co ANALYSIS • . . . '-,.,·X.·:;··-.··, . .... ",;" .. • "',' ' .'. . ..-, . International Series In Pure and Applied Mathematics S,..,nie,. G. S",.,.... a .... E. B. eon...'ting Bdj...,. AlIlfar.: Complex Analysis BeMer and Oruag: Advanced Mathematical Methods for Scientists and Engineers Bvc/c: Advanced Calculus B1l.tJClcn and lMaly: Finite Graphs and Networks CMnsy: Introduction to Approximation Theory Ch£rler: Techniques in Partial Differential Equations ('odtli",ton and Uvln••", Theory of Ordinary Differential Eq".t;ons Conle and de B_: Elementary N1!muical Analysis: An Algoritbmic Approach : Introduction to Partial Dilterential Equations and Boundary Value Problems DeUman: Mathematical Methods in Physics and Engineering Golomb and Shanka: Elements of Ordinary DilI'erentiai Equations Hammi",: Numuical Methods for Scientists and Engineers Hildebrand: Introduetion to Numerical Analysis Hl1IC1elwld#r: The Numerical Treatment of a Single Nonlinear Equation Kal_. fi'alh. aM Arbib: Topics in Mathematical Systems Theory LaB... Vector and Tensor Analysis McCtmll: Topology: An Introduction with Applications to Topological Groups Monk: Introduction to Set Theory Moure: Elementa of Linen Algebra and Matrix Theory M01lI'-a aM I>uria: Elementary Theory and Application of Numerical Analysis PUJf'I: Matrix Theory and Finite Mathematics Pip"" and Haruill: Applied Mathematics for Engineers and Physicists RalaIon and RepIOdueed, stored ill a reIrieval OYI""', or tnnSIPitled, in ....y for. M by ....y - . .,Inlrollic, m""',anical, pho~ """".Iin.. or othenrioo, witbout doe paior .,i"on ponDiosiao 01. tile publiober. 161718192021 BRBBRB 969876543210 Tbia bonk ..... ""' in Modo, n SA by MollO&Y," Compooition Compony, Inc. Tbe eclillDll Wec6 Carol Nap. and 8\opben WocIoy; the ..........ction IUponioor _100 (};Pmp...'b. (In_tiobal _ _ in pwe ....d applied "'a\lwm,"ti.,.) IncJudea index. 1. A.IIaIyt\c functioDl. I. Title. QAlI31.A411 urn 511;'.93 '18-17078 IBM .... .---..,--,"~ -~,.'- ..•.. .. " ,'~ ....-.... _: .-,..:,. ".,.: " '.-:"~:,,-;,,"-."•-'1""'..-, ToEma , ,'. " -' .. , Contents . • • • ... xm 1 COMPLEX 1 The AlI.bra of Comp'- Number. 1 .. 1.1 Arithmetio OperatiODB 1 .•.. 1.2 Square Rook a .. 1.3 Just.;fieat.ion 4 ...... 1.4 Conjuption, Absolute V&lue 6 Inequalitiee 9 Geometric .Repre.."tation of Comples Number. 12 Addition IIDd Multiplication 12 The BiDomi·1 Equation Iii Analytic Geometry 17 • 'l'he Spherical Representation 18 2 COMPLEX FUNCTIONS . . .' ; -, . .'.. ......' . : .._..._,;.,......,.,.. : ..' .-.'... .. -_.. ,' ... .. '- 21 21 22 24 28 30 33 •. 33 · • 36 . ..· ... . CONTENTS 2.3 Ulrifonn Convergence 35 2.4 Power Series 38 2.5 Abel's Limit Theorem 41 The Esponentiol and Trigonometrie Functions 3.1 The Exponential 42 3.2 The Trigonometric Functions 43 3.3 The Periodicity. 44 3.4 The Ingsrithm 46 CHAPTER 3 ANALYTIC FUNCTIONS AS MAPPINGS 49 I Elementary Point Set Topology 50 1.1 Sets and Elements 50 1.2 Metric Spsces 51 1.3 Connectedness Ii( 1.4 Compsctness 59 1.5 Continuous Functions 63 1.6 Topological Spaces 66 2 Con/or_lity 67 2.1 Arcs and Closed Cw ves 67 2.2 Analytic Functions in Regions 69 2.3 Confonnal Mapping 73 2.4 Length and Area 75 3 Linear T,..~ormations 76 3.1 The Linear Group 76 3.2 The Cross Ratio 78 3.3 Symmetry 80 3.4 Oriented Circles 83 3.5 Fa.milies of Circles 84 4 Elementary Conformal Mappings 89 4.1 The Uee of Level Curves 89 4.2 A Survey of Elementary Mappings 93 4.3 Elementary Riemann Surfaces 97 CHAPTER 4 COMPI EX INTEGRATION 101 1 Fundamental Tlworemll 101 1.1 101 1.2 104 1.3 105 1.4 109 1.5 112 COIiTIEIiTI .. Z Cauchy'" IRtegral Formula 114 2.1 The Index of a Point with Roolpect to a Closed Curve 114 2.2 The Integral Formula 118 2.8 Higher Derivatives 120 a LocOI Propertie" of Analytical Funetiona 124 3.1 Removable Singularities. Taylor'. Theorem 124 3.2 Zeros and Poles 126 3.3 The Local Mapping 130 3.4 The Maximum Principle 133 4 The GeneFUI Form of Cauchy's Theorem 137 4.1 Chain. and Cycles 137 4.2 Simple Connectivity 138 4.3 Homology 141 4.4 The General Statement of Cauchy's Theorem 141 4.5 Proof of Cauchy's Theorelii 142 4.6 Locally Exact Diftereotial. 144- 4.7 Multiply Connected Regions 146 5 The Calculua of Rellidlle. 148 5.1 The Residue Theorem. 148 5.2 The Argument Principle 152 5.3 Evaluation of Definite Integrals 154 Harmonic Functiona 162 6.1 Definition and Basic Properties 162 6.2 The Mean-value Property 165 6.3 POll!oon'. Formula 166 6.4 Schwara's Theorem 168 . 6.5 The RelIection Principle 172 • CilHAPTER 5 SERIES AND PRODUCT DEYELOPMENIS 175 -tc.--~._-'oUler Se,ia £"JHln.iona 175 1.1 Weierotrass'a Theorem 176 ".;. 1.2 The Taylor Suies 179 l '-'"., " "•',•.,"..-. .1. 3 "., -Pa r The Laurent tial FracCio Sa na ies an d FacforiHtioR 184 187 :Partia..l,F-r-a.ct.iooa 187 Infinite PrOduct.. 191 O'D'lI1ioaI Prodnota 193 The Gem.... Fun.ction IG8 SIiIrIiq'. Formula 201 ,. . . . . .