Table Of Contentabout the book
I I I
Completew ithcl ear explanationnsu,m erous examplese,x ercises, illustrativfieg ures,a nd
selectivbei bliographthisies ,i mportantre rermce/text focuses on thet heory off unctions
ofone complexv ariable,refl ecting theres earch and teaching experience of4 0year s ofa
leading authorityo nthe subjeUcntiq.uin e i tsco mprehensivaep proacht,h ivso lume
combinesf undametnoptiacsli nt het heory ofa nalytifucn ctionwsi tvha luabdlies
cm&ons ofno nanalytfunctiiocns and generalized analyfutnctiico ns as developedi nt he
last few decades.
Suitafbolrae widev arieotfcoy u rses int hes ubjecCtlas,si calC ompleAxn alysis
maintahiingssh t anrddsa ofm athematiricaglor and presentsm ateriinaa l r eadable,
insttruicvfeo rmat givesc leaarn dd etaileexdp lanataicoconmsp anibedy m any
. . •
illust rationsan d examplest haetl ucidaitmepo rtancotnce pts includese xtevnseise ts of
••.
exercises thatp rovidset udewnittsah w orking knowledge oft hes ubject. .co.n tains
historical background that makest hseub ject more interesantindg relatetso pitcsoea ch
otherf ora broado verview andope m thew ayf or further rbeseayrs cuhp plying
• . .
bibliographciicatla tionsea chf oirm portant theorem and formula.
Classical ComplexA nalysis serves as ani nformatirevfeere nce forp ure, applied, and
industrimaalt hematiciansand � as uperbtext forad vanced undergraduanadt grade uate
oourses inoo mpleaxn alysis.
about the author
.•.
MARio 0. GoNZALEz is Prof�or Emerituso fM athemataitcst hUe nive1SoiftAl ya bama
inTuscal oma, Alabama, where heta ughfrotm 1961t o1 978P.r iotrot habte he ldt he
chair inm athmeaticala nalyastith se U niversiotfyHa vana, Cuba, 19f44rt om o1 960.
Thea uthro ofn umreoupsa perosna nalysbies ,ai sm embero ft heA merican
MathematiScaocli etanyd .M atheatmicalAs soci atioonf AmeriHcae.w � aG uggenheim
Felloiwnm athemataittcs h eM assachutssIe ntstiotfuT teec hnol(o1g39y9 )a nd
PrincetonU niversi(t1y9 40-41H)e.serv ed as presidenotft hFe ederacidoenD octoenr es
aencias y enF ilosoffya Letras (1945-46) and Sociedad Qlbana deC iencFifsasi cas y
Matem�lie$( 1952-60P)ro.f� Gonz4lez received the doctorate( 193i8n)p hysiacsn d
mathematfroimcs theU niversiotyf Hav ana.
Prin1ed in the United States ofAme rica ISBNo-82:4 7-8415-4
lrc./newyork
marcedle kker, basal· hongk ong
•
ClassiCcoamlp leAxn alysis
Mari0o. G onzalez
ProfeEsmseororiMf ta utsh ematics
UniveorAfsl iatyb ama
Marcel DeIknkce.r , New York•• HBoansegKl o ng
11
LibraorfCy o ngreCsast aloging-in-DPautbal ication
GonzalMeazr,i 0o.
Classiccoamlp laenxa lyIsM iasr i0o. G onzalez.
p. cm.(Monograanpdth esx tboionpk usr aen da pplimeadt hematics)
Inclubdiebsl iograrpehfiecraealnn cdie nsd ex.
ISBN0 -8247-8415-4
1.F unctioofcn osm plveaxr iabIl.Te ist.lI eIS..e ries.
QA331.7.G651 991
515.'9 --dc20 91-22128
CIP
Thibso oiks p rinotnea dc id-pfarpeeer .
