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FunctionsofComplexVariableandIntegralTransforms
GaiYunyingDepartmentofMathematicsHarbinInstitutesofTechnologyPreface
Therearetwopartsinthiscourse.ThefirstpartisFunctionsofcomplexvariable(thecomplexanalysis).Inthispart,thetheoryofanalyticfunctionsofcomplexvariablewillbeintroduced.
Thecomplexanalysisthatisthesubjectofthiscoursewasdevelopedinthenineteenthcentury,mainlyby
AugustionCauchy(1789-1857),laterhistheorywasmademorerigorousandextendedbysuchmathematiciansasPeterDirichlet(1805-1859),KarlWeierstrass(1815-1897),and
GeorgFriedrichRiemann(1826-1866).Complexanalysishasbecomeanindispensableandstandardtooloftheworkingmathematician,physicist,andengineer.Neglectofitcanprovetobeaseverehandicapinmostareasofresearchandapplicationinvolvingmathematicalideasandtechniques.ThefirstpartincludesChapter1-6.
ThesecondpartisIntegralTransforms:theFourierTransformandtheLaplaceTransform.
ThesecondpartincludesChapter7-8.1Chapter1
ComplexNumbersandFunctionsof
ComplexVariable1.Complexnumbersfield,complexplaneand
sphere1.1Introductiontocomplexnumbers
Asearlyasthesixteenthcentury
CeronimoCardanoconsideredquadratic(andcubic)equationssuchas
,whichissatisfiedbynorealnumber
,forexample
.Cardanonoticedthatifthese“complexnumbers”weretreatedasordinarynumberswiththeaddedrulethat
,theydidindeedsolvetheequations.Theimportantexpressionisnowgiventhewidelyaccepteddesignation.Itiscustomarytodenoteacomplexnumber:Therealnumbersandareknownastherealandimaginarypartsof,respectively,andwewriteTwocomplexnumbersareequalwhenevertheyhavethesamerealpartsandthesameimaginaryparts,i.e.
and.
Inwhatsensearethesecomplexnumbersanextensionofthereals?
Wehavealreadysaidthatifisarealwealsowritetostandfora
.Inotherwords,wearethisregardingtherealnumbersasthosecomplexnumbers
,where
.
If,intheexpression
theterm
.Wecallapureimaginarynumber.
Formally,thesystemofcomplexnumbersisanexampleofafield.Theadditionandmultiplicationofcomplexnumbersarethesameasforrealnumbers.If1.2Fourfundamentaloperations
Thecrucialrulesforafield,statedhereforreferenceonly,are:
AdditivelyRules:i.;ii.
;iii.
;
iv..
MultiplicationRules:i.;ii.
;iii.
;iv.
for
.DistributiveLaw:
Theorem1.Thecomplexnumbers
formafield.
Iftheusualorderingpropertiesforrealsaretohold,thensuchanorderingisimpossible.1.3Propertiesofcomplexnumbers
Acomplexnumbermaybethoughtofgeometricallyasa(two-dimensional)vectorandpicturedasanarrowfromtheorigintothepointin
givenbythecomplexnumber.
Becausethepoints
correspondtorealnumbers,thehorizontalor
axisiscalledtherealaxistheverticalaxis(theaxis)iscalledtheimaginaryaxis.Figure1.1
Vectorrepresentationofcomplexnumbers
Thelengthofthevector
isdefinedasandsupposethatthevectormakesanangle
withthepositivedirectionoftherealaxis,where.Thus.Sinceand,wethushaveThiswayiswritingthecomplexnumberiscalledthepolarcoordinate(triangle)representation.Figure1.2
Polarcoordinaterepresentationofcomplexnumbers
Thelengthofthevector
isdenotedandiscalledthenorm,ormodulus,orabsolutevalueof
.Theangleiscalledtheargumentoramplitudeofthecomplexnumbersandisdenoted
.Itiscalledtheprincipalvalueoftheargument.WehavePolarrepresentationofcomplexnumberssimplifiesthetaskofdescribinggeometricallytheproductoftwocomplexnumbers.
