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COLLECTION ENSEIGNEMENT DES SCIENCES HERMANN ADIWES INTERNAT IONAL SERIES N MA THEMATICS A. J. Lohwater, ConsultEidnigto r HENRI CARTAN UniversitoyfP aris Elementatrhye ory ofa nalytfiucn ctioonfos n eo r severcaolm lexv ariables p EDITIONS SCIENTIFIQUES HERMANN, PARIS ADDISON-WESLEY PUBLISHING COMP ANY, INC. Reading,M assachusetts- PaloA lto- London This ist ranslatfreodm T HEORIE ELEMENT AIRE DES FONCTIONS ANALYTIQUES D'UNE OU PLUSIEURS VARIABLES COMPLEXES Hermann, Paris © 1963 HermPaanrni,s TABLE OF CONTENTS CHAPTERI .P OWERS ERIEISNO NEV ARIABLE 1. Formaplo wesre ri.e. s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.C onvergepnotw esre rie. s.. . . . . . . . . . I.6 . . . . . . . . . . . . . . . . . . . 3.L ogarithamnidce xponentfuinaclt ion.s. . . . . . .2 8. . . . . . . . . . 4.An alytifucn ctioonfso nev ariab.l. e . . . . . . . 3.6 . . . . . . . . . . . . . Exercis.e. .s . .. . . .. .. . . .. . ... . .. . . .. . . . . .. .. . . .. . .. . . .. 43 CHAPTEIRI.H oLOMORPHFIUNCC TIONSC;A UCHY'ISN TEGRAL I.C urvilineianrt egrparlism;i toifva ec losfoerdm .. . . . .4 9. . . . . . 2.H olomorphfuincc tionfusn;d amenttahle orem..s .. . . .6 6. . . . . . Exercis.e s. . . . . . . . . . . . . . 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTEIIRI. T AYLORAN D LAURENT EXPANSIONS I.C auchyi'nse qualiLtiioeusv;i ltlhee'osr e.m . . . . . 7.9 . . . . . . . . . 2.M eanv aluper operatnydt hem aximumm odulupsr incip. .l e 8I 3.S chwarlze'm ma. . . . . . . . . . . . . . 8.3 . . . . . . . . . . . . . . . . . . . . . . . 4.L aurenetx'psa nsi.o n. . . . . . . . . . . .8 4. . . . . . . . . . . . . . . . . . . . . . 5.I ntroducotfit ohnep oinatti nfinity. Residue. . .t .h .e. o re.89m 6.E valuatoifointn egrablyts h em ethoodf resid.u. e s. . . 9.9 . . . . . Exercis.e. s . . . . . . . . . . . . . . . 08. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I CHAPTER IV. ANALYTICF UNCTIOONFSS EVERAL VARIAHBALREMSO;N IC FUNCTIONS 1.P owesre riienss e vervaalr iabl.e s. . . . . . . . I. I. 8 . . . . . . . . . . . . . . 2.A nalytfiuncc tions. . . . . . . . . . . . .I 2 I. . . . . . . . . . . . . . . . . . . . . . . 3.H armonifucn ctioonfst wor eavla riab.l. e s. . . . . 1.2 2. . . . . . . . . . 4.P oissofonr'msu laD;i richlperto'bsl em.. . . . . . .I 2.7 . . . . . . . . . . 5.H olomorphfuincc tioonfss evercaolm plevxa riabl.e. s. . . I.3 2. . Exercis.e. s . . . . . . . . . . . . . . 1.3 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 CARTAN TABLEO F CONTENTS CHAPTER V. CONVERGENCEO F SEQ.UENCEOSF HOLOMORPHICO R MERO· MORPHICF UNCTIONSSE; RIEISN,F INIPTREO DUCTSN;O RMAL FAMILIES 1.T opoloogfyt hes pacee( D). . . . . . . . . . . . .1 4.2 . . . . . . . . . . . . . . . 2.S erioefsm eromorphfuincc tio.n s. . . . . . . . . .1 .4 8. . . . . . . . . . . . . 3 Infiniptreo ducotfhs o lomorpfuhnicct io.n. s. . . . . . I5. 7 . . . . . . . . 4.C ompacstu bseotfs� (D). .. ... . . . ... .. .. ... . . . .. ... .. 1.6 2 Exercis.e s. . . . . . . . . . . . . . . . . . 1.6 8. . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER VI. HoLOMORPHICT RANSFORMATIONS I. Genertahle oreyx;a mple. .s .. . . . . . . . . . . I7 .2 . . . . . . . . . . . . . . . . 