September Cauchys theorem Cauchys formula corollarie - PDF document

PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014) iscontinuousthroughout[a;b],and[a;b]breaksintonitely-manysubintervalsoneachofwhich iscontinuouslydierentiable.[1.2.1]Proposition:Forcontinuous,complex-valuedfon andniceparametrizedpath :[a;b]! ,thepathintegralR foffalong expressedasalimitofRiemannsums,isexpressibleintermsoftheparametrization,asZ f=Zbaf( (t)) 0(t)dt(where 0(t)=d dt (t)asexpected)Proof:ThepointisthatRiemannsumsdirectlyonthecurveareequaltoRiemannsumson[a;b],viatheparametrization.Thefactor 0(t)isthelimitingcaseofthemultiplicationbydierenceswj�wj�1inthedirectRiemannsumversion.Consider (t)=x(t)+iy(t)withonce-continuously-dierentiablereal-valuedfunctionsx(t);y(t).Given�0,choosea=t1t2:::tn=bon[a;b]suchthatj (tj)� (tj�1)j,usingtheuniformcontinuityof ontheboundedinterval[a;b].Themeanvaluetheoremappliedtofunctionsx(t);y(t)showsthat (tj+1)� (tj)iswellapproximatedby 0(tj)(tj+1�tj): (tj)� (tj�1) tj+1�tj� 0(tj)!0(astj+1�tj!0)Againbecause[a;b]isbounded,thislimitbehaviorisuniform:given"�0,thereis�0suchthat (t)� () t�� 0()"(fort6=in[a;b]withjt�j)Thus,thedirectRiemannsumiswellapproximatedbyamodiedform:Xjf( (tj))( (tj+1)� (tj))�Xjf( (tj)) 0(tj)(tj+1�tj)"
Xjf( (tj))
(tj+1�tj)ThemodiedRiemannsumPjf( (tj)) 0(tj)(tj+1�tj)isexactlyaRiemannsumfortheparametrized-pathintegral.Theright-handsideintheinequalityis"timesaRiemannsumforthereal-valuedfunctiont!jf( (t))j,andtheseRiemannsumsconvergetoanitenumber.Since"isassmallasdesired,thelefthandsidegoesto0.Thus,Riemannsumsfortheparametrized-pathintegralconvergetothesamelimitastheRiemannsumsforthedirectly-denedpathintegral.===[1.2.2]Remark:Thepreviousdiscussionalsoshowsthatthepathintegraldoesnotdependontheparametrization.Independenceofpathparametrizationcanalsobeprovendirectlybychangingvariables,fromthechainrule:Let 2:[a2;b2]! and':[a2;b2]![a;b]dierentiablesuchthat '= 2.Unwindingthedenitions,andusingthechainrule,withu='(t),Z 2f=Zb2a2f( 2(t)) 02(t)dt=Zb2a2f( '(t))( ')0(t)dt=Zb2a2f( '(t)) 0('(t))d'(t)=Zbaf( (u)) 0(u)du=Z fprovingindependenceofparametrization.2 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014)Byuniformcontinuityoffonanopensetwithcompactclosurecontainingthepath,given"�0,forsmallenough,jf(z)�f(wj�1)j"forallzonthestraightlinesegment`jfromwj�1towj,soZ`jf�f(wj�1)
Z`j1"
jwj�wj�1jandZ f�XjZ`jf"
Xjjwj�wj�1jObviously,thestraightlinesegments`jassembletoapolygonapproximating .Thesituationsuggeststhatthelimitas!0+ofPjjwj�wj�1jisthelengthof .Thiswillfollowfromthenitely-piecewisecontinuousdierentiabilityof .Itsucestoconsideroneofthenitely-manycontinuouslydierentiablepiecesof ,thus,wetake :[a;b]! continuouslydierentiablewithoutlossofgenerality.Weclaimthatlim!0Xjjwj�wj�1j=Zbaj 0(t)jdtWith (tj)=wj,bytheuniformcontinuityofthederivative,wj�wj�1 tj�tj�1� 0(tj�1)�!0(uniformly,as!0)Thus,forgiven"�0,forsmallenough�0,Xjjwj�wj�1j�Xjj 0(tj�1)j
