Analytic function.docx
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Analytic function.docx
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Analyticfunction
Analyticfunction
2010MathematicsSubjectClassification:
Primary:
30-XXSecondary:
32-XX[MSN][ZBL]
Afunctionthatcanbelocallyrepresentedbypowerseries.Suchfunctionsareusuallydividedintotwoimportantclasses:
therealanalyticfunctionsandthecomplexanalyticfunctions,whicharecommonlycalledholomorphicfunctions.Thisentryconcernsthelatter:
thereaderisreferredtoRealanalyticfunctionforthefirstclass.
Theexceptionalimportanceoftheclassofanalyticfunctionsisduetothefollowingreasons.First,theclassissufficientlylarge;itincludesthemajorityoffunctionswhichareencounteredintheprincipalproblemsofmathematicsanditsapplicationstoscienceandtechnology.Secondly,theclassofanalyticfunctionsisclosedwithrespecttothefundamentaloperationsofarithmetic,algebraandanalysis.Finally,animportantpropertyofananalyticfunctionisitsuniqueness:
Eachanalyticfunctionisan"organicallyconnectedwhole",whichrepresentsa"unique"functionthroughoutitsnaturaldomainofexistence.Thisproperty,whichinthe18thcenturywasconsideredasinseparablefromtheverynotionofafunction,becameoffundamentalsignificanceafterafunctionhadcometoberegarded,inthefirsthalfofthe19thcentury,asanarbitrarycorrespondence.Thetheoryofanalyticfunctionsoriginatedinthe19thcentury,mainlyduetotheworkofA.L.Cauchy,B.RiemannandK.Weierstrass.The"transitiontothecomplexdomain"hadadecisiveeffectonthistheory.Thetheoryofanalyticfunctionswasconstructedasthetheoryoffunctionsofacomplexvariable;atpresent(the1970's)thetheoryofanalyticfunctionsformsthemainsubjectofthegeneraltheoryoffunctionsofacomplexvariable.
Contents
[hide]
1Analyticfunctionsofonecomplexvariable
1.1Complexdifferentiability
1.2Cauchy-Riemannequations
1.3Powerseriesexpansion
1.4Cauchyintegralformula
1.5Uniquecontinuation
1.6SingularitiesandLaurentseries
1.6.1Residues
1.6.2Meromorphicfunctions
1.7Entirefunctions
1.7.1WeierstrassandMittag-LefflerTheorems
1.8ConformalityandRiemann'smappingtheorem
1.9Harmonicfunctions
1.10Analyticityonnon-opendomains
1.11AnalyticcontinuationandRiemannsurfaces
2Analyticfunctionsofseveralcomplexvariables.
2.1Complexdifferentiability
2.1.1Cauchy-Riemannsystem
2.1.2Hartogs'theorem
2.2Powerseries
2.3Weierstrasspreparationtheorem
2.3.1Analyticvarieties
2.4Cauchy'sintegraltheorem
2.5Integralrepresentations
2.5.1Martinelli-Bochnerformula
2.5.2Lerayformula
2.6Hartog'sextensiontheorem
2.7Multi-dimensionalresidues
2.8Domainsofholomorphy
2.8.1Plurisubharmonicfunctions
2.8.2Envelopeofholomorphy
2.9Steinmanifolds
2.10Cousinproblems
▪3Furtherdevelopments
▪4References
Analyticfunctionsofonecomplexvariable
Therearedifferentapproachestotheconceptofanalyticity.Onedefinition,whichwasoriginallyproposedbyCauchy,andwasconsiderablyadvancedbyRiemann,isbasedonastructuralpropertyofthefunction—theexistenceofaderivativewithrespecttothecomplexvariable,i.e.itscomplexdifferentiability.Thisapproachiscloselyconnectedwithgeometricideas.Anotherapproach,whichwassystematicallydevelopedbyWeierstrass,isbasedonthepossibilityofrepresentingfunctionsbypowerseries;itisthusconnectedwiththeanalyticapparatusbymeansofwhichafunctioncanbeexpressed.Abasicfactofthetheoryofanalyticfunctionsistheidentityofthecorrespondingclassesoffunctionsinanarbitrarydomainofthecomplexplane.