~., . ~, . • . .. " "'.' .. CONTENTS Entire F..nc:tiona 206 3.1 Jensen'. Formula 'JJIl 3.2 Hadam.ro'. Theorem 208 4 The Riemann Zeta Function 212 4.1 The Product Development 213 4.2 Extension of t(.) to the Whole Plane 214 4.3 The Functional Equation 216 4.4 The Zeros of the Zeta Function 218 $ Normm 219 5.1 Equicontinuity 219 5.2 N()lwality ....d 220 5.3 Arsel·'s Theorem 222 5.4 Families of Analytic Functions 223 5.5 The Classical Definition 225 CHAPTER 6 CONFORMAL MAPPING. DIRICHLET'S 229 1 TIw Riemann Mapping Tlworem 229 1.1 Ststement ....d Proof 229 1.2 Bouvdary Behavior 232 1.3 Use of the Ralection Principle 233 1.4 Analytic Ares 234 Z Conformal Mapping oj PoIyg01l3 2.1 The Behavior at &n Angle 235 2.2 The Schwan-Christoffel Formula 236 2.3 Mapping on & Rectangle 238 2.4 The Triangle Functions of Schwars 241 A Closer Loolc at Harmonie Functiona 241 3.1 Functions with the Mean-value Property 242 3.2 Harnack'. Principle 243 4 TIw Diriehlet Problem 245 4.1 Subharmonic Functions 245 4.2 Solution of Dirichlet'. Problem 248 5 Canonical Mapping. of Multiply Connected Regiana 251 5.1 Harmonie Moosures 5.2 Green'. Function , 5.3 Parellel Slit Regions 252 257 , , 259 CONTENTS • 7 ELLIPTIC FUNCTIONS 263 1 Simply Periodic Functio.... 263 1.1 Repr""""tation by Exponentials 263 1.2 Th.. Fourier Development 264 1.3 FunctioDB of Finite Order 264 2 Doubly Periodic FunctionB 265 2.1 Tbe Period Module 266 2.2 Unimodular Transformations 266 2.3 Tbe Canonical B8l!is 268 2.4 Geneml PropeJ1ies of Elliptic Functions 270 :1 The Weier8t.....8 Theory 272 3.1 The Weierstrass p-fuootion 272 3.2 The Functions t(z) and q(.) 273 3.3 The Dillerential Equation 276 3.4 The Modular Functi<>n X(T) 277 3.1i The Conf"rmal Mapping by MT) 279 CHAPTER 8 GLOBAL ANALmC FUNCTIONS 283 1 Analytic Continuotion 283 1.1 The WeierstnIM Th=y 283 1.2 Germs and Sbeaves 264 1.3 Sections and Riemann Surfaces 287 U Analytic Continuations ahlDK Arcs 289 1.6 H<>mooopic Cum.. 291 1.6 The M"nodromy Theorem 295 1.7 Branch Points 297 .. J Algebroic FunctioR8 300 • 2.1 Tbe Resultant mTwo Polyn<>mials 300 2.2 Definition and Properties of Algebro.ic Functions 301 2.3 Behavior at tbe Critical Points 304 ."·· Picard'. Theorem - 306 • "-' .~. ' - 3.1 hounlry Valuee 307 0- Linear Dige,../itio! Eq....cloru 308 ',.,. 4.1 Ordinary Points 309 ,.2 Rcgn],r BiOl'dpr Points 311 ;"" 4.3 Solutions at Infinity 313 4.4 The Hyperge<>metric DilI_ntial Equation 3l1i ·.'.' 4.6 Riemann's P<>int "f View 318 328 Preface • Complu A1I(llyaia has successfully maintained its place as the sta.ndsrd elementary text on functions of one complex varisble. There is, never- theless, need for a new edition, pa.rtly because of changes in current mathe- matical terminology. partly because of differences in student preparedness · and aims. There aTe no radical innovations in the new edition. The author still believes strongly in a geometric approach to the b!lSics. and for this reason the introduetory chapters are virtually unchanged. In a few places, :throughout the book, it was desirable to clarify certain points thst ex- f'perience has shown to have been a source of possible misunderstanding or • • ,-difficulties. Misprints and minor errors that have come to my attention "have been corrected. Oth~I wise, the main differences between the second r'~d third editions can be summarized as follows; •:. 1. Notations and terminology have been modernized, but it did not necessary to cha.nge the style in any significant way. Z. In Chapter 2 a brief section on the change of length and area under mapping has been added. To some degree this infringes on the self-contained exposition, for it forces thc reader to fall back on for the definition and manipulation of double integrals. The I•S mm• or. 4 there is a new and simpler proof of the general form of theorem. It is due to A. F. Beardon, who has kindly ~I witted to reproduce it. It complements but does not replace the old proof. has been 1etained and improved. ,. A short .. on the Riemann zeta function he" been included. . . PRI!FACE This always fascinates students, and the proof of the functional equation illustrates the UBe of in a less trivial situation than the mere computation of definite integrals. 5. Large parts of Chapter 8 have been eompletely rewritten. The main purpose WIlB to introduce the reader to the terminology of genllB and sheaves while emphllBizing all the classical concepts_ It goes without saying that nothing beyond the basic notions of sheaf theory would have heen compatible with the elementary nature of the book. S. The author hIlB successfully resisted the temptation to include Riemann surfaces IlB one-dimensional complex manifolds. The book would lose much of its usefulness if it went beyond its purpose of being no more than an introduction to the basic methods and results of complex funetion theory in the plane. It is my pleasant duty to thank the many who have helped me by pointing out misprints, weaknesses, and errors in the second edition. I am partieularly grateful to my eolleague Lynn Loomis, who kindly let me share student reaction to a recent based on my book. LaTif V. AM/Drs COMPLEX ANALYSIS • 1 PLEX NU • L THE ALOE.ItA OF COMPLEX NU• •EItI It is fundamental that real and complex numbers obey the sarne basic laws of arithmetic. We begin Our study of complex fWle- tion theory by stressing and implementing this analogy. 