuadraticFormsandTheirTransformations .................... 127 M5.4.2. CanonicalandNormalRepresentations ofaQuadraticForm ...... 129 M5.5. LinearSpaces ....................................................... 131 M5.5.1. ConceptofaLinearSpace. ItsBasisandDimension ............. 131 M5.5.2. SubspacesofLinearSpaces ................................... 134 M5.5.3. CoordinateTransformationsCorrespondingtoBasisTransformations inaLinearSpace ............................................ 135 M5.5.4. EuclideanSpace ............................................ 136 Bibliography forChapterM5 ............................................... 138 M6. LimitsandDerivatives ............................................... 139 M6.1. BasicConceptsofMathematical Analysis .............................. 139 M6.1.1. NumberSets. FunctionsofRealVariable ....................... 139 M6.1.2. LimitofaSequence ......................................... 141 M6.1.3. LimitofaFunction. Asymptotes .............................. 144 M6.1.4. InfinitelySmallandInfinitelyLargeFunctions .................. 146 M6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind ....................................................... 147 M6.1.6. ConvexandConcaveFunctions ............................... 148 M6.1.7. Convergence ofFunctions .................................... 149 M6.2. DifferentialCalculusforFunctionsofaSingleVariable .................. 150 M6.2.1. Derivative and Differential: Their Geometrical and Physical Meaning ................................................... 150 M6.2.2. TableofDerivativesandDifferentiation Rules ................... 151 M6.2.3. TheoremsaboutDifferentiable Functions. L’HospitalRule ........ 153 M6.2.4. Higher Order DerivativesandDifferentials. Taylor’sFormula ..... 154 M6.2.5. ExtremalPoints. PointsofInflection ........................... 156 M6.2.6. QualitativeAnalysisofFunctions andConstruction ofGraphs ..... 158 M6.2.7. Approximate Solution of Equations (Root Finding Algorithms for Continuous Functions) ....................................... 160 K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pagevii K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pageviii viii CONTENTS M6.3. FunctionsofSeveralVariables. PartialDerivatives ....................... 162 M6.3.1. PointSets. Functions. LimitsandContinuity ................... 162 M6.3.2. Differentiation ofFunctions ofSeveralVariables ................ 163 M6.3.3. Directional Derivative. Gradient. Geometrical Applications ....... 166 M6.3.4. ExtremalPointsofFunctionsofSeveralVariables ............... 167 M6.3.5. Differential OperatorsoftheFieldTheory ...................... 170 Bibliography forChapterM6 ............................................... 170 M7. Integrals ............................................................ 171 M7.1. IndefiniteIntegral ................................................... 171 M7.1.1. Antiderivative. IndefiniteIntegralandItsProperties .............. 171 M7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral. ExamplesofIntegration ...................................... 172 M7.1.3. Integration ofRationalFunctions .............................. 174 M7.1.4. Integration ofIrrational Functions ............................. 177 M7.1.5. Integration ofExponential andTrigonometric Functions .......... 179 M7.1.6. Integration ofPolynomialsMultiplied byElementaryFunctions ... 181 M7.2. DefiniteIntegral ..................................................... 183 M7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical MeaningoftheDefiniteIntegral ............................... 183 M7.2.2. PropertiesofDefiniteIntegrals andUsefulFormulas ............. 184 M7.2.3. AsymptoticFormulasfortheCalculation ofIntegrals ............ 185 M7.2.4. Mean Value Theorems. Properties of Integrals in Terms of Inequalities ................................................. 187 M7.2.5. GeometricandPhysicalApplications oftheDefiniteIntegral ...... 188 M7.2.6. ImproperIntegralswithInfiniteIntegration Limits ............... 190 M7.2.7. ImproperIntegralsofUnbounded Functions .................... 193 M7.2.8. Approximate (Numerical) Methods for Computation of Definite Integrals ................................................... 194 M7.3. DoubleandTripleIntegrals ........................................... 195 M7.3.1. DefinitionandProperties oftheDoubleIntegral ................. 195 M7.3.2. Computation oftheDoubleIntegral ............................ 197 M7.3.3. GeometricandPhysicalApplications oftheDoubleIntegral ...... 199 M7.3.4. DefinitionandProperties oftheTripleIntegral .................. 