Copyri©g 1h9t9b2y M ARCELD EKKERIN,C .A llR ighRtess erved
Neithtehrib so onko ra nyp armta yb er eproduocrte rda nsmiitnat neyfd o romr
bya nym eans, eleocrmt ercohnainci icnacll,u dpihnogt ocopmyiicnrgo,fi lming,
andr ecordoirnb gya, n yi nformasttioornaa gnedr etriseyvsatlew mi,t hopuetr mission
inw rintgif rotmh peu blisher.
MARCELD EKKERI,N C.
270M adisAovne nuNee,w Y orkN,e wY or1k0 016
Currepnrti nt(ilnadgsi tg it):
109 8 7 6 5I 4 3 2
PRINTEIDN T HEU NITEDS TATESO FA MERICA
111
Tom yw ifLeo la,
tom yd aughtVeircska yn dL auraan,d
tom ys on-inH-alraowlC .dR aley,
witdhe eapff ection.
v
Preface
Thiissa b oookn t hteh eoorfyf unctioofon nsec omplveaxr iaIbtcl oen.t awiinstm,ha nye laborations,
thseu bjeofct th ceo ursaensds eminoafrfse rbeytd h aeu thdourr ianp ge rioofds ome4 0y earSso.m e
topiwchsi cdhu,et ot imlei mitatwieornpesr ,e senbtreide ifnlth ye c lassrhoeormae r gei veanm ore
leisuarnedel xyt enstirveea tment.
Ina ddittioto hnbe a stiocp iicnts h cel asstihceaolor fya nalyftuincc tiuosnusa,lt layut gi hnf irst
coursoents h seu bjetchtte,e xcto ntasionmsem ateroinan lo nanalfyutnicct iaonndas b riaecfc ouonft
variotuysp eosfg eneralainzaeldyf tuincc tiaosdn iss cusbsyse edv earuatlh oirnts h lea sfte wd ecades.
Int hirse speocutbr o odki fffreormsa lslt andwaorrdk osn t hseu bject.
Becauosfet haem ounotfm ateriinavlo lavneddt hdee taimlaendn eirnw hicsho met opiacrste r eated,
thibso oskh oubledc onsidaes roeudr fcreo wmh icahv arioeftc yo ursceasnb ed rawdne,p endoinn g
thper efereonftc heiesn struTchteco lre.e axrp lanatniuomnesr,oe uxsa mpleexse,r ciasneds ,
illustfriagtuiarvleess o mtahkbeeo oikn valufaobsrle el f-satnudad sya r eferewnocrek .
Whenevpeors siIbh laev een deavotroge idv aes pecirfiecf erefnoecrae c ihm porttahneto roerm
formula diisnct uhtsees xeAtdl. s oi,nt hbei bliogartat phheeyn do fe ach chIah patveiern clutdheed
mossti gnifbiocoaknastn dp apecrosn sulitnte hdpe r eparaotfti hocenh apter.
VI
Ia m especiaglrlayt eftoum ly brothLeuri Rs. G onzalfeozr h ise xpertiinsc eo mputer programming
whicmha dep ossibtlheea ccuractoen strucotfia o nnu mbeorf f iguraessw ,e lals t hee laboratoifao n
tabloeft heg eneralitzaendg efnutn ctitoobn e f ounidn t heA ppenditxot hec ompaniovno lume
CompleAxn alysiSse:l ectTeodp ic(sM arceDle kkerI,n c.r,ef errteoad s S electTeodp ics).
AlsoI,a cknowledwgiet ghr atittuhdeae s sistaonfmc ey formegrr aduastteu denatnsdf rienDdrss .
JerraAlbde rcrombJioes,e pCha briB,a rbarFa.C hamberCsh,r istofer HoLroslfaKi iesledEr,l, e na
MedinWai lsonG,a stoSnm itha,n dS araWhi lliamwsh,o r eaodn eo rm orec haptearnsdm adea
numbeorf v aluabslueg gestions.
