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Accurate Computation of Mathieu Functions Synthesis Lectures on Computational Electromagnetics Editor ConstatineA.Balanis,ArizonaStateUniversity SynthesisLecturesonComputationalElectromagneticswillpublish50-to100-pagepublicationson topicsthatincludeadvancedandstate-of-the-artmethodsformodelingcomplexandpractical electromagneticboundaryvalueproblems.Eachlecturedevelops,inaunifiedmanner,themethod basedonMaxwell’sequationsalongwiththeboundaryconditionsandotherauxiliaryrelations, extendsunderlyingconceptsneededforsequentialmaterial,andprogressestomoreadvanced techniquesandmodeling.Computersoftware,whenappropriateandavailable,isincludedfor computation,visualizationanddesign.eauthorsselectedtowritethelecturesareleadingexperts onthesubjectthathaveextensivebackgroundinthetheory,numericaltechniques,modeling, computationsandsoftwaredevelopment. eseriesisdesignedto: • Developcomputationalmethodstosolvecomplexandpracticalelectromagnetic boundary-valueproblemsofthe21stcentury. • Meetthedemandsofaneweraininformationdeliveryforengineers,scientists,technologists andengineeringmanagersinthefieldsofwirelesscommunication,radiation,propagation, communication,navigation,radar,RFsystems,remotesensing,andbiotechnologywho requireabetterunderstandingandapplicationoftheanalytical,numericalandcomputational methodsforelectromagnetics. AccurateComputationofMathieuFunctions MalcolmM.BibbyandAndrewF.Peterson 2013 SelectedAsymptoticMethodswithApplicationstoElectromagneticsandAntennas GeorgeFikioris,IoannisTastsoglou,andOdysseasN.Bakas 2013 Double-GridFinite-DifferenceFrequency-Domain(DG-FDFD)MethodforScattering fromChiralObjects ErdoganAlkan,VeyselDemir,AtefElsherbeni,andErcumentArvas 2013 iii MultiresolutionFrequencyDomainTechniqueforElectromagnetics MesutGokten,AtefElsherbeni,andErcumentArvas 2012 ScatteringAnalysisofPeriodicStructuresUsingFinite-DifferenceTime-DomainMethod KhaledElMahgoub,FanYang,andAtefElsherbeni 2012 IntroductiontotheFinite-DifferenceTime-Domain(FDTD)Methodfor Electromagnetics StephenD.Gedney 2011 AnalysisandDesignofSubstrateIntegratedWaveguideUsingEfficient2DHybrid Method XuanHuiWuandAhmedA.Kishk 2010 AnIntroductiontotheLocally-CorrectedNyströmMethod AndrewF.PetersonandMalcolmM.Bibby 2009 TransientSignalsonTransmissionLines:AnIntroductiontoNon-IdealEffectsandSignal IntegrityIssuesinElectricalSystems AndrewF.PetersonandGregoryD.Durgin 2008 ReductionofaShip’sMagneticFieldSignatures JohnJ.Holmes 2008 IntegralEquationMethodsforElectromagneticandElasticWaves WengChoChew,MeiSongTong,andBinHu 2008 ModernEMCAnalysisTechniquesVolumeII:ModelsandApplications NikolaosV.KantartzisandeodorosD.Tsiboukis 2008 ModernEMCAnalysisTechniquesVolumeI:Time-DomainComputationalSchemes NikolaosV.KantartzisandeodorosD.Tsiboukis 2008 ParticleSwarmOptimization:APhysics-BasedApproach SaidM.MikkiandAhmedA.Kishk 2008 iv ree-DimensionalIntegrationandModeling:ARevolutioninRFandWirelessPackaging Jong-HoonLeeandManosM.Tentzeris 2007 ElectromagneticScatteringUsingtheIterativeMultiregionTechnique MohamedH.AlSharkawy,VeyselDemir,andAtefZ.Elsherbeni 2007 ElectromagneticsandAntennaOptimizationUsingTaguchi’sMethod Wei-ChungWeng,FanYang,andAtefElsherbeni 2007 FundamentalsofElectromagnetics1:InternalBehaviorofLumpedElements DavidVoltmer 2007 FundamentalsofElectromagnetics2:QuasistaticsandWaves DavidVoltmer 2007 ModelingaShip’sFerromagneticSignatures JohnJ.Holmes 2007 Mellin-TransformMethodforIntegralEvaluation:IntroductionandApplicationsto Electromagnetics GeorgeFikioris 2007 PerfectlyMatchedLayer(PML)forComputationalElectromagnetics Jean-PierreBérenger 2007 AdaptiveMeshRefinementforTime-DomainNumericalElectromagnetics CostasD.Sarris 2006 FrequencyDomainHybridFiniteElementMethodsforElectromagnetics JohnL.Volakis,KubilaySertel,andBrianC.Usner 2006 ExploitationofAShip’sMagneticFieldSignatures JohnJ.Holmes 2006 SupportVectorMachinesforAntennaArrayProcessingandElectromagnetics ManelMartínez-RamónandChristosChristodoulou 2006 v eTransmission-LineModeling(TLM)MethodinElectromagnetics ChristosChristopoulos 2006 ComputationalElectronics DragicaVasileskaandStephenM.Goodnick 2006 HigherOrderFDTDSchemesforWaveguideandAntennaStructures NikolaosV.KantartzisandeodorosD.Tsiboukis 2006 IntroductiontotheFiniteElementMethodinElectromagnetics AnastasisC.Polycarpou 2006 MRTD(MultiResolutionTimeDomain)MethodinElectromagnetics NathanBushyagerandManosM.Tentzeris 2006 MappedVectorBasisFunctionsforElectromagneticIntegralEquations AndrewF.Peterson 2006 Copyright©2014byMorgan&Claypool Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin anyformorbyanymeans—electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotations inprintedreviews,withoutthepriorpermissionofthepublisher. AccurateComputationofMathieuFunctions MalcolmM.BibbyandAndrewF.Peterson www.morganclaypool.com ISBN:9781627050852 paperback ISBN:9781627050869 ebook DOI10.2200/S00526ED1V01Y201307CEM032 APublicationintheMorgan&ClaypoolPublishersseries SYNTHESISLECTURESONCOMPUTATIONALELECTROMAGNETICS Lecture#32 SeriesEditor:ConstatineA.Balanis,ArizonaStateUniversity SeriesISSN SynthesisLecturesonComputationalElectromagnetics Print1932-1252 Electronic1932-1716 Accurate Computation of Mathieu Functions Malcolm M. Bibby and Andrew F. Peterson GeorgiaInstituteofTechnology SYNTHESISLECTURESONCOMPUTATIONALELECTROMAGNETICS #32 M &C Morgan &cLaypool publishers ABSTRACT islecturepresentsamodernapproachforthecomputationofMathieufunctions.esefunc- tionsfindapplicationinboundaryvalueanalysissuchaselectromagneticscatteringfromelliptic cylindersandflatstrips,aswellastheanalogousacousticandopticalproblems,andmanyother applicationsinscienceandengineering.eauthorsreviewthetraditionalapproachusedforthese functions, show its limitations, and provide an alternative “tuned” approach enabling improved accuracy and convergence. e performance of this approach is investigated for a wide range of parameters and machine precision. Examples from electromagnetic scattering are provided for illustrationandtoshowtheconvergenceofthetypicalseriesthatemployMathieufunctionsfor boundaryvalueanalysis. KEYWORDS ellipticcylinder,electromagneticscattering,boundaryvalueproblems