200 M7.3.5. Computation of the Triple Integral. Some Applications. Iterated IntegralsandAsymptoticFormulas ............................ 201 M7.4. LineandSurfaceIntegrals ............................................ 203 M7.4.1. LineIntegraloftheFirstKind ................................. 203 M7.4.2. LineIntegraloftheSecondKind .............................. 205 M7.4.3. SurfaceIntegral oftheFirstKind .............................. 206 M7.4.4. SurfaceIntegral oftheSecondKind ........................... 207 M7.4.5. IntegralFormulasofVectorCalculus ........................... 208 Bibliography forChapterM7 ............................................... 209 M8. Series ............................................................... 211 M8.1. NumericalSeriesandInfiniteProducts ................................. 211 M8.1.1. Convergent Numerical Series and Their Properties. Cauchy’s Criterion ................................................... 211 M8.1.2. Convergence CriteriaforSerieswithPositive(Nonnegative) Terms 212 M8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional Convergence ..................................... 213 K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pageviii K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pageix CONTENTS ix M8.1.4. Multiplication ofSeries. SomeInequalities ..................... 215 M8.2. FunctionSeries ..................................................... 215 M8.2.1. PointwiseandUniformConvergence ofFunctionSeries .......... 215 M8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent Series ........................................... 216 M8.3. PowerSeries ........................................................ 218 M8.3.1. RadiusofConvergence ofPowerSeries. Properties ofPowerSeries 218 M8.3.2. TaylorandMaclaurinPowerSeries ............................ 220 M8.3.3. Operations with Power Series. Summation Formulas for Power Series ...................................................... 221 M8.4. FourierSeries ....................................................... 223 M8.4.1. Representation of 2π Periodic Functions by Fourier Series. Main Results ..................................................... 223 M8.4.2. Fourier Expansions of Periodic, Nonperiodic, Even, and Odd Functions .................................................. 225 M8.4.3. CriteriaofUniformandMean SquareConvergenceofFourierSeries 226 Bibliography forChapterM8 ............................................... 227 M9. FunctionsofComplexVariable ....................................... 229 M9.1. ComplexNumbers .................................................. 229 M9.1.1. Definition of a Complex Number. Arithmetic Operations with ComplexNumbers .......................................... 229 M9.1.2. Trigonometric FormofComplexNumbers. PowersandRadicals .. 230 M9.2. FunctionsofComplexVariables ....................................... 231 M9.2.1. BasicConcepts. DifferentiationofaFunctionofaComplexVariable 231 M9.2.2. Integration ofFunctionsofComplexVariables .................. 238 M9.2.3. TaylorandLaurentSeries .................................... 241 M9.2.4. ZerosandIsolatedSingularities ofAnalyticFunctions ............ 243 M9.2.5. Residues. Calculation ofDefiniteIntegrals ..................... 245 Bibliography forChapterM9 ............................................... 249 M10. IntegralTransforms ................................................ 251 M10.1. GeneralFormofIntegralTransforms. Inversion Formulas ............... 251 M10.2. LaplaceTransform ................................................. 251 M10.2.1. LaplaceTransformandtheInverseLaplaceTransform ......... 251 M10.2.2. MainPropertiesoftheLaplaceTransform. Inversion Formulasfor SomeFunctions ........................................... 252 M10.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series ......................................... 255 M10.3. VariousFormsoftheFourierTransform ............................... 255 M10.3.1. FourierTransformandtheInverseFourierTransform .......... 255 M10.3.2. FourierCosineandSineTransforms ......................... 257 M10.4. MellinTransformandOtherTransforms ............................... 258 M10.4.1. MellinTransform andtheInversion Formula .................. 258 M10.4.2. Main Properties of the Mellin Transform. Relation Among the Mellin,Laplace,andFourierTransforms ..................... 259 M10.4.3. SummaryTableofIntegralTransforms ....................... 259 Bibliography forChapterM10 .............................................. 260 K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pageix