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SOME PROPERTIES OF HOLOMORPHIC CLIFFORDIAN FUNCTIONS IN COMPLEX CLIFFORD ANALYSIS

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SOME PROPERTIES OF HOLOMORPHIC CLIFFORDIAN FUNCTIONS IN COMPLEX CLIFFORD ANALYSISOF,IN,in,SOME,some,Some

    SOME PROPERTIES OF

    HOLOMORPHIC CLIFFORDIAN

    FUNCTIONS IN COMPLEX CLIFFORD

    ANALYSIS

    Availableonlineatwww.sciencedirect.com

    ScienceDirectbcien

    ActaMathematicaScientia2010,30B(3):747-768

    数学物理

    http://actams.wipm.aC.an

    SoMEPR0PERTIES0FHoL0MoRPHIC

    CLIFFoRDIANFUNCT10NSINCoMPLEX

    CLIFF0RDANAISIS

    KuMinDuJinyuan2(杜金元)WangDaoshun(王道顺)

    1.TsinghuaNationalLaboratoryforInformationScienceandTechnology, DepartmentofComputerScienceandTechnology,TsinghuaUniversity,Beijing10008~,China

    2.SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,China E-mail:kumin0844@126.com;jydu@whu.edu.cn;daoshun@mail.tsinghua.edu.cn AbstractInthisarticle,wemainlydevelopthefoundationofanewfunctiontheory ofseveralcomplexvariableswithvaluesinacomplexCliffordalgebradefinedonsome subdomainsofC+1.

    SO-CalledcomplexholomorphicCliffordianfunctions.Wledefinethe complexholomorphicCliffordianfunctions,studypolynomialandsingularsolutionsofthe equationDAmf=0,obtaintheintegralrepresentationformulaforthecomplexholo- morphicClifrordianfunctionswithvaluesinacomplexClifrordalgebradefinedonsome submanifoldsofC"+1.deducetheTaylorexpansionandtheLaurentexpansionforthem

andproveaninvarianceunderanactionofLiegroupforthem.

    KeywordsComplexCliffordalgebra;holomorphicCliffordianfunctions;Taylorexpan

    sion;Laurentexpansion;invaxiance

    2000MRSubjectClassification22E30;30G35;31C10;32A10

    1Introduction

    Theclassicaltheoryofanalyticfunctionsofonecomplexvariablewasgeneralizedinthree directions.Thefirstdirectionisthetheoryofholomorphicfunctionsofseveralcomplexvari

    ables:inthiscasethecomplexnumberfieldCiskeptandthesystemofpartialdifferential operators,whereJ=1,2,,n,isconsidered.Theseconddirectionisthetheoryofregular functions:inthiscasetherealCliffordalgebraandthegeneralizedCauchy--Riemannoper-? atorD=?e,whereeiistheorthogonalbasis,wereconsidered.In1,2,R.Delanghe,

    F.Sommen,F.Brackxintroducedthemonogenic(regular)functionswithvaluesinarealClifford

    algebradefinedonanonemptysubsetofRandobtainedmanyimportantmonogenic(regular) functiontheoreticresults,suchastheCauchyintegralformula,theTaylorexpansion,andthe LaurentexpansionandSOon,whicharetheextensionsofthewel1.knownclassicaltheorems. ThethirddirectioniSthetheoryofcomplexregularfunctions:inthiscasethecomplexClifford ReceivedJuly10,2007;revisedMarch25,2008.SupportedbyNNSFofChina(6087349,10871150),863

    ProjectofChina(2008AA01Z419),RFDPofHigherEducation(20060486001),andPost

    DoctorFoundationof

    China(20090460316)

    静静

    一劳

    帮孽

    

    748ACTAMATHEMATICASCIENTIAVo1.30Ser.B

    algebraandtheoperatorD?ci0wereconsidered.In[as],JohnRyanintroducedthe

    =U

    complexleft(right)regularfunctionswithvaluesinacomplexCliffordalgebradefinedonsome

    subdomainsofC+andfor礼三1mod2establishedtheintegra1representationforC(1).

    functionsdefinedonsomesubdomainsofC

    1.theanalogousCauchy'stheorem.theCauchy

    integralformula,andtheTaylorexpansionforthecomplexleft(right)regularfunctionsdefined

    onsomesubmanifoldsofC"+1.In481hefurtherdiscussedthesingularities,theLaurentex- pansions,andtheRungeapproximationtheoremforthecomplexleft(right)regularfunctions.