1.1. ..4ritiunetJe Operations. From elementary algebra the reader is acquainted with the imaginlJf1l unit i with the property ," = -1. U the imaginary nnit is combined with two real num- bel8 a, fl by the proce:aaes of addition and multiplication, we obtain B compw number a + ifl. a and II are the r~al and '; pari of the complex number. If '" = 0, the number il to be """,Zy imaginlJf1l; if fl - 0, it is of C01l1Be real. Zero i8 the only number which is at once real and purely imaginary. Two complex numbers are equal if and only if they have the same res] part and the 88me imaginary part. Addition and multiplication do not lead out from the system of oomplex nnmbers. Ae,nming that the ordinary rules or . arithmetic apply to complex numbers we find indeed (1) (a + ill) + ('"( + sa) - (a + '"() + i(fl + /I) and (2) + + + + (a iII}(., i.) - (..., - (l6) i(a. fl.,). In the IU, identity have made lI'e of the relation " - -1. It is . . obvioua that division ill We wi&h to ...., ..,,- .-, ".--' :, ..;- -- . . ";~<,"'-'" -___ '~-::'""., __~-',:" ",,-', __ -I __ '-. ':.<',. ," . ••:,~':; __-_,.' '.._. -'_,.'.'.•.-.;.:.'-' 1 ,'" ... - - -,'~ -~- '-'~-.~ 2 COMPLEX "NA~ VSI. + + + ahow that (.. ifJ)/h if) is a complex number.• provided that 'Y + if ¢ O. If the quotient is denoted by x i1/, we must have a + ifJ = + + (-y i6)(x iy). By (2) this condition can be written .. + ifJ - ('Yz - 6y) + i(1x + 'VY), and we obtain the two equations a~'Yz-6y fJ = 3x + 'VY· This system of simultaneous linear equations has the unique solution . x = .'Y..",++fJ0"i fJ'V -ao 11-'1"+3" + for we know that '1' 6· is not zero. We have thus the (3) + + + a i(j _ (JI.'Y fJ3 i fI'Y - ai. + + + 'Y i6 - 'Y. 01 '1' 01 Once the existence of the quotient has been proved, its value can be found in a simpler way. If numerator and denominator are multiplied with '1 - io, we find at once + + + + (JI. i(j (a ifJ)('Y - i3) (a-y (ja) i(fJy - ai) + 'Y+ ii = h ia)('Y - io) = 'Y' + 6" .. As a special the reciprocal of & complex number ¢ 0 is given by 1 a-i/J a+ifJ=a'+fJ" We note that i" has only four possible valuee: 1, i, -1, -i. They correspond to values of 11 which divided by 4 leave the remainders 0, 1, 2,3. EXERCISES I. Find the values of (1 + 2i)', I) -3 + 4i' 2 +i I 3 - 2i ' + + (1 i)' (1 - ,)-. + Z. If Z 0= Z i1l (z and 11 real), find the real and imaginary parts of z-1 ' •- 1 z + l' _1 . Z" So Sbow that -1 ± i 3 • ~1 and 2 • =1 for all combinations of signs. 1.%. Squa.... Roots. We she!) now show tbai the square root of II + complex number can be found explicitly. If the given number is a i/J. + we are looking for a number x ill such that + + (x i1/)' = a ip. Thill is equivalent to the ByBtem of equatioll8 <") ZI - y~ = a 2zy = /J. From theee equations we obtain · . • • + + + (x' 1/')' - (x' - y')' ","'y' ~ a' P'. we must have x' + II' = Va' + pI, · tbe square root is positive or zero. Together with the first equa.· . (") we lind z· = l(a + V Ot' + p') 1/' = l( -Ot + Va' + P·)· that tbeee qnantities are positive or zero regardJeIl8 of the sign • • ... •.,The equations (5) yield, in general, two opposite values for :t and two . . .Butth .Be values cannot be combined arbitrarily, for the aeoond (4) is not II cppeequence of (5). We must therefore be careful andy BO that their product bas the Bign of p. This leads to the -± +. •.jfjJj -a va . . (fitlo;;' For' .;. 0' theviilues are ± if a ;;: 6, ±i v::; .. - -' •.·'_. ,.' " .• ". 'C', a). It is most eMily defined in terms of the set R+ of poMtiDe real numbers: ex < (J if and only if fJ - a e R+. The set R+ is characterized by the following properties: (1) 0 is not a posi- tive number; (2) if ex '" 0 either ex or -a is positive; (3) the 8UD\ and the product of two positive numbers are positive. From theoe conditioDl! one den ves all the usual rules for manipulation of inequalities. In particular one finds that every square a' is either positive or zero; therefore 1 .. l' is a positive number. By virtue of the order relation the sums 1, 1 + 1, 1 + 1 + 1, ... are all different. Hence R contains the natural numbers, and aince it is a field it must contain the subfield formed by all rational numbers. Finally, R satisfies the following eomplolene88 ctmdition.: every incre8&- ing and bounded sequence of real numbers has a limit. Let a, < al < a. < . . . < a. < . . . ,and "'''Slime the existence of a real number B such that ex. <: B for all... Then the completeness condition reqnires the existence of a number A .. lim. •• a. with the following property: given any. > 0 there exists a natural number ... such that A - • < a. < A for all .. > .... Our disell"8ion of the reaI-number systsm is incomplete inasmuch as we have not proved the existence and uniqueness (up to isomorphisms) of . a Bylltem R with the postulated properties. t The student who is not thoronghly (amiliar with one of the constructive procesees by which real numbers can be introduced should not fail to fill this gap by consulting any textbook in which a full axiomatic treatment of real numbers is given. + + The equation ",I 1 ~ 0 has no solution in R, for a l 1 is always >positive. ':luppose now that a field F can be found which cont'inB R as a ',aubfieId, and in which the equation:r" + 1 .. 