Manyt hankasr ea lsdou et ot hes tafoffM arceDle kkerI,n ci.n,p articutloMa sr.,M ariAal legarnad t o
Mr.A ndrew BeWriint.hp atienucned,e rstandainndcg o,m petentchee ayi demde consideradbulryi ng
thep reparatoifto hni bso okf ort hep ress.
MARIO 0.G ONZALEZ
Contents
Preface v
Intrdouction 1
0.1 Sets 1
0.2 Mappings 3
6
0.3 Notations
1 ComplexN umbers 7
1.1 TheC ompleNxu mber System 7
1.2 Reala ndI maginaCroym pleNxu mbersT.h eC ompleUxn its 11
1.3 CompleCxo njugates 12
Exercis1e.s1 14
1.4 Orderinogft heC ompleNxu mbers 15
1.5 TheC ompleSxy steams a LineaSry steamn da sa nA lgebra 17
1.6 AbsoluVtael ueo fa Complex Number 19
Exercis1e.s2 24
1.7 GeometrRiecp resentaotfi oCno mplNeuxm bers 37
1.8 PolaFro rmo ft heC ompleNxu mber 30
1.9 ExponentFioarlm o ft heC ompleNxu mber 32
Exercis1e.s3 34
vii
viii Contents
1.10 GeometrRiecp resentaotfit ohneO peratiownist hC omplex 35
Numbers
Exercis1e.s4 40
1.11 Powerosf C ompleNxu mbers 42
1.12 ExtractoifoR no ots
Exrcis1e.s5 44
1.13 Powerwsi thF ractioEnxaplo nents 48
1.14 Powerwsi thB asee anda CompleExx ponent 51
1.15 Logarithtmots h eB asee 53
1.16 Powerwsi tha CompleBxa sea nda CompleExx ponent 54
Exercis1e.s6 56
1.17 Infiniitnyt heC omplePxl aneT.h eR iemanSnp here 58
Exercis1e.7s 58
1.18 TheG eneraBli narCyo mpleNxu mber 65
1.19 HypercomplNeuxm bersQ:u aternions 66
1.20 AxiomatFiocu ndatiooftn h eC ompleNxu mberS ystem 69
Bibliography 73
2 Topologoyf P laneS etsof P oints 76
2.1 Introduction 76
2.2 AdditioDneafiln itios 79
2.3 TheC alculoufsS etsB.o oleaAnl gebra 80
Exercis2e.s1 81
2.4 Setosf C ompleNxu mbers 89
Exercis2e.s2 90
2.5 MetriScp aces 93
Exercis2e.s3 93
2.6 Neighborhoods.a:nOdpC elno seSde ts 96
2.7 SetAss sociatweidt ha GivenS et 97
2.8 IsolatPeodi ntLsi,m iPto intasn,d C ontacPto intDse.r iveSde t.
Densea ndP erfecSte ts 98
2.9 Distanfrcoem a Pointto a Seta ndD istanBceet weeTnw o Sets.
Diametoerfa SetB.o undeSde ts
Exercis2e.s4 99
2.10 ConnectSeedt s 100
Exercis2e.s5 105
2.11 CompacSte ts 106
Exercis2e.s6 114
2.12 Completeness 115
Exercis2e.7s 120
2.31 TopologiScpaalc es 120
Contents ix
Bibliography 123
3 FunctionLsi,m itasn dC ontinuiAtryc.s a ndC urves 125
3.1 CompleFxu nctions 125
3.2 TheC omponenotfs Cao mpleFxu nction 126
3.3 GeometrRiecp resentaotfCi oomnp leFxu nctions 127
3.4 FunctioAnsss ociawtietdh a GFiuvnecnt ion 130
3.5 Limit oCfo map leFxu nctiaotn Pao int 132
3.6 FiniLtiem it Inafitn ity 133
3.7 PropertoifeF si niLtiem its 135
3.8 InfiniLtiem it Faitn iaPt oei nt 137