    Alltheseresultsarefurtherextensionsofthewell

    knownclassicaltheorems.RJelatedresults

    oncomplexCliffordalgebracanbeseenin[1,9l2].Moreover,thereisanimportantfact

    thatinthetheoryofonecomplexvariabletheidentity(that,x)anditspowers(that,x)are holomorphicandplayverycrucialrolewhiletheyarenotmonogenicandregularinCliff0rd analysisandcomplexClifrordanalysis.Onthebasisofthis,in[13underthesettingofthe

    realCliff0rdalgebra.GuyLavilleandIvanRamadanofffollowedadifferentpathtointroduce holomorphicClifrordianfunctionswithvaluesinrealClifrordalgebradefinedonanonempty subsetofR,whicharetheextensionsofmonogenicfregular1functions,anddiscussedsome propertiesforthem.Morerelatedresultscanbealsofoundin1418].

    Inthisarticle,onthebasisoftheideascontainedin{13l5l,ourmotivationistothink

    oftheseveralvariableszanditspowersz"underthesettingofcomplexClifrordalgebra.We shallstudythecomplexholomorphicCliffordianfunctionswithvaluesinacomplexClifford algebradefinedonsomesubdomainsofC+1.whichistheextensionofholomorphicCliffcIrdian

    functionsinf13,16,18]andreferencestherein.BecausethestructureofthespaceC+is morecomplicatedthanthatofthespaceR"+1.wewillgetdifierentresultsfromthoseofthe holomorphicCliffordianfunctionsinI13,161.Inthefollowingsections,wewilldiscusssome propertiesofthecomplexholomorphicCliffordianfunctions.InSection3.wewillintroduce thenotionofcomplexholomorphicCliffordianfunctionstakingvaluesinthecomplexCliffor

d

    algebram+1=CR0,

    2m+ldefinedonsomesubdomainsofC+,whereRo,

    2m+listhe

    realClif1ordalgebrawithaquadraticformofnegativesignature,thatis,DAmf=0,where D=?eioand?一?巷.InSection4,wewilldiscusssomesimplepropertiesofthe

    complexholomorphicCliffordianfunctions.InSection5,wewillgiveanimportantexampleof

    thecomplexholomorphicCliffordianfunctions,whichplaysaverysignificantroletodeducethe

    CauchykernelofthecomplexholomorphicClifrordianfunctionsinthelatersections.Then, inSections7l_10,onsomesubmanifoldsofC+,wewillstudytheintegralrepresentation,

    polynomialandsingularsolutionsoftheequationD?.,=0,andobtaintheTaylorexpansion

    andtheLaurentexpansionforthecomplexholomorphicCliffordianfunctions.Inthelast section,wewillproveaninvarianceunderanactionofaLiegroupforthecomplexholomorphic

    Cliffordianfunctions.TomakethearticleselgcontainedinSection2.aquickintroductionis giventothenotionsoftherea1andcomplexClifrordalgebra.Theresultspresentedinthis articlegiveimprovedinformationoverthecorrespondingresultsof13,3,16,18,4,8,2,11.

    2Notations

    Inthissection,weshallintroduceprimarilytherealandcomplexCliffordalgebra.More detailscanbefoundin[1,1923,15,9,10,24].

    No.3Kuetal:SOMEPROPERTIESOFHOLOMORPHICCLIFFORDIANFUNCTIONS

    749

    LetCm+1bethecomplexCliffordalgebra,thatis,C2m+1:C~Ro

    ,

    2m+1,whereR0,

    2m+1is

    therealCliffordalgebraofarealvectorspaceVofdimension2m+l,providedwithaquadratic formofnegativesignature,mEN.Thisalgebrahasidentitye0=1andbasiselements

0,el,e2,.,e2m+l,,e1'''e2m+l,

    denotedby{eAIAEPN},P?={(1,,ik)lli1<i2m+1andforeachJ,kwith 1J,k2m+1andJ?k,onehasejek=ekejande2=o.Ageneralbasiselementsof m+1isdenotedby1J,withlr2rn+1andJ1<<J.