Complexdifferentiability
LetDbeadomain(thatis,anopenset)inthecomplexplaneC.Iftoeachpointz∈Dtherehasbeenassignedsomecomplexnumberw,thenonesaysthatonDa(single-valued)functionfofthecomplexvariablezhasbeendefinedandonewrites:
w=f(z),z∈D(orf:
D→C).Thefunctionw=f(z)=f(x+iy)mayberegardedasacomplexfunctionoftworealvariablesxandy,definedinthedomainD⊂R2(whereR2istheEuclideanplane).Todefinesuchafunctionistantamounttodefiningtworealfunctions
u=ϕ(x,y),v=ψ(x,y),(x,y)∈D(w=u+iv).
Havingfixedapointz∈D,onegivesztheincrementΔz=Δx+iΔy(suchthatz+Δz∈D)andconsidersthecorrespondingincrementofthefunctionf:
Δf(z)=f(z+Δz)−f(z).
IfΔf(z)=AΔz+o(Δz)
asΔz→0,orinotherwords,iflimΔz→0Δf(z)Δz=A
exists,thefunctionfissaidtobecomplex-differentiableatz;A=f′(z)isthecomplexderivativeoffatz,andAΔz=f′(z)dz=df(z)
isitscomplexdifferentialatthatpoint.Afunctionfwhichiscomplex-differentiableateverypointofDiscalledholomorphicinthedomainD.
Cauchy-Riemannequations
Onemaycomparetheconceptsofdifferentiabilityoff,consideredasafunctionoftworealvariablesvariables,anditscomplexdifferentiability.Intheformercasethedifferentialdf,whichisalinearmapfromR2toChastheform
∂f∂xdx+∂f∂ydy,
where∂f∂x=∂ϕ∂x+i∂ψ∂x,∂f∂y=∂ϕ∂y+i∂ψ∂y,
arethepartialderivativesoff.Passingfromtheindependentvariablesx,ytothevariablesz,z¯,whichmayformallybeconsideredasnewindependentvariables,relatedtotheoldonesbytheequationsz=x+iy,z¯=x−iy(fromthispointofview,thefunctionfmayalsobewrittenasf(z,z¯))andexpressingdxanddyintermsofdzanddz¯accordingtotheusualrulesofdifferentialcalculus,onecanwritedfinitscomplexform:
df=∂f∂zdz+∂f∂z¯dz¯
where
∂f∂z=12(∂f∂x−i∂f∂y),∂f∂z¯=12(∂f∂x+i∂f∂y),
arethe(formal)derivativesoffwithrespecttozandz¯,respectively.Itisseen,accordingly,thatfiscomplexdifferentiableifandonlyifitisdifferentiableinthesenseofR2anddfturnsouttobealinearmapfromC→C.Thisisthecaseifandonlyiftheequation∂f/∂z¯=0issatisfied,whichinexpandedformmaybewrittenas
∂ϕ∂x=∂ψ∂y∂ψ∂x=−∂ϕ∂y.
(1)
fisthenholomorphicinthedomainDifandonlyfisdifferentiableasareal-variablefunctionandtheequations
(1)(whicharecalledCauchy-Riemannequations)aresatisfiedatallpointofthedomain.Theseequationsoccurredalreadyinthe18thcenturyinJ.L.d'Alembert'sandL.Euler'sstudiesonfunctionsofacomplexvariable.