0 can be solved. Denote a (BOlution by i. Then ,,' + 1 .. (:r + .)(" - .), and the equation + !.:r' 1 .. 0 has exactly two roots in F, i and -i. Let C be the subset of + ': consisting of all elements which can be expreosed in the form a i(J + + . real a and (J. This reprnentation is unique, for ex ifJ .. eI i(J' a - eI .. -i({J - ,8'); hence (ex - eI)' .. - «(J - ,8')', and this is only if a .. a', fJ .. ,8'. The 8ubeet C is a subfield of F. In f&Clt, except for trivial verifica- the Ieader is asked to earry out, this is exactly what was shown . Sec. 1.1. What is more, the strueture of C is independent of F. For if .is another field containing R and a root i' of the eqnation ,,' + 1 - 0, , , '... ' tAn u.m'p'lIAi.tm. belw!eq two 6.elda is a one-to 0"8 aouwpondenoe which pte .. '01''''' alid . .. The waid ia !lIed quite. puraIJy to indicate a ........ v 'e S;;d . .. - . .. ., . . . d'leIath·.1h·" .... OopAidered ilDpocl.nt .. ". . . """', • . , • _,_-._,:-.'",,•,"._-..••:'\-,"- ':.1.-(,-,,,,.._.•...-",'..,_...."-.._._"._..,,, ,,'"' . . .-.... "_."v<._--_, '_.'.,.;..,,.~.:.;.-...~-~_..".-,,_.. ..'..,."-".-':.---,._-~....'. .... ,,','.~-,-_,,_•••• , ........- . . . - . ' , ' ,- , ..... :;..,__ "._ .. ,,_.', .. ,.·~··.-·:;:.,;-','-.,i',;·::·,~:·'·- , _~-< - ,. " ,:-," ..:."(,',.,,. • •" • COIIPLEX ANALYS'S the + subset C' is formed by all elements .. i'/J. There is a one-to-one correspondenee between C and C' which .. + ifJ and .. + i'fJ, and this correspondence is evidently a field isomorphism. It is thus demonstrated that C and C' an) isomorphic. We now define the field of compln numbers to be the subfield C of an a.rbitrarily given F. We have just BOOn that the choice of F makes no difference, hut we have not yet shown that there exists a field F with the required properties. In order to give our definition a meaning it remains to exhibit a field F which contains R (or a subfield isomorphic with R) and in which the equation ",I + 1 ~ 0 bas a root. There an) many ways in which such a field can be constructed. The simplest a.nd most direct method is the foUowing: Consider all expre:::ions + + of tbe form a ifJ where .., fJ arere·l numbws while theBigns and i are pure aymbols (+ does not indicate addition, and i is not an element of a field). These expressions an) elements of a field F in which addition and mwtiplication &nl defined by (1) and (2) (observe the two dift'erent mean- i!J&ll {If the sign +). The elements of the pa.rticula.r form a + 10 are seen + to constitute a subfield isomorphic to R, and the element 0 i1 satisfies + + + the equation x" 1 = 0; we obtain in fact (0 i1» - - (1 10). The field F bas thus the required properties; moreover, it is identical with ~ conuponding subfield C, for we can write a + ifJ = (.. + 10) + fJ(O + il). The existence of the complex-number.field is now proved, and we can go + + baCk to the simpler notation a ifJ where the indicates addition in C and i is a root of the equation ",' + 1 = O. -- . ' · EXIERC:ISIES (For students with a h.okground in algebra) :L Show that the Bylltem of all matrices of the special f6rm • a fJ , -fJ a • . . G!l!Jibined by matrix addition and matrix multiplication, is isomorpbie to .the field of complex numbers. •. 2. Show that the complex-number system can be thought of as the ~d of all .polynomials with real coefficients modulo the irreducible + polynomial ",' 1. 1.4. Co.vugation, Ab"olute Yahle. A complex number can be denoted either by a single letter a, representing an element of the field C, or + in the fornl a ifl with real .. ....d fl. Other standa.rd notations &nl + + + z = X iv, r - ~ 1'1, 1.11 = u iv, and when ,.,ed in this conneotion it • - -- . COIIPLEX NUII.EIIS 7 is tacitly undel'lltood that z, y, ~, 'I, ", V are real numbers. The real and ime.giuary part of a complex number a will also be denoted by Re a, 1m a. In deriving the rules for complex addition and multiplication we used only the fact that i! ~ -1. Since -i has the same property, all rules must remain valid if i is evelY where replaced by - i. Direct verification + shows that this is indeed so. The trlVlBformation which replaces .. i/3 by a - ifl is called complex Clmjugation, and a - ill is the clmjugale of .. + ifl. The conjugate of a is denoted by ii. A number is real if and only if it is equal to its conjugate. The conjugation is an involulory transformation: this meaDS that " ~ a. The formulas Re II = a+4 2 t A-a Ima - 2i the real and imaginary part in terms of the complex number and its conjugate. By systematic use of the notatioDS a and Ii it is hence possible to dispense with the use of separate letters for the real and imaginary part. It is more convenient, though, to make free lI'e of both notations. The fundamental property of conjugation is the one already referred to, namely, that a+b~ii+ii (iij = a. Ii. The corresponding property for quotients is a coll8equence: if 4Z = b, then Iii = Ii, and hence (bla) - Ii/a. More generally, let R(a,b,c, .••) stand for any rational operation applied to the complex numbers a, b, c, .... Then R(a,b,c, . • .) = R(4,Ii,c, . • .). As an application, consider the equation + + . . . + coli" CIZ"-I c.-~ + c. = o. U t' is a root of this equation, thtn f is a root of the equation cor + c¢' I + . . . + c._~ + c. = O. r . tn particIIJ8I', if the coefIicients are reo/, rand are roots of the same equa- li$D, and we have the familiar theorem that the noDnlal roots of an eque - ~OD with real ooeflicients occur in paire ofoonjugate roots. + . The product 44 - a l fJ' is alays positive or zero. Its Donnega- .. the modllZua or of the DUID- •.a; it - ne't&t1Iljnoloi,i" bJ' • CO.~LE. ANALYSIS the fact that the modulus of a real m.mber coincides with its numerical value taken with the positive !lign. We repeat the definition . ad ~ lal', where lal ~ 0, and observe that 141 ~ lal. For the absolute value of a product we obtain and hence • labl' ~ ab· iili ~ abclii ~ a4bb ... lal'lb\", labl = la\ • Ibl since both are ~ o. In words: TM absolute value of a product is equal to tM product of tM abeol~ of tM f~tor8. It is clear that this property extends to arbitrary finite products: la,a, .•• a,,1 = la.1 . 11121 ••• 1a,,1· The quotient alb, b F 0, satisfies b(a/b) = a, and hence we have also Ibl . la/bl GO lal. Of The formula for the absolute value of a Bl1m is not as simple. la + bl" = (a + b)(<< + b) = ad + (ab + 1>4) + bb or (7) la + bl' = lal" + Ib\" + 2 Re abo The conesponding formula for the difference is (7') la - bl" = lal" + Ib\" - 2Re ab, and by addition we obtain the identity (8) 10 + bl' + la - hl" = 2{lal' + Ibl'). We find EXERCISIS L Verify by calculation that the values of z . . z'+ 1 + for Z = :E. ilJ and II = :E - ilJ are conjugate. 2. Find the absolute values of + + .) -2i{3 .)(2 + 4J)(1 and (3+41)(-1 +2.) (-1-1)(3-1) . COIIPLEX NUMBERS • L Provetb&t a-b 1-a/) =1 + + if either lal - lor Ihl = 1. What exception must be made if lal - Ibl = 1? 4. Find the eonditions under which the equation az hi c = 0 in one complex unknown has exaetiy one SDlution, and compute that SDlution. L Prove lAgrange's identity in the complex form 1.$. Ineq..aliti.... We shall now prove SDme important inequalities which will be of consta.nt use. It is perhaps well to point out that there is nO order relation in the complex-number system, and henee all inequalities must be between real numbers. From the definition of the absolute value we deduce the ineq"alities (9) -Ial ;l!! Re a ;l!! lal -Ial :ii 1m a :;; lal· The eqnality Re a = lal holds if and only if a is real and ;;:; O. If (9) ill applied to (7), we obtain , and hence , " (10) la + bl" ~ (Ial + lb/)' la + bl ;l!! lal + Ibl· i This ill called the triang~ iMqUGlitll for reMOns which will emerge later. , By induction it can be extended to arbitrary BUmB: ;' Ii.. '(11) la, + a. + . . . + aal ~ la,1 + laol + .'. . + laal. " "I:.. _ The abeolute 1IGl~ 0/ a &'Um Ut at tI163t equal to IAe 114m o/1Ae ab,olute ;~ , o/Ihe It:rmo. , The reader ill well aware of tbe importance of (11) in the .,: ~ case, and we .haJJ find it no Ie. imporlant in the theory of complex t:;.-. bnmbu t', Let\JJl detbjmine a.ll of equality in (11). In (10) the equality r holds if and only if ali ~ 0 (it is convenient to let c > 0 indicate that f ill is real and~). If b pi 0 , oondition can be written in the : form Ibll(a/b) ~ 0, and it ill bellee!!CtiliWlleDt to alb ;; O. In ' ' "1 ' . ", ", ," , . ' .. . .- .'