(tj�tj�1)"
Xjjtj�tj�1j="
jb�aj�!0TheRiemannsuminvolving 0goestoRbaj 0(t)jdt.===[1.6.1]Remark:Moregenerally,curves forwhichthelimitofthesumPjjwj�wj�1jexistsarerectiable.Theredoexistcontinuousbutnot-rectiablecurves.Weneedat-worstnitely-piecewisecontinuouslydierentiablecurves,soworryaboutfurtherpossibilitiesisnotnecessary.[1.7]ThetrivialestimateonpathintegralsThisgivesasimple,usefulupperboundforR f:[1.7.1]Claim:Z fsupz2 jfj
length( )Proof:Thisarisesfromthecorrespondingassertionforreal-variablescalculus:with :[a;b]!C,approximatingacomplexintegral'sabsolutevaluebytheintegraloftheabsolutevalue,Z f=Zbaf( (t)) 0(t)dtZbajf( (t))j
j 0(t)jdtsupz2 jfj
Zbaj 0(t)jdt=supz2 jfj
length( )asclaimed.===4 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014)Proof:Picksomepointzoin ,andformtrianglesfromzoandeverypairofconsecutiveverticesofP.Thesetriangleslieinside sinceitisconvex.Thesumofthe(counter-clockwise)integralsoverthesetrianglesistheintegralover�,sincetheadded-onpathsaretracedinbothdirections,socancel.Theintegralovereachtriangleis0,byCauchy'stheorem.===[2.0.3]Corollary:Foranitely-piecewisecontinuouslydierentiableclosedcurve inaconvexset and thepathintegralaroundPtracedcounter-clockwise,forfcomplex-dierentiableon ,R f=0.Proof:Approximate bypolygonsinside .===[2.0.4]Remark:Unsurprisingly,thesameargumentworksunderaweakerhypothesisthanconvexity:for and starlikeaboutz,meaningthatthelinesegmentconnectingztoanyotherpointof liesentirelyinside ,andthelinesegmentconnectingztoanypointof meets onlyatthatpoint.[2.0.5]Remark:Ever-morecomplicated,weakerhypothesesonthetopologyof and stillallowtheconclusionR f=0.Asimpleusefulcaseisthat iscontractiblein ,meaningthatitcanbe(piece-wisesmoothly!)shrunkdowntoapointwithoutpassingoutside . 3.Cauchy'sformula/integralrepresentationAgain,thebasecaseinvolvestheverysimplestpaths,forexample,triangles:[3.0.1]Theorem:Forfcomplexdierentiablenearz,for acounter-clockwisepatharoundatrianglehavingzinitsinterior,f(z)=1 2iZ f(w) w�zdwProof:ThefunctionF(z)=f(z)�f(zo) z�zoiscomplex-dierentiablewherefis,exceptpossiblyzo.Let 0bethepathcounterclockwisearoundasmalltriangleT0aboutzo,entirelyinsidethelargertriangleT.ConnecttheverticesofT0tothoseofT.Asinearlierepisodes,thesumofpathintegralsovertheboundariesofthethreeresultingquadrilateralsandtheboundary 0ofT0istheintegralover ,becausetheinteriorpathsaretraversedinbothdirections,socancel.Cauchy'stheoremfornicepolygonsaboveshowsthattheintegralovereachquadrilateralis0.Thus,Z F=Z 0FUsingcontinuityofFatzo,given"�0thereis�0suchthatjF(z)�F(zo)j"forjz�zoj.WithT0chosensmallenoughtobeinsidethediskofradiusatzo,Z 0F(z)�F(zo)dz"
length 0"
