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1、ComplexAnalysis2002-2003cK.Houston20031ComplexFunctionsInthissectionwewilldefinewhatwemeanbyacomplexfunc-tion.Wewillthengeneralisethedefinitionsoftheexponential,sineandcosinefunctionsusingcomplexpowerseries.Todealwithcomplexpowerserieswedefinethenotionsofconver-gentandabsolutelyconvergent,andseehowto
2、usetheratiotestfromrealanalysistodetermineconvergenceandradiusofconvergenceforthesecomplexseries.Westartbydefiningdomainsinthecomplexplane.Thisrequirestheprelimarydefinition.Definition1.1Theε-neighbourhoodofacomplexnumberzisthesetofcom-plexnumbers{w∈C:
3、z−w
4、<ε}whereεispositivenumber.Thustheε-neigbourh
5、oodofapointzisjustthesetofpointslyingwithinthecircleofradiusεcentredatz.Notethatitdoesn’tcontainthecircle.Definition1.2Adomainisanon-emptysubsetDofCsuchthatforeverypointinDthereexistsaε-neighbourhoodcontainedinD.Examples1.3Thefollowingaredomains.(i)D=C.(Takec∈C.Then,anyε>0willdoforanε-neighbourhood
6、ofc.)(ii)D=C{0}.(Takec∈Dandletε=1(
7、c
8、).Thisgivesa2ε-neighbourhoodofcinD.)(iii)D={z:
9、z−a
10、0.(Takec∈Candletε=1(R−
11、c−a
12、).Thisgivesaε-neighbourhoodofcinD.)2Example1.4ThesetofrealnumbersRisnotadomain.Consideranyrealnumber,thenanyε-neighbourhoodmustcontainsomecomplexnumbers,i.e.theε-neighbou
13、rhooddoesnotlieintherealnumbers.Wecannowdefinethebasicobjectofstudy.1Definition1.5LetDbeadomaininC.Acomplexfunction,denotedf:D→C,isamapwhichassignstoeachzinDanelementofC,thisvalueisdenotedf(z).CommonError1.6Notethatfisthefunctionandf(z)isthevalueofthefunctionatz.Itiswrongtosayf(z)isafunction,butsome
14、timespeopledo.Examples1.7(i)Letf(z)=z2forallz∈C.(ii)Letf(z)=
15、z
16、forallz∈C.Notethatherewehaveacomplexfunctionforwhicheveryvalueisreal.(iii)Letf(z)=3z4−(5−2i)z2+z−7forallz∈C.Allcomplexpolynomialsgivecomplexfunctions.(iv)Letf(z)=1/zforallz∈C{0}.ThisfunctioncannotbeextendedtoallofC.Remark1.8Functionss
17、uchassinxforxrealarenotcomplexfunctionssincethereallineinCisnotadomain.Laterweseehowtoextendtheconceptofthesinesothatitiscomplexfunctiononthewholeofthecomplexplane.Obviously,iffandgarecomplexfunctions,thenf+g,f−g