- 10 COMPLEX ANALYSIS we proceed 88 follows: Suppose that equality holds in (11); tben la.1 + la,l + ... + 1...1= I(a. + a,) + a, + ... + ...i :;; la. + a,l + la,l + . . . + 1...1:;; la.1 + la,1 + . . . + la.l. Hence la. + a.1 = la.1 + 11101, and if a, ~ 0 we conclude that a./a, iii: O. But the nnmbering of the terms is arbitrary; thus the ratio of any two nonzero terlll8 must be positive. Suppose conversely that this condition is fulfilled. ABsuming that a, ~ 0 we obtain . . . + ...1= la.l· 1 + ~ + . . . + ~ 41 o. 1 + a, + . . . + ~ a, a, = la,l 1 + llaa..11 + . . . + I... a. = la.1 + la.1 + ... + la.l. To sum up: 1M sign of equalitylwlds in (11) if and mUy if 1M ratio oj any huo nonzero terms is pOBitive. By (10) we have also lal = I(a - b) + bl ~ la - bl + Ibl or la! - Ibl ~ la - bl· For the Same reason Ibl - lal :;; la - bl, and these inequalities can be combined to (12) la - bl .. lIal - Ibll· Of course the same estimate can be applied to la + bl. A special case of (10) is the inequality .613) la + i.61 ~ lal + IIlI which expressee that the absolute value of a complex number is at most equal to the sum of the absolute values of the real and imaginary part. Many other inequalities whose proof is less immediate are a1AO of fre ' .ent llBe. Foremost is Caudal!3 inequality which states that ,da,b, + ... + a.b.I' :;; (ja,l' + ... + 1...I')(lb1I' + ... + Ib.l") n,. in shorter notation, • t t J. a eonven;eot s..mmation index and, nsed sa & subscript, cannot. be O()nfqzed :"ith the bnaFnary unit.. It. e--,.,ms poin~lesa f,o bcm ita 13". , ,,' , ," ,,' " '".,,:,.~,: COIIPLEX NUIIBERS 11 To prove it, let;>. denote an arbitrlll'Y complex number. We obtain by (7) (15) . . " " k k k,-1 '-I 140 - ;>.ii,I' = 1401· + 1;>.1' Ib,I' - 2 Re}; }; 11,'1>,. i-I ~_1 This expreBBion is ~ 0 for all;>.. We can choose for if the denominator should vanish there is nothing to prove. This choice is not arbitrary, but it is dictated by the desire to make the (15) as small as possible. Substituting in (15) we find, after Bimplifieations, which is equivalent to (14). From (15) we conclude further that the sign of equality holds in (14) if and only if the a. are proportional to the ii,. Cauchy's inequality can aIao be proved by means of Lagrange's '. identity (Sec. 1.4, Ex. 4). EXERCISES , I. Prove that , , , a-b 1- db <1 \' If lal < 1 and 1"1 < 1. I' " 2. Prove Cauchy's inequality by induction. t , " U 1a.1 < 1. >-. ii:; Ofori = 1, •..• nand;>., +;>.. + •.• +}... = I, Callow that ,: J ,"',. !}...a, + }..... + . . . + }...o.! < l. .. ,, 4. Show - t '!. ) a '- t '. t b ml ' -'. are complex 111!mberu satisfying - '. '." -., .. ,." . , . .•.. ,,- : .~, '.' , , .,.i,:,·,;,,:I· .....,-t+,!-+oI-2Iol· " '. '" .,•"~-.-.. . >..' - .. --,: ... , , - '--' _'-:;','.: ..;, -:' -..-,.,,<-: .-:";',:: ' '::,<:_::.,:::~":J::.:('~.~:-:;7~,~~::L-:. ::.'~.:~ 'i~'- ~,,~,;,~ ,..: , - , -...":".....-, , ,- ..'.:: . -,~,-",'- ;-,.-' 12 COIIPLEX ANALYS .. if and only if lal ;:;; lei. If this condition is and values of Izl? what am the smallest 2. THE CEOMETRIC REPRESENTATION OF COMPLEX NUMBER. With respect to a given rectangular coordinate system in a plane, the + complex number a = a ill can be represented by the point with coordi- nates (a,Il). This representation is constantly and we shan often speak of the point a as a synonym of the number a. The first coordinate axis (z-axis) takes the name of real aai8, and the second coordinate axis (y-axis) is called the imaginary 1IriB. The plane itself is referred to as the compln plane. The geometric representation derives its usefnlness from the vivid mental pictures 1W3ociated with a geom~tric language. We take the point of view, however, that all conclusions in analysis should be derived from the properties of real numbers, and not from the axioms of geometry. For this reason we shall nee geometry only for descriptive and not for valid proof, unIe8B the language is so thinly veiled that the analytic interpretation is self-evident. This attitude relieves us from the exigencies of rigor in cODllectioD with geometric considerations. 2.1. C.ometric Addidora orad The addition of com- ple:J: nlJmbers MIl be visualized as vector addition.. To this end we let a . number be not only by a point, but also by a vector from the origin to the point. The Dumber, the point, and the vector will all be denoted by the same letter IJ. As usual we identify all vectors which can be obt.ained from each other by parallel displacements. Place a second vector b so that its initial point coincides with the end point of a. Then a + b is represented by the vector from the initial point 01 a to the end point of b. To cOnstruct the difference b - a we draw + bo~ vectors a and b from the I!&me initial point; then b - a points from the end point of a to the end point of b. that a b and a - b are the diagonals in a parallelogram with the sides a and b (Fig. 1-1)• . An additional advantage of the vector repreeentation is that the length of the vector a is equal to lal. Hence the distance between the points a and b is la - bl. With this interpretation the triangle inequality la + bl ;:;; tal + Ibl and the identity la + bll + la - bl' ~ 2(lal' + Ibll ) become familiar geometric theorems. The point a and its conjugate «lie By I ometrically with to the real The symmetric point of a wi~ respect to the imaginary exis is P". 