Chapter-1-复变函数与积分变换英文版课件

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