    DenotebySthesetofthescalarsinC2m+1,whichcanbeidentifiedtoC.ADoint

    z=(zo,z1,,Z2m+1)ofC+2couldbealsoconsideredasanelementofS0V,where

    denotethesetof1-vectorsinC2m+1

    ,namely,z=?eizi.Soz,beinginS0V,isinthe

    algebraC2m+1.WeactbyoneoftheprincipalinvolutionsinC2 m+1,

    =

    ?A?=?A,VZ?m+lAEPNAEPN

    w_hereA=(1)eAwithsdenotingthenumber.fe1emeI1ts.f(see[1Jpp.85s6)

    Especially,

    2m+1

    z=e0

    ?e,Vz?s.i=1

    ThebarofzEm+1isdefinedby

    =

    ?eAZA,?Cr.A?PN

    Theinnerproductonthealgebram+lisdefinedby (a,b)=[00,Va,b?C21

    wherea]0denotesthescalarpartofa.Especially, Izl=(?).,?.\AEPN/

    whichcoincideswiththecaseofonecomplexvariable. Itisremarkablethatzz::++++1,名?0V.Thisisdifferentfrom thatofrealCliffordanalysis.

    Wecalltheset

    =

    {EC+::0)

    asingularconein.WeobservethatallvectorsintheopensetCr.m+.\areinvertible inthealgebra.TheinverseofavectorinC+\sbiszz)一?c2m+2\S0.Similar1

    ^s={?C+.:(?)()=0}.

    LetQbeaspecialsubmanifoldofC+.

    Functionsf(z)definedinQandwithvaluesin

    C2~+iwillbeconsidered,thatis,f:Q_.?m+1,moreprecisely,,(z)=?SA(z)eA,z:

    (z.,-,z2+1)?Qand,()istheeAc.mp.neI1t.f,().0bvi.u81yAEP(N)isc.mp1eX-

    valuedinQ,whichiscalledtheeAcomponentfunctionof,().Wheneverapropertysuch ascontinuity,holomorphic,andsoonisascribedto,(),itisclearthatinfactallofthe

    componentfunctions(z)possesthecitedproperty.So,f?c(+1)isveryclear,wherethe

    symbolf?C(+)isabbreviatedfromf(z)?c(2m+1(Q,G2m+1).

    750ACTAMATHEMATICASCIENTIAV

    o1.30Ser.B

    3GeneralDefinitions

    LetUbeadomainofS0V.Weareinterestedinthefunctionsf:U__+m+1.Particu. 1arly,onemightconsideronlythefunctions,:__?SOV.Itiswellknownthatthefollowing operator,namedgeneralizedCauchyRiemannoperatorin

    al,liesatthebasisofthetheoryof

    complexleftregular(mon0genic)functions

    a

    i=0

    (1)

    Aholomorphicfunctionf:Um+1issaidtobecomplex(1eft)regularfunctioninUif andonlyif

    Df(z)=0

    foreachz?ItisimportanttonotethattheoperatorDpossessesaconjugateoperator D:D:e.0

    OZo

    Aholomorphicfunctionf:U__?m+1issaidtobecomplexharmonicinUifandonlyif

foreachz?U

    Inthefollowing,weshalldenotetheoperator

    (2)

    byA.Then,(3)isequivalenttoaf(z)=0,whereDD=DD=A.

    Generally,aholomorphicfunction/issaidtobecomplexpolyharmonicofordermifand onlyif

    N. A/(z)=0,m?

    Remark1Allcomplex(1eft)regularfunctionsarecomplexharmonicfunctions.Con. versely,if/:U-_?C2m+liscomplexharmonic,then,Df(z)iscomplex(1eft)regular. Now,letUSstatethefollowing

    Definition1Aholomorphicfunctionf:U}C2m+1issaidtobeCOmplex(1eft1

    holomorphicc1irdianinUifandonlyifDAmf(z1=0,foreachofU.