2.C onformarle presentaatuitoonm;o rphiosfmt sh ep lane, the Riemansnp herteh,eo pend is.c .. . . . . . . . . . 1.7 8. . . . . . . . . . . . . . 3.F undamenttahle oreomfc onformarle present.a.t.i.o.n....1.8 4 4.C oncepoftc omplemxa nifolidn;t egraotfid oinff erenfotrmisa.l I8 8 5.R iemann sur.f a.c e.s . . . . . . . . . . . . .1 9.6 . . . . . . . . . . . . . . . . . . . . Exercises 207 CHAPTER VII.H OLOMORPHICS YSTEMOSF DIFFERENTIALE Q.UATIONS 1.E xisteanncdeu niquentehseso re.m . . . . . . . . . 2.1 0. . . . . . . . . . . . 2.D ependenocnep arametaenrsd o ni nitcioanld iti.o n.s . . 2.1 6. . . 3.H igheorr dedri fferenetqiuaalt io.n s. . . . . . . . 2.1 .8 . . . . . . . . . . . Exercis.e s. . . . . . . . . . . . . . . . . .2 I .g . . . . . . . . . . . . . . . . . . . . . . . . . . SOME NUMERICALO R Q.UANTITATIVAEN SWERS• . . . . • • • . • • • • • • • 2• 2. 2• • • TERMINOLOGICAL INDEX . • • • . . . . . . ...• • • • • . • . • • . . • • . . • • • •• •• • • 223 NOTATIONALI NDEX• • . . . • . • • • • • • . . . • • . • • • . . • • • . • • . 2• 2. 8. . . • • • • • • PREFACE The presenvto lumec ontaitnhse s ubstanwciet,h s omea dditioonfsa , coursoefl ecturgeisv eant t heF aculotyfS cienicneP arifosr ther equire mentso ft hel icedn'censee ignedmuernitn tgh ea cademisce ssi1o9n5s 7-1958, 1958-19a5n9d 1 959-196I0t.i sb asicaclolnyc ernewdi tht het heoroyf analytic funocfta i coonmsp levxa riablTeh.e c asoef a nalytfuincc tions ofs everraela olr c omplevxa riabilseh,so wevetro,u cheodn inc haptIeVr ifo nlyt og ivaen i nsigihntt toh eh armonifucn ctioonfst wor eavla riables asa nalytfuincc tioanns dt op ermitth et reatmeinntc haptevur oft he existentchee orefomr thes olutioofnd si fferenstyisalt eimnsc asewsh ere thed atai sa nalytic. Thes ubjemcatt teorf t hibso okc overtsh apta rotf t he"M athematiIcI" s certificsaytlel abguisv etno a nalytfuincc tionsT.h iss ames ubjemcatt ter wasa lreadiyn cludiendt he" Differentainadli ntegrcaall cul"u cse rtifi catoef t heo ldl icence. As thes yllabuosfec se rtificfoart etsh el iceanrceen otfix ed ind etaitlh,e teacher usuallay coennjsoiydse radbelger eoef freedomi n choosing thes ubjemcatt teorf h isc ourse.T hisfr eedomi sm ainllyi mitebdy traditainodn ,i nt hec aseo fa nalytic funcotfai ocnosm plex variable, thet raditiinoF nr anciesfa irlwye lels tablishIetwd i.l tlh erefore perhaps beu sefutlo i ndicahteer eto w hate xtenIt h aved epartfreodm thitsr adi tion.I nt hefi rsptl acIe d ecidetdob egibny o fferinngo tC auchyp'osi nt ofv iew( differentfuinacbtlieo annsd C auchyi'nst egrbaultt) h eW eierstrass poinotf v iewi,. et.h et heoroyf c onvergepnotw ers erie(sc hapt1e)r. Thisi si tseplrfe cedbeyd a brieafc counotf fo rmalo peratioonn pso wer serieis.,ew .h ati sc allneodw adaytsh et heoroyffo rmals erieIsh .a vea lso mades omethionfga ni nnovatbiyod ne votitnwgop aragraopfhc sh aptVeIr to a systemattihco ughv erye lementaerxyp ositoifot nh et heoroyf abstraccotm plemxa nifoldosf o nec ompledxi mensioWnh.a ti sr eferred toh erea sa complemxa nifolidss implwyh atu setdo b ec allae Rdi emann surfaacned i so ftens tiglilv etnh atn ame;fo r ourp artw,e decidetdo keept het ermR iemann surfofarc teh