6Again,theintegraloftheconstantF(zo)aroundaclosedpathis0.Thus,theintegralofFitselfissmallerthanevery"&#x-556;0,andisnecessarily0.Thus,relabellingthevariablestobetterexpressourintent,0=1 2iZ f(w)�f(z) w�zdw=1 2iZ f(w) w�zdw�f(z)
1 2iZ 1 w�zdw6 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014)Ash!0,forzuniformlyboundedawayfromw,thisgoesto0uniformlyinz;w.Sincefiscontinuous,itisuniformlyboundedonthecompactsetconsistingofthecurve .Thus,thedenitionofthederivativebeinggivenbytheexpectedformulaisveried.===Weakeningaconvexityhypothesis:forthefollowingcorollary,aregion isstarlikeaboutzwhenthelinesegmentconnectingztoanyotherpointin liesentirelyinside .Apath insidestarlike isstarlikeaboutzwhenthelinesegmentconnectingztoanypointwon meets onlyatw.[3.0.3]Corollary:For astarlikepolygonalpathaboutz,tracedcounter-clockwise,andfcomplex-dierentiableonastarlike containing ,Z f(w)dw w�z=2i
f(z)Proof:PutasmalltriangleTaroundz,smallenoughsothat,bycontinuity,reasonablechoicesoflinesegmentsconnectingtheverticestotheverticesofthepolygonlieinside .ThesumoftheintegralsovertheresultingtrianglesotherthanTare0,byCauchy'stheorem,andtheintegralaroundTgives2if(z),asjustproven.===[3.0.4]Corollary:For astarlikenitely-piecewisecontinuouslydierentiablepathaboutz,tracedcounter-clockwise,andfcomplex-dierentiableonastarlike containing ,Z f(w)dw w�z=2i
f(z)Proof:Approximate byconvexpolygonalpaths.===[3.0.5]Remark:Thesameargumentsshowinquitegeneralsituationsthat1 2iZ f(w)dw w�z=f(z)
1 2iZ dw w�zThelatterintegralisproven,invariousways,tobeaninteger,andisthewindingnumberof aroundz,whichismeanttobethenumberoftimes goesaroundz.Thisisimpreciseasitstands,butcanbemadepreciseinvariousways,thebestonesinvolvingalittlealgebraictopology.Asimpleclosedcurve aboutzisonesuchthat1 2iZ dw w�z=1 4.Powerseriesexpansions,Morera'stheorem[4.0.1]Theorem:AfunctionadmittingaCauchyintegralrepresentationf(z)=1 2iZ f(w) w�zdwforsomexedsimpleclosedpath aboutztracedcounter-clockwise,hasaconvergentpowerseriesexpansionforzneareveryzoinside :f(z)=1Xn=0f(n)(zo) n!
(z�zo)n8 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014)Ifdesired,thisiseasilyrearrangedtogive"ratherthan3",provingcontinuityofthelimitfthroughout .Withcontinuityinhand,wecancertainlyintegratefoverboundariesoftrianglesandothersimpleclosedcurves inside .SinceR fj=0forallj,andsince(theimageof) iscompact,theintegralsRgamfjgotoR f,whichistherefore0.ByMorera'stheorem,fisholomorphic.=== 5.Identityprinciple[5.0.1]Theorem:Ifholomorphicfunctionsf;gonaconnectedopenset takethesamevaluesatdistinctpointsz1;z2;z3;:::in ,andlimjzj=zo2 ,thenf=gthroughout .Proof:First,onenaturallineofargumentcanbefollowedtoitslogicalend:bycontinuityoff;g,f(zo)=limjf(zj)=limjg(zj)=g(zo)Equalityoftherstderivativeatzofollowssimilarly:f0(zo)=limjf(zo)�f(zj) zo�zj=g(zo)�g(zj) zo�zj=g0(zo)Perhapsitisfeasibletoexpresshigherderivativesatzoasmorecomplicatediterateddierencequotients,butthisisbestdoneinasomewhatrepackagedform:considerh=f�g,aholomorphicfunctionwithh(zj)=0andzj!zo.Thatis,insideeverypunctureddisk0jz�zojthereisazeroofh.Ifhwerenotidentically0,itwouldhaveaconvergentpowerseriesh(z)=cN(z�zo)N+cN+1(z�zo)N+1+:::(withcN6=0)Theideaisthatforzveryclosetozothe(z�zo)Ndominatesthepowerseries,butisnot0forz6=zo,contradictingtheassumptionthathisnotidentically0.Indeed,forsomer&#x-296;0thepowerseriesisabsolutelyconvergentforjz�zojr,socertainlycnrn!0,sothesenumbersareboundedinabsolutevalue,saybyC.Forjzj�zoj=,with0rand1 2,XnN+1cn(zj�zo)nCXnN+1nC
N+1 1�2C
N+1Meanwhile,jcN(zj�zo)Nj=jcNj
NTakingjlargeenoughsuchthatissmallenoughsothatjcNj
N�2C
N+1,wehavejh(zj)j=Xncn(zj�zo)njcNj
N�2C
N+1�0contradictingh(zj)=0.Thus,h=f�gmusthavebeenidentically0.===[5.0.2]Example:Euler'sintegralfortheGammafunctionis�(s)=Z10tse�tdt t(forRe(s)�0)Forx�0,bychangingvariables,Z10tse�txdt t=x�sZ10tse�tdt t=x�s
�(s)10 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014) 7.LaurentexpansionsaroundisolatedsingularitiesThebasicCauchytheoryisdisk-oriented,sincepowerseriesconvergeindisks.Thenextsimplestregionfromthisviewpointisanannulusofinnerradiusr,outerradiusR,aboutapointzo: =fz2C:rjz�zojRg[7.1]Laurentexpansionsonanannulus[7.1.1]Theorem:Aholomorphicfunctionfintheannulus hasaLaurentexpansionf(z)=Xn2Zcn(z�zo)n=:::+c�2(z�zo)�2+c�1(z�zo)�1+c0+c1(z�zo)+c2(z�zo)2+:::absolutelyconvergentintheannulus,uniformlyoncompactsubsets.TheLaurentcoecientscnaregivenbycn=8&#x-278;&#x-278;&#x-278;&#x-278;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:1 2iZ Rf(w)dw (w�zo)n+1(forn0)1 2iZ rf(w)dw (w�zo)�n+1(forn0)where RanyacircleaboutzoofradiusslightlylessthanR,and risacircleaboutzoofradiusslightlymorethanr.Thesecoecientsareunique,forrjz�zojR.[7.1.2]Remark:Thepositive-indextermsgiveapowerseriesconvergentatleastinjz�zojR,andthenegative-indextermsgiveapowerseriesin(z�zo)�1convergentatleastinjz�zoj&#x-296;r.Proof:Let bethepaththatrsttraverses R,thento ralongaradialsegmenttowardzo,traverses rbackward,thenbackoutto Ralongthesameradialsegment.Thetwointegralsalongtheradialsegmentareinoppositedirections,socanceleachother,givingZ Rf�Z rf=Z fFurther,forzbetween Rand r,Cauchy'sintegralformulagivesf(z)=1 2iZ f(w)dw w�z=1 2iZ Rf(w)dw w�z�1 2iZ rf(w)dw w�zTheintegralover RcanberearrangedjustaswasdoneinthediscussionofCauchy'sformulaforderivativesandpowerseriesexpansionsonadisk,producingthenon-negative-indextermsintheLaurentexpansion:1 2iZ Rf(w)dw w�z=Xn01 2iZ Rf(w)dw (w�zo)n+1
(z�zo)nHowever,herethoseintegralsarenotassertedtohaveanyrelationwithderivativesoff.Theintegralover rcanberearrangedinasimilarway,butnowusingjz�zoj�jw�zoj:�1 2iZ rf(w)dw w�z=�1 2iZ rf(w)dw (w�zo)�(z�zo)=1 z�zo
1 2iZ rf(w)dw 1�w�zo z�zo=1 z�zo
1 2iZ rf(w)1+z�zo w�zo+�z�zo w�zo
2+:::dw=X�n01 2iZ rf(w)dw (w�zo)n+1