Powerseriesexpansion
Remarkably,withoutanyfurtherassumptionsthandifferentiabilityandusingjustthefactthattheidentities
(1)holdseverywhere,itispossibletoshowthatholomorphicfunctionsareextremelyregular.Inparticularthecomplexderivativef′canbeprovedtobeitselfanholomorphicfunction.Thisfact,appliedrecursively,impliesthatfisinfinitelydifferentiable,infactinfinitelycomplex-differentiable,andjustifiesthenotationf(n)(z0)forthen-thcomplexderivativeoffatthepointz0.Moreover,foreverypointz0initsdomainofdefinitionthereisaneighbourhoodUofthispointinwhichfmayberepresentedbyapowerseries:
f(z)=∑nan(z−z0)n∀z∈U
(2)
(whereweareusingtwoconventionswhichwillholdthroughtherestofthisentry:
00issettobe1andwhenwewrite
(2)weimplicitelyassumethattherighthandsideof
(2)convergesateverypointwheretheidentityholds).Itcanindeedbeshownthat
(2)istheTaylorseriesoffatthepointz0,namelythatan=f(n)(z0)n!
andhencethatthepowerseriestakesthewell-knownform
f(z)=∑nf(n)(z0)n!
(z−z0)n.(3)
Thus,theholomorphyofafunctionfinadomainDimpliesthatfisinfinitelydifferentiableatanypointinDandthatitsTaylorseriesconvergestoitinsomeneighbourhoodofthispoint.
Viceversa,ifthefunctionfiscomplexanalyticatz0,i.e.itcanbeexpandedina(complex)powerseriesintheneighborhoodUofapointz0(namelyiftheidentity
(2)holdsforsomesequenceofcomplexnumbers{an}),thenfiscomplex-differentiableeverywhereinUandindeeditscomplexderivativef′(z)equalsthepowerseriesobtainedbydifferentiatingthelefthandsideof
(2)termbyterm,namely
f′(z)=∑nnan(z−z0)n−1.
Inparticularthetwonotionsofholomorphyandcomplexanalyticityareequivalent.
Cauchyintegralformula
Oneothercharacteristicofananalyticfunctionisconnectedwiththenotionofpathintegration.Theintegralofafunctionf=ϕ+iψalongan(orientedrectifiable)arcγparametrizedbyz:
[α,β]→Cmaybedefinedbytheformula:
∫γf(z)dz:
=∫βαf(z(t))z′(t)dt
or(equivalently)bymeansofacurvilinearintegralofadifferentialform(seealsoIntegrationonmanifolds):
∫γf(z)dz:
=∫γ(ϕdx−ψdy)+i∫γ(ψdx+ϕdy).
AkeyresultinthetheoryofanalyticfunctionsisCauchy'sintegraltheorem:
IffisholomorphicinadomainDthen
∫γf(z)dz=0(4)
foranyclosedcurveγboundingadomaininsideD(henceforanyclosedcurvewhenDissimplyconnected).Theconverseresult,Morera'stheorem,isalsotrue:
Iff:
D→CiscontinuousonanopendomainDandif(4)holdsforanycurveγwhichboundsadomaininD,thenfisholomorphicinD.Inparticular,inasimply-connecteddomain,thoseandonlythosecontinuousfunctionsfareanalytic,whoseintegralalonganyclosedcurveγiszero(or,whichisthesamething,theintegralalonganycurveγconnectingtwoarbitrarypointspandqdoesdependonlyonthepointspandqthemselvesandnotontheshapeofthecurve).Thischaracterizationofanalyticfunctionsformsthebasisofmanyoftheirapplications.
Cauchy'sintegraltheoremyieldsCauchy'sintegralformula,whichexpressesthevaluesofananalyticfunctioninsideadomainintermsofitsvaluesontheboundary.Moreprecisely,ifDisanopendomainwhoseboundaryconsistsofafinitenumberofnon-intersectingrectifiablecurves(orientedpositivelywithrespecttoD)andf:
D→Cisholomorphic,then
f(z)=12πi∫∂Df(ζ)ζ−zdζ∀z∈D.(5)
Thisformulamakesitpossible,inparticular,toreducethestudyofmanyproblemsconnectedwithanalyticfunctionstothecorrespondingproblemsforave
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