1-' Veetc>r addition. -4. The four points a, -ii, -a, 4 are the vertices of a rectangle which is BY rometnc with re8pect to both &Xes. In order to deri"" a geometric infAlrjlletation of the product of two complex numbers we introduce polar coordinafAls. Ifthe polar coordinafAls of the point (a,/J) are (r,'P), we know that a=rCOll'P fl ~ rain 'P. + Hence we can write a = a + ifl = • r{c08 'P i RiD ",). In thia trigo- nometric fO''IIl of a complex nllmber r is alway8 £; 0 and equal to the modulua lal. The polar angle", is called the argument or amp/iltMie of the complex number, and we denote it by arg a. + Consider two complex numbera II, ~ r,(coB '1" i sin tpJ and + a, - r,{c08 '1'1 i sin 'P')' Their product can be written in the from + + II,/It = T,rll{coa "', cos 'PI - ain 'P,8in 'PI) .(ain '1',_ 'PI C08 '1" sin '1'.)]. By meana of the addition tbeoreI1l8 of the cosine and the sine this lion can be simplified to ,,. (Ui) We recogoize that the product haa the modulus "T, and the argument + ,, '1'1 '1'" The latter result is new, and we expreBB it through the equation (17) f· '.' It is clear that. this formula can be extended to arbitn.ry products, and L," -- We can therefore state: :1; . - TM lD'IIumem of II product ia eqUIIllo lite aum of lite argument8 of lite '. fat:tma. : Tbia is fundamental. The rille that we have just formwated give/! a ; deep and IIne*pcdted justification of the geometric rep_tation of com- i }>lex numhera. We must be fully aware, however, that the manner in ;:.·wLioh we have ..tlIe!OImula (l7).vioIatee our prineiplell. ,~":.,\¥... ' 't:,i. ~;":. . . . .' " .-:.... . . -- '. - .. -:':"~' , .'", .. . ... - ," . . ." ' --.: ,' '..'-~". "..' ' ', . .,."...,'".". . ..' , ". ' . '- ,- '-' , In the .' " ... .' " first place the equation (17) is between tlnglu rather than between num- bers, and secondly its proof rested on the III!e of trigonometry. Thus it remains to define the argument in analytic terms and to prove (17) by purely analytic means. For the moment we postpone this proof and shall be content to dismlf", the of (17) from a less critical standpoint. We remark first that the argument of 0 is not defined, and hence (17) has a meaning only if til and a. are #- O. Secondly, the polar angle is deterillined only up to multiples of 360°. For this reason, if we want to interpret (17) numerically, We must ag.ee that multiples of 360° shall not count. By Dltl&llB of (17) a simple geometric construction of the product ala. r.a.o be It folloWD indeed that the with the vertices 0, 1, til is similar to the triangle whoae vertices are 0, a., ala.. The points 0, 1, tI" &lid a. being given. this similarity detbzmines the point ala. (Fig. 1-2).ln the case of division (17) is replaced by (18) argaa-., = arg a. - arga,. The geometric construction is the eame, except that the similar triangles are now 0, 1, tI, and 0, tI./tll, al. Reliltlrk: A perfectly acceptable way to define angles and arguments would be to apply the familiar methods of calculns whioh permit US to e:xpreA8 the length of a circular arc SII a definite integral. This leads toa correct definition of the trigonometric functions, and to a computational proof of the addition theorems.- The reason we do not follow this path is that complex analysis, as PlO. w, vector multiplication. COMPLEX NUMBERS l' opposed to real analysis, offers a much more direet approach. The clue lies in a direet oonneetion between the exponential function and the trigonometrie functions, to be derived in Chap. 2, Sec. 5. Until we reach this point the reader is asked to subdue his quest for complete rigor. EXIRelns 1. Find the symmetric points of a with respect to the lines which bisect the angles between the coordinate axes. 2. Prove that the points 4., /It, 4. are vertices of an equilateral triangle 4: a: a: + + + + if and only if = a./It a,a. aall•. 1. Suppoee that II and b are two vertices of a square. Find the two other vertices in all poesible cases. 4. Find the center and the radius of the circle which eircuID8Cribes a., a.. the triangle with vertices /Is, ExprMl the result in symmetric fom•. 11.11. The Binomial Equation. From the preceding results we derive + that the powers of a = r(cos'P i sin 'P) are given by (19) + a" = r>(COII "" i sin Ikp). • This formula ;8 triyially valid for .. = 0, and 8ince , it holds also when 11 is a negative integer. ..,.,,'', For r = 1 we obtain de M oivr6'8 !I1rnl1l1a (20) , . which provides an extremely simple way to "teI.,r.ms of cot! 'P and sin 'P. . cos Ikp and 8in 1Itp in '. To find the nth root of a complex number a we have to solve the + :., Biipposing that a ¢ 0 we write a = ,(cos 'P i sin "') and z = p(008 B + iBin B}. (21) takes the fonn p"«()011 ,,/} + fain.,) - r{cos tp + i sill 'P). , , " , """""-', -",--, . - , ".''-,-'" , '"-".-"'-,,".c.'..'.--" ,"- .' , .... - - ,'.;. ," :.',. 18 COM PLE!X ANALYSIS This equation is certainly obtain the root A' if p. = r and = 'P. Hence· we 00.8'nP- + I•S.lnnt-p I where .y;: denotes the positive nth root of the positive number r. But this is not the only solution. In fact, (22) is also fnJfiJ1ed if n8 diJfelll from " by & multiple of the full anp. If ang\eB are eJ:preBBed in the full angle is 2Ir, and we find that (22) is Il&tisfied if and only if 8 _ ! + k. 2Ir, n n k is any . However, only the values k = 0, 1, • • . • n - 1 give values of r. Hence the complete solution of the equation (21) is given by + + i sin ! k ~ I k .. 