(z�zo)n=Xn01 2iZ rf(w)dw (w�zo)�n+1
(z�zo)�n12 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014)Proof:Let�Nbethemost-negativeindexsothattheLaurentcoecientisnon-zero.Asz!zo,themonomial(z�zo)�NeventuallydominatestherestoftheLaurentexpansion,andinabsolutevaluegoesto+1.Ontheotherhand,ifjf(z)j!+1atzo,thenj1=f(z)j!0,sohasaremovablesingularity,andisoftheform1 f(z)=(z�zo)N
h(z)forhholomorphicandnon-vanishingatzo.Inverting,bythequotientrule1=h(z)iscomplex-dierentiableatzo,sohasaconvergentpowerseriesexpressionthere.Thenf(z)=(z�zo)�N
1 h(z)givesaLaurentserieswithnitely-manynegative-indexcoecients.===[7.6]EssentialsingularitiesIsolatedsingularitieswhichareneitherremovablenorpolesarecalledessentialsingularities.Unlikepoles,atwhichthevaluesofafunctionbecomelarge,atanessentialsingularitythebehaviorismorechaotic:[7.6.1]Corollary:(Casorati-Weierstrass)Letzobeanessentialsingularityofotherwise-holomorphicf.Then,givenw12C,given"�0and�0,thereisz1satisfyingjz1�zojandjf(z1)�w1j".Proof:Theideaistoproveaconverse:ifthereissomevaluew1whichf(z)staysawayfromthroughoutsomepunctureddisk0jz�zoj,thenzoiseitherremovableorapole.Thus,considerg(z)=1 f(z)�w1Thehypothesisthatfstaysawayfromw1assuresthatthedenominatorisboundedawayfrom0,sog(z)isboundednearzo,sohasaremovablesingularitythere.Thus,f(z)=1 g(z)+w1.Ifg(zo)6=0,thenfhasaremovablesingularitythere.Ifg(zo)=0,sinceg(z)isnotidentically0,theng(z)=(z�zo)N
h(z)forsomehholomorphicandnon-vanishingatzo.Then1=h(z)isagainholomorphicatzo(bythequotientrule!),sohasaconvergentpowerseriesexpansionthere.Thenf(z)=w1+(z�zo)�N1 h(z)givestheLaurentexpansionoff,withnitely-manynegative-indexterms.=== 8.ResiduesandevaluationofintegralsTheproofofuniquenessofLaurentexpansionsusedtheeasybutprofoundfactthatZ (w�zo)Ndw=Z20(eit)Nieitdt=iN+1Z20e(N+1)itdt=8:0(forN6=�1)2i(forN=�1)for acirclegoingcounterclockwisearoundzo.Asinearlierdiscussions,thesameoutcomeholdsfor anyreasonableclosedcurvegoingoncearoundzocounterclockwise.Thiswasusedabovetoshowthat,forfholomorphicon0jz�zojRwithLaurentexpansionf(z)=:::+c�2(z�zo)�2+c�1(z�zo)�1+c0+c1(z�zo)+c2(z�zo)2+:::(on0jz�zojR)and asmallcirclearoundzotracedcounterclockwise,Z f=2i
c�1Itistraditionaltonamethe�1thLaurentcoecient:c�1=Resz=zof=residueoffatzo14 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014)with1=(z+i)holomorphicatz=i,thepropositionjustabovegivesResz=i1 1+z2=1 z+iz=i=1 2iThus,byresidues,Z Tdz 1+z2=2i
Resz=i1 1+z2=2i
1 2i=ItmayseemstrangethatforT�1theintegralsover TdonotchangeasT!1,butthatisaclearconsequenceofthebehaviorofintegralsoverclosedpaths.Thenexttrickistoseethattheintegraloverthesemi-circlesTofradiusTgoto0asT!1.Theusualcrudeestimatesuces:ZTdz 1+z2(lengthT)
maxz2T1 j1+z2j=T