    Remark2ThesetofthecomplexholomorphicCliffordianfunctionsiswiderthanthe setofcomplexregularfunctions.Infact,allcomplexregularfunctionsarecomplexholomorphic

    Cliffordian,buttheconverseisfalse.Because,ifof=0,then,DAmf=AmDf:0fAmisa scalaroperator).Thesimplestexampleofafunction,whichiscomplexholomorphicCliffordian

    butnotcomplexregular,istheidentity}Z,forwhich

    .z:

    rnq-1

    ei

    t~/2mq-1

    e

    )

    

    卅?

    2

    O

ll

    ?

    一亏—cc)

    ?

    NO.3竺竺!竺兰!!ERTIESOFHOLOMORPHICCLIFFORDIANFUNCTIONS751 andclearly

    DA:?Dz=A(?2m)=0

    we.will

    rprovethatallentirep.wers.f2arec.mplexhol.m.rphicCliffordian,whilenoneotthem1Scom

    plexregular.

    Remark3,isc.mplexh.1.morphicCliffordianifand.nlyifAmfisc.

    mpiexregu1ar.

    4SimpleProperties

    ..

    sen'wewillbeginwithSection3ofthecomplexholomhicCliff

    ordianfunc.

    .anaoDtainsomesimplepropertiesofthem. Pr0osition1AllcomponentsofacomplexholomorphicCliffordianfunctionarecom.

    ..

    'ri??IfDA,()=0,then,actingDonbothsidesoftheequa1ityjone

    .binD?,(z)=0?Thati8,?+,=0.Hence,

    theresuItfollowsbecauseAm+li

    sa

    +1

    P

    .

    r

    1

    oposition2

    .

f

    ,

    f

    ,

    (z)isapolyharm.nicfuncti.noforderm+l,thatis,Am+,():0,fe n,theunctionD,()iscomplexholomorphicCliffordian.

    

    A+,(2):DDA,(z)=DAm(J[),)):0 Remark4Thispr.pertypr.Videsanimp.rtant

    wayt.c.n8tructthecompleXh.J.m.r

    phicCliffordianfunctions. Wewillu8ethislater.

    ………一

    

    P.p.ii.n3,)iscomplexh.j.m.rphicCliffordianifand.nIyif,(2)

    ,z)arebothpolyharmonicoforderm+1.

    一一

    Proof

    thati8

    Inotherwords

    ?c=2m+c=[(),+)]

    (et,()+oif())

    =

    2m+

    (e,+b~zf(z)+z,)

    =

    2m+

    (-f(z)+ei0,)+2m+lz,())+z)=

    2Df(z)+zAf(z),

    ?(,()):2Df(z)+zAf(z) 2Dr(z)=A(zf(z))一?,(),

    z?,():?(t,())2Df(z)

    旦眺

    ?

    lJ

    752ACTAMATHEMATICASCIENTIAVb1.30Ser.B

    Therefore,

    So.

    2DAf(z):?(?,())一名(?(?.(z)))

    =

    A(zAf(z))一?f(z)

    =

    A(~zf(z)2Df(z))zAf(z)

    =

    A(z,())2DAf(z)zA.f(z)

    Byrecursion,oneobtains 4DAf(z)=A.(,(z))zA.f(z)

    2+1)DAPf(z)=Ap+(,())zAp+lf(z)

    foreachP?N.

    LetP=m.onededuces

    2(m+1)DAf(z)=Arn+l(.,())zAm+ly(z) ""If.,(),zf(z)arebothpolyharmonicoforderm~l,then,wehaveDAf(z)=0

    ""Iff(z)iscomplexholomorphicCliffordian,then,DAf(z)=0.

    Applying(4),wehave

    Ara+l(,(z))=zAm+lf(z) Incontrast,zAm+lf(z)=zD(DA,())=

    polyharmonicoforderm+1. Therefore,using(5),weobtain Itfollowsthat

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