ed oubldea tumo fa complemxa ni folda nda holomorphmiacp pinogf t himsa nifolidn ttoh ec omplex plane 7 PREFACE (orm,o reg eneralilnyta,on othecro mplemxa nifold)I.n t hiwsa ya d istinc tioinsm adeb etweetnh et woi deawsi tahc lariutnya ttainwaibtloher thodox terminoloWgiyt.h a subjeacstw elels tabliassht ehde t heoroyfa nalytic functioonfsa complevxa riabwlhei,c hh asb eeni nt hep astt hes ubject ofs om anyt reatiasnedss tiilsil n a llc ountritehse,rc eo ulbde n oq uestion ofl ayincgl aitmo o riginalIiftt yh.e p resetnrte atdiisffee risna nyw ay from itsfo rerunneirnsF rancei,td oess op erhapbse causietc onforms toa recenptr actiwchei cihs b ecominign creasipnrgelvya lean mta:t hema ticatle xmtu stc ontaipnr ecisstea temeonftp sr opositoirot nhse ore-ms statemewnhtisc ha rea dequaitnet hemselavnedst ow hichr eferecnacne be made ata ll timWeist.h a veryf ewe xceptiwohnisc ha rec learly indicatceodm,p letper oofasr eg iveonf a llt hes tatemeinntt sh et ext. Thes omewhatti cklpirsohb leomfsp lanteo poloignyr elatitooCn a uchy's integraanldt hed iscussoifmo ann y-valufuendc tioanrsea pproachqeudi te openliync hapt1e1r. H erea gaiintw as thoutghhaattf ewp recisstea tements werep referatbolv ea guei ntuitiaonndsh azyi deasO.n thesper oblems ofp lanteo pologIy d,r ewm y inspiratfrioomn t hee xcellebnoto kb y L.A hlfors( CompleAxn alysiwsi)t,h ouhto wevecro nformincgo mpletely witht hep ointosf v iewh e developTsh.e basicco ncepotfs g eneral Topologayr ea ssumetdo b e familiatro t her eadearn da ree mployed frequentilnyt hep resewnotr ki;n fa ctt hicso ursei sa ddressteosd t udents of' MathematIiIc' sw hoa ree xpectteodh avea lreasdtyu ditehd'e M athe maticIs 's yllabus. I expremsys h earttyh anktso M onsieuRre ijTia kahashwih,oa refr om experiegnacien eidn d irecttihnegp ractiwcoarlko fs tudenhtassc, o nsen tedt os upplemetnhte v arioucsh apteorfst hibso okw ithe xersicaensd problemsI.t i sh opedt hatt her eadewri ltlh usb e ina posititoon makes uret hath e hasu nderstooadn da r ;imilattheedt heoretiidceala s seto uti nt het ext. HENRI CARTAN Die( Drome)A,u gus4t1 h,1 960 8 CHAPTER I PowerS erieisnO ne Variable 1.FormaPlo wer Series I. ALGEBRA OF POLYNOMIALS LetK be a commutatifiveel d.W e considtehre fo rmalp olynomials ino nes ymbo(lo r' i ndetermi'n)a tewX i thc oefficienitnsK (fort he X). momentw e do notg ivae valuteo Thel awso fa dditioofnt wop oly nomialasn do fm ultiplicoafta i opno lynomibayl a s'c al'a mra kest he setK [X]o fp olynomiianltsao v ectospra ce overK witht hei nfinibtaes e r,X.,., .X ".,. . X• Eachp olynomiiasal fi nitlei near combionfat thieo nw ithc oefficients inK andw e writiet � .Xa",w herei ti su nderstootdh aotn lya finite n�O an numbero ft hec oefficientasr en on-zeirnot hei nfinisteeq uenocfet hese coefficientTsh.em ultiplication table XP.qX XPH = defineas multiplicaintK i[oXn] ;t hep roduct (I. I) c.= p+�q= na pq.b Thism ultpilicatiisoc no mmutatiavned a ssociatIitvi esb. il ineairnt he senset hat (r2.) 9 POWER SERIEISN ONE VARIABLE for alplo lynomiPa,lPs 1 P,2 Q, a nda lslc ala1r.s I ta dmitass u nietl e ment( denotbeyd 1)t hep olynomi�al an Xn sucht hata 0= I and n�O an= o for n >o . We expreaslstl h esper opertbiyes sa yintgh aKt[ X], provided iwtivste hc tsopra cset ructaunrdei tmsu ltipliciasta i coonm,m u tatailvgee wbirtaha unietl emenotv ert hefi eldK ; iti si,n p articual ar, commutatirvien wgi tha unietl ement. 