1 T(T�1) T�1!0Thus,Z1�1dz 1+z2=limTZ Tdz 1+z2�ZTdz 1+z2=�0=asweprobablyalreadyknewforotherreasons.[8.0.4]Example:Letbereal,andconsider[1]Z1�1eixdx 1+x2(withreal)Asinthepreviousexample,where=0,wewouldliketocomputethisbyresidues,bylookingatintegralsfrom�TtoTandthenoverasemi-circle.Indeed,for0,theexponentialisdecreasinginsizeintheupperhalf-plane,sinceei(x+iy)=eix
e�yThus,anearlyidenticalargumentgivesZ1�1eixdx 1+x2=2i
Resz=ieiz 1+z2=2i
eiz z+iz=i=2i
e�xi 2i=e�(for0)However,for0,theexponentialblowsupintheupperhalf-plane.Fortunately,theexponentialgetssmallerinthelowerhalf-plane.Thus,weuseasemi-circleinthelowerhalf-plane.Notethatthewholecontourisnowtracedclockwise,sotherewillbeasign:Z1�1eixdx 1+x2=�2i
Resz=�ieiz 1+z2=�2i
eiz z�iz=�i=�2i
e�xi �2i=e(for0)Accommodatingbothsigns,Z1�1eixdx 1+x2=e�jj[8.0.5]Remark:Itisinfeasibletosurveyalltheimportantexamplesofintegrationbyresiduesintheliteratureinthisbriefintroduction. [1]ThisintegralisessentiallyanormalizationoftheFouriertransformof1=(1+x2),sohassomesignicanceinthatcontext,forexample.16 PaulGarrett:Cauchy'stheorem,Cauchy'sformula,corollaries(September17,2014)[10.0.2]Remark:Forholomorphicf(z)withpowerseriesc0+c1(z�zo)+c2(z�zo)2+:::withc0=c1=:::=cN�1=0,themultiplicityofthezerozooffisN.Asimplezerohasmultiplicity1,adoublezerohasmultiplicity2,andsoon.Proof:Thefunctionf0(z)=f(z)isholomorphicawayfromthezerosoffinside ,so,asusual,wereducetosmallcirclesaroundthezerosoff,andaddupthesecontributions.Withf(z)=cN(z�zo)N+cN+1(z�zo)N+1+:::withcN6=0,f(z)=cN(z�zo)N1+:::Also,f0(z)=NcN(z�zo)N�1+(N+1)cN+1(z�zo)N+:::=NcN(z�zo)N�1
1+:::)Thus,f0(z) f(z)==NcN(z�zo)N�1
(1+:::) cN(z�zo)N
(1+:::)=N z�zo
(1+:::)=N z�zo+holomorphicatzoThus,integratingcounterclockwisealonasmallcircle oaroundzo,Z of0(z)dz f(z)=Z oN z�zo+holomorphicatzodz=2iN+0Addingtheseupoverallthezeroszogivestheargumentprincipleformula.===[10.0.3]Remark:Animportantvariantoftheargumentprinciplecomesfromf0(z) f(z)=d dzlogf(z).TheideaisthattraversingasmallcirclearoundazerozooforderNoffcausesthevaluef(z)togoaroundzeroNtimes,increasingtheargumentoff(z)byN
2inthecourseofreturningtotheoriginalpoint. 11.Re ectionprincipleThereiscertainlynogeneralpromisethataholomorphicfunctiondened(forexample)onahalf-plane,canbeextendedtoaholomorphicfunctiononthewholeplane,butundersomestraightforwardhypotheseswecanextendacrossaline:[11.0.1]Theorem:Let beanopensubsetofthecomplexupperhalf-plane,suchthattheclosure meetsRinanon-empty(necessarilyclosed)interval[a;b]Rofpositivelength.Thenanyholomorphicfon extendingtoacontinuousfunctionon [(a;b),real-valuedon(a;b),extendstoaholomorphicfunctionefontheunionof ,(a;b),andtheimageof re ectedacrosstherealaxis,thatis,on [(a;b)[f z:z2 gTheextensionisdenedonthere ectedimagebyef( z)= f(z)(forz2 )Proof:Fromitsdenition,theextensionefisreadilyseentobeholomorphiconthere ectedimageof :limh!0ef(z+h)�ef(z) h=conjugateoflimh!0f( z+h)�f( z) h=conjugateoflimh!0f( z+h)�f( z) h= f0( z)18