Let
and
.Then
Theorem3.and
Asaresultoftheprecedingdiscussion,thesecondequalityinTh3shouldbewrittenas.“
”meaningthattheleftandrightsidesoftheequationagreeafteradditionofamultipleof
totherightside.
Theorem4.
(deMoivre’sFormula).If
andisapositiveinteger,then.
Theorem5.Let
beagiven(nonzero)complexnumberwithpolarrepresentation
,Thenthethrootsof
aregivenbythecomplexnumbers
Example1.
Solve
for.
Solution:
If
,then,thecomplexconjugateof
,is
definedby.Figure1.3
ComplexconjugationTheorem6.
i.
iv.
andhenceis
,wehave
.ii.iii.forvii..vi.andv.ifandonlyifisrealTheorem7.i.
vii.vi.v.iv.thatis,
and
.iii.and;ii.If,thenFigure1.4Triangleinequality1.4Riemannsphere
Forsomepurposesitisconvenienttointroduceapoint“
”inadditiontothepoints
.Figure1.5Complexsphere
Formallyweaddasymbol“”to
toobtaintheextendedcomplexplane
anddefineoperationswith
bythe“rules”
2.ComplexnumberssetsFunctionsofcomplexvariable2.1Fundamentalconcepts(1)neighborhood
ofapoint
:(2)Adeletedneighborhoodofapoint:
(3)Apoint
issaidtobeaninteriorpointof
.Ifthereexists
.(4)Aset
isopen
iffforeach,
isaninteriorpointof
.
2.2Domain
Curve
Anopenset
isconnectedifeachpairofpoints
andinitcanbejoinedbyapolygonalline,consistingofafinitenumberoflinesegmentsjoinedendtoend,thatliesentirelyin
.Anopensetthatisconnectediscalledadomain.
Acurve,if,theniscontinuousandifthen
iscalledasimplecurve.
If
andiscalledasmoothcurve(apiecewisesmoothcurve).
Adomain
iscalledthesimplyconnectediff,foreverysimplyclosedcurve
in,theinsideof
alsoliesin
,orelseitiscalledthemultipleconnecteddomain.2.3Mappingsandcontinuity
Let
beaset.Werecallthatamappingismerelyanassignmentofaspecificpoint
toeach,
being
thedomainof
.Whenthedomainisasetin
andwhentherange(thesetofvalues
assumes)consistsofcomplexnumbers,wespeakof
asacomplexfunctionofacomplexvariable.
Wecanthinkof
asamap
;therefore
becomesavector-valuedfunctionoftworealvariables.Thus
andaremerelythecomponentsofthoughtofasavectorfunction.Hencewemaywriteuniquely,whereandarereal-valuedfunctionsdefinedon.For,wecanletanddefineand.
Def1.Let
bedefinedonadeletedneighborhoodof
.Themeans
thatforevery
,thereisa
suchthat
,andimplythat.
Wealsodefine,forexample,
tomeanthatforany,thereisan
suchthatimpliesthat.Figure1.6
iscloseto
when
iscloseto
Thelimitas
istakenforan
arbitraryapproachingbutnotalonganyparticulardirection.ii.
Thelimit
isunique.Thefollowingpropertiesoflimitshold:
If
and
,theni.iv.iii.if.Also,ifisdefinedatthepointsand
,then
Th1.Let
thenandProof:Itiseasybyusingthefollowinginequalities
Def2.Letbeanopensetandlet
beagivenfunction.Wesay
iscontinuousat
iffand
iscontinuouson
is
iscontinuousateach
.
From(i),(ii),and(iii)wecanimmediatelydeducethatif
andarecontinuouson,thensoarethesum
andtheproduct,andsoisif
forall
.Alsoif
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