3.9 InfiniLtiem it Inafitn ity 138
3.10 InfiniteSlmya ll aInndfi nitLealry gFeu nctions 139
Exercis3. e1s 141
3.11 ContinuoiftC yo mpleFxu nctions 142
3.12 PropertoifeC so ntinuoFuusn ctions
Exercis3 e.s2 144
3.13 OrientAerdc s and Curves 150
Exercis3 e.s3 158
3.14 Chainasn dC ycles 160
3.15 DeformationA rocfsa nd Curves. Homotopy 161
3.16 TheW indinNgu mbero f Cau rve 164
3.17 Homology: CTohnen ectiovfi tRyae gion 167
Exercis3.e4s 169
Bibliography 171
4 Sequencaensd Series 173
4.1 SequencoefsC omplex Numbers 173
4.2 ConvergencSee qoufe nces 174
4.3 PropertoifeC so nvergent Sequences 175
4.4 Limit oRfe aalS equencLei:m iStu periaonrdL imiItn ferior 180
4.5 CauchCyo nditifoorn C onvergence 182
Exercis4.e1s 183
4.6 SerioefsC ompleNxu mbers 185
4.7 Criteforri aC onvergencSee roifoe fsC ompleNxu mbers 186
4.8 SomeP ropertoifeS se rioefsC omplex Terms 189
4.9 AbsoluCtoen vergenTcees ts 193
4.10 Sequencaensd S erioefs Functions 197
4.11 PowerS eries 203
Exercis4.e2s 210
Bibliography 213
x Contents
5 ElementarFuyn ctions 214
5.1 TheTr anslatiw o=n z + b 214
az( af. )
5.2 TheS imilitwu d=e 0 215
5.3 TheL ineaFurn ctiown = az+ b a f.(0 ) 216
/
5.4 TheR eciproFucnaclt iown = 1 z 218
5.5 TheB ilineFaru nctiwo n= (az + b)/+( dc)z 220
5.6 FixedP ointosft heB ilineTarra nsformation 225
5.7 Multiploifte hre B ilineTarra nsformation 226
5.8 Classificatoifto hne B ilineTarra nsformations 228
5.9 Symmetrwyi thR especttoa Circle 236
5.10O rieanttioonfa Circle 240
5.11T heP oincaMroed elo fL obachevskNyo n-Euclidean
Geometry 241
Exersceis5 .1 248
5.12T heC onjugaBtiel ineFaurn ction 251
5.13T heG eneraBli lineFaurn ction 255
Exersceis5 .2 259
5.14T heP olynomiFaulnc tiown = P(z= )a 0+ a1+z + a zn 260
· · · n
5.15T heFu nctiown = (z -a)", n 1> 262
P(z)/Q(z)
5.16T heR ationFauln ctiwo n= 264
5.17T heJ oukowsFuknic tiown = 1/.i+ 1(/zz) 268
Exersceis5 .3 270
5.18T heE xpoenntiFuanlc tion 272
5.19T heC irculaanrdH yperboFluincc itons 274
Exersceis5 .4 283
V'z.
5.20T heF unctiwo n= • Intruocdtioonft heR iemann
Surfaces 285
�
5.21T heR iemannS urfaocfew = • 289>
5.22T heR iemannS urfaocfew = •Vz° + •J z 2- 294
5.23T heL ogarithFmuincc tion 296
5.24T he GeneProawle rF unction 299
5.25T heI nverosfet heC irculaanrdH yperboFluincc tions 300
Exersceis5 .5 305
Bibliography 307
6 Differentiation 308
6.1 TheC oncepotft heD erivatiMvoen.o gcnanidc Analytic
Functions 308
6.2 ContinuaintdyD iffeerntiability 311
6.3 DifferentiaRtuilsoe n 311
6.4 Differentiaobfia l RietaylF unctioofnT wo RealV ariables 313
