Ellipsoidal harmonics: theory and applications

476 Pages·2012·4.155 MB·English

by  Dassios, George

ELLIPSOIDAL HARMONICS
TheoryandApplications
Thesphere,becauseofitshighsymmetry,iswhatmightbecalledaperfectshape.
Unfortunatelynatureisimperfectandmanyapparentlysphericalbodiesarebetter
representedbyanellipsoid.Consequentlyincalculationsaboutgravitational
potential,forexample,sphericalharmonicshavetobereplacedbythemuchmore
complexellipsoidalharmonics.Theirtheory,whichwasoriginatedinthenineteenth
century,couldonlybeseriouslyappliedwiththekindofcomputationalpowerthat
hasbecomeavailableinrecentyears.This,therefore,isthefirstbookcompletely
devotedtoellipsoidalharmonics.
Afteracompletepresentationofthetheory,appliedtopicsaredrawnfrom
geometry,physics,biosciences,andinverseproblems.Thebookcontainsclassical
resultsaswellasnewmaterial,includingellipsoidalbiharmonicfunctions,the
theoryofimagesinellipsoidalgeometry,geometricalcharacteristicsofsurface
perturbations,andvectorsurfaceellipsoidalharmonics,whichexhibitaninteresting
analyticalstructure.Extendedappendicesprovideeverythingoneneedstosolve
formallyboundaryvalueproblems.End-of-chapterproblemscomplementthe
theoryandtestthereader’sunderstanding.
Thebookservesasacomprehensivereferenceforappliedmathematicians,
physicists,engineers,andforanyonewhoneedstoknowthecurrentstateoftheart
inthisfascinatingsubject.Specificchapterscanserveasteachingmaterial.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercises
attheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
ENCYCLOPEDIAOFMATHEMATICSANDITSAPPLICATIONS
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
93 G.Gierzetal.ContinuousLatticesandDomains
94 S.R.FinchMathematicalConstants
95 Y.JabriTheMountainPassTheorem
96 G.GasperandM.RahmanBasicHypergeometricSeries,2ndedn
97 M.C.PedicchioandW.Tholen(eds.)CategoricalFoundations
98 M.E.H.IsmailClassicalandQuantumOrthogonalPolynomialsinOneVariable
99 T.MoraSolvingPolynomialEquationSystemsII
100 E.OlivieriandM.EuláliaVaresLargeDeviationsandMetastability
101 A.Kushner,V.LychaginandV.RubtsovContactGeometryandNonlinearDifferentialEquations
102 L.W.BeinekeandR.J.Wilson(eds.)withP.J.CameronTopicsinAlgebraicGraphTheory
103 O.J.StaffansWell-PosedLinearSystems
104 J.M.Lewis,S.LakshmivarahanandS.K.DhallDynamicDataAssimilation
105 M.LothaireAppliedCombinatoricsonWords
106 A.MarkoeAnalyticTomography
107 P.A.MartinMultipleScattering
108 R.A.BrualdiCombinatorialMatrixClasses
109 J.M.BorweinandJ.D.VanderwerffConvexFunctions
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmannetal.TheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLévyNoise
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributed
ParameterSystems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiricGeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopological
GraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModelling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,Computer
Science,andEngineering
135 V.BerthéandM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Kristály,V.D.RadulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,
Geometry,andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland
DifferentialEquations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
GabrielLamé(1795–1870)
FrenchengineerandMathematician

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Ellipsoidal Harmonics
Theory and Applications
GEORGE DASSIOS
UniversityofPatras,Greece
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521113090
(cid:2)c GeorgeDassios2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Dassios,G.(George)
Ellipsoidalharmonics:theoryandapplications/GeorgeDassios.
pagescm.–(Encyclopediaofmathematicsanditsapplications;146)
Includesbibliographicalreferencesandindex.
ISBN978-0-521-11309-0(Hardback)
1.Lamé’sfunctions. I.Title.
QA409.D37 2012
(cid:3)
515.53–dc23 2011051233
ISBN978-0-521-11309-0Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
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Contents
Prologue pagexi
1 Theellipsoidalsystemanditsgeometry 1
1.1 Confocalfamiliesofsecond-degreesurfaces 1
1.2 Ellipsoidalcoordinates 8
1.3 Analyticgeometryoftheellipsoidalsystem 13
1.4 Differentialgeometryoftheellipsoidalsystem 17
1.5 Sphero-conalandellipto-sphericalcoordinates 22
1.6 Theellipsoidasadyadic 29
1.7 Problems 35
2 Differentialoperatorsinellipsoidalgeometry 39
2.1 Thebasicoperatorsinellipsoidalform 39
2.2 EllipsoidalrepresentationsoftheLaplacian 42
2.3 ThethermometricparametersofLamé 43
2.4 SpectraldecompositionoftheLaplacian 45
2.5 Problems 48
3 Laméfunctions 49
3.1 TheLaméclasses 49
3.2 Laméfunctionsofclass K 51
3.3 Laméfunctionsofclasses L and M 54
3.4 Laméfunctionsofclass N 58
3.5 DiscussionontheLaméclasses 60
3.6 Laméfunctionsofthesecondkind 66
3.7 Problems 67
4 Ellipsoidalharmonics 70
4.1 Interiorellipsoidalharmonics 70
4.2 Harmonicsofdegreefour 73
4.3 Exteriorellipsoidalharmonics 77
4.4 Surfaceellipsoidalharmonics 78
viii Contents
4.5 Orthogonalityproperties 79
4.6 Problems 86
5 ThetheoryofNivenandCartesianharmonics 89
5.1 TherootsoftheLaméfunctions 89
5.2 ThetheoryofNivenharmonics 94
5.3 Thecharacteristicsystem 97
5.4 FromNivenbacktoLamé 102
5.5 TheKlein–Stieltjestheorem 104
5.6 Harmonicsofdegreefourrevisited 106
5.7 Problems 108
6 Integrationtechniques 109
6.1 Integralsoveranellipsoidalsurface 109
6.2 Thenormalizationconstants 113
6.3 Thenormalizationconstantsrevisited 117
6.4 Problems 121
7 Boundaryvalueproblemsinellipsoidalgeometry 124
7.1 Expansionofthefundamentalsolution 124
7.2 Eigensourcesandeigenpotentials 127
7.3 Theclosurerelation 129
7.4 Green’sfunctionanditsimagesystem 131
7.5 TheNeumannfunctionanditsimagesystem 143
7.6 Singularitiesofexteriorellipsoidalharmonics 156
7.7 Problems 160
8 Connectionbetweenharmonics 163
8.1 Geometricalreduction 163
8.2 Sphero-conalharmonics 169
8.3 Differentialformulaeforharmonicfunctions 173
8.4 Sphero-conalexpansionsofinteriorellipsoidalharmonics 180
8.5 Integralformulaeforharmonicfunctions 190
8.6 Sphero-conalexpansionsofexteriorellipsoidalharmonics 195
8.7 Problems 200
9 Theellipticfunctionsapproach 203
9.1 TheWeierstrassapproach 203
9.2 TheJacobiapproach 207
9.3 TheWeierstrass–Jacobiconnection 210
9.4 IntegralequationsforLaméfunctions 213
9.5 Integralrepresentationsforellipsoidalharmonics 219
9.6 Problems 224
10 Ellipsoidalbiharmonicfunctions 226
10.1 Eigensolutionsoftheellipsoidalbiharmonicequation 226