2. THE ALGEBRAO F FORMALS ERIES A formalp owesre rieisn X is foar male xpress�iao nnX n,w herteh itsi me n�O · Wf!n ol ongerre quitrhea/Qt n lya finitneu mbero ft hec oefficieannta sr e non-zerWoe. definet hes umo ft wofo rmals eriebsy where Cn= an+ bn, and tphreo ducotfa formals eriweist ha scalbayr Thes etK [[X]o]f fo rmals eritehse n foram vse ctosrp acoev eKr. The neutral eloeftm heena td ditiiosdn e notebdyo ;i ti st hefo rmals eriewsi th alilt cso efficieznetrso . Thep roduocftt wofo rmals eriiesds e finebdy t hefo rmul(a1 .1 ), which stihlalsa meaninbge cautshee s umo nt her ighhta nds idieso vear finite numbero ft ermsT.h e multipliciasts itoincl olm mutatiavses,o ciative andb ilinewairt hr espetcott hev ectosrt ructuTrheu.s K [[X]]i sa n algeborvae rt hefi eldK witha unietl emen(td enotbeyd 1 ), whichi s thes eri�esa n Xn sucht hata 0= I and an= o for n >o . n�O The algebKr[aX ] isi dentifiweidt ha subalgebofr Ka[ [X]]t,h e subalgeobfrfo ar mals eriwehso sec oefficieanrtesa llz eroe xcepfotr a finitneu mbero ft hem. 3·T HE ORDER OF A FORMALS ERIES Denot�e anXnb yS (X)o,r ,m oreb rieflbyy, S . Theo rdwe(rS )o ft his n�9 . seriiesas n i ntegwehri cihs o'n ldye finewdh enS =I= o;i ti st hes mallens t sucht haatn =I= o. We sayt haat fo rmals eriSe hsa so rde)>r ki fi ti so ori fw (S))> k. By abusd el angaweg wriet,e w(S)>) ke venw henS = o althouwg(hS )i sn otd efineidn t hicsa se. IO FORMALP OWERS ERIES I.1.3 NoteW.e canm aket hec onventthiaowtn( o=) + oo. TheS such thawt( S:;);;,k ,.( fora giveinn tekg)ear r sei mptlhyse e riLie ansX n such n�O thaatn o forn < k.T hefoyr ma vectsourb spoafcK e[ [X]]. = DefinitiAo fanm.i l(yS ;(X)w)hieerI1e d >e noats eesot fi ndiicses sa,it do bes ummaibffol,re a nyi ntekg,we r( S:;;;;,k,.)fo ra.l blu at fin�ntuem bero f thien diic.eB syd efinittihosenu o,mf a s ummabfalmei loyffo rmal series S;(=X )Li n,ainX n�O ist hsee ries S(X) =Li anXn, n�O wherfoer,e acha n=Li a ,niT·h imsa kesse nbseec aufosrfie x,e da ll n, n, i buta finintuem beorft .h ea, n arez erboy h ypotheTshieos p.e ration 1 ofa dditoiffoo rnm asle riwehsi cfohr ms ummabfalmei ligeesn erathlei zes finiatded itoifto hnve e ctsotrr ucotfuKr[e[ X]T]h.eg eneraaldidzietido n isc ommutaatnidav ses ociianat s ievnews hei cthh ree adsehro uslpde cify. Thefo rmanlo tatLii aonnnX c atnh ebnej ustibfiywe hda fotl lowLest. n�O a monomoifda elg rpe bee a formasle riLie ansX n sucthh aatn= o for n;::.o n=I= pa ndl eatp XdPe nostuec ahm onomiaTlh.efa miloyfm onomials (ann)Xne(:'!N b eintgh es eotf i nteg:;e;;,ors,.)i so bviosuusmlmya banlde , itssu mi ss imptlhyefo rmasle riLie sa ,.X". n�O NoteT.h ep roduocftt w ofo rmasle ries ism eretlhyes umo ft hes ummabfalmei lfoyr mebdy a ltlh ep roducts oia monomioaftl h fiers ts erbiyeo sn eo ft hsee cond. PROPOSITI3O.1N. Threi Kn[g[ X]i]sa ni ntedgormaa(lit nhm iesa nthast S=I= o nadT =I= o implSyT =I= o). ProoSfu ppotshea St( X=) Lia pXaPn dT (X)= �b qXaqr en on-zero. p Letp = w(Sa)n dq = w(Tl)e,t S(X)·T(=X�)C nnX; n II