0, 1, ... , A-I. n n Tllenl (Ire n nth roots of any compiez number ¢ O. Tiley 1r.avs Ute modulus, and their arguments arB equally epaced. Geometrically, the nth roote are the vertices of a regnlar polygon with n sides. The (I = 1 is particularly important. The roots of the equation z" - 1 are called nth roots of unity, and if we set (23) Cd = C.OB2 r + ,.8.1D -2Ir n A all the roote can be by 1, (oJ. (oJ', ••• , ",_1. It is alAO quite evident that if Va denotes any nth root of (I, then all the nth roots can be exp..-d in the fOlln ",. . Va, k = 0, 1, . . . , n - 1. EXERCI. ., L Eqlreee C08 31', COB 4." and sin 5., in terms of COB ., and sin .,. + + + . . . + Z. Simplify 1 C08 I' 008 2., + . . . + ain 2" sin RIp. COB RIp sin ., + I. the fifth and tenth roots of unity in algebraic form• .. If til is given by (23), prove that + ... + + . . . + 1 ",tA ",<_1)1 = 0 for any integer It. which is not a multiple of n. COMPLEX NUMBERS 17 S. What ill the valUll of 1 - ... + .... - ... + (-1)-",(0-1)'1 1.3. A.nalytic GfH1_try. In classical anaJytic geometry the equation of a loous ill as a relation between:l: and y. It can juat as well be in tenruJ of z and i, sometimes to distinct advantage. The thing to remember is that a complex equation ill ordinarily equivalent to two rea1 equations; in order to obt.ain a genuine locua these equations should be the same. For instance, the equation of a circle is III - 01 ~ r. In algebraic form it can be rewritten as (z - a){l! - 4) ~ r". The fact that this equa- tion is invariant under complex conjugation is an indication that it repreeents a single real equation. A Bkaight line in the complex plane can be given by a parametric equation z '" a + bt, where a and b are complex numbers and II ,& 0; the + parameter t nIDS through all real values. Two equations II '" II bt and + II ~ a' b't represent the same line if and ouly if a' - a and h' are real multiples olb. The Ii"'" are parallel wbenever II' ja a ree ' multiple of b, and they are equally directed if b' ill a pOBitive multiple of b. The direc- tion of a directed line can be identified with arg b. The sngle between + + • = a bt and It ... a' b't is a.rg b'/b; observe that it depends on the order in which the line. are named. The lines are orthogonal to each other if b'/II is purely imaginary. Problema of finding int..reeetioDS between lines and circles, parallel or orthogonal Jines, tangents, and the like usually become exceedingly' simple when expreaaed in complex form. An ineqlla1ity I_ - al < r the inside of a circle. Similarly, + a d irooted line II - a bt determines a right half plane consisting of all pointuwithIm {. - a)/b < oand a left half plane with 1m (11- a)/b > O. · An easy argument shows that this distinction is independent of the · .' p.,.,.metric representation. .... aXERC.SES · • L When + + to its outside Izl > 1. In function theory the sphere S is referred to as the Riemann sphere. If the complex plane is identified with the (%.,x.)-plane with the %,- and %.-axis to the real and imaginary axis, respectively, the transCormation (24) takes on 'I simple geometric ml'8.ning. Writing + ,. = Z iI/ we caD verify that (27) z:I/:-l = Z,:ZI:Z. - 1, and this means that the points (z,I/,O) (z"z.,x.), and (0,0,1) are in a straight line. Hence the is a central projection from the center (0,0,1) 88 shown in Fig. 1-3. It is called a lllereograpkic pr(}jeetion. The context will make it clear whether the stereographic projection is regarded 88 a mapping from S to the extended complex plane, or via _Ba. In the spherical representation there is no Himple interpretation of addition and multiplication. Its advantage lies in the fact that the point at infinity is no longer distinguished. It is geometrically evident.that the stereographic projection traQ8- forms every straight line in the z-plane into a circle on S which through the pole (0,0,1), and the is also true. More generally, any circle on the sphere corresponds to a~le or straight line in thez-plane. To prove this we observe that a circle on the aphere liee! in .. plane + + + a,%, a,xl a,xl - ao, where we can 8'sume that at a~ + ai - 1 ° and ;:i! ao < 1. In terms of z and i this equation takes the form + + + a,(,. i) - ati(,. - t) a.(W - 1) = ao(W 1) or + (a. - a.)(z· + y') - 2..,z - 2a.y + ao a. - O. a. a. a, a. For ~ this is the equation of .. circle, and for = it represents a straight line. CODveISely, the equation of any circle or straight line N • • • r, , • !9-:,'- . . .~W, ~> iI<' __ • • • • . . . " , . . . . .""" .' ' . . . '. - .- , cen be written in this f()ml. The ooi1espondence is coll8equently one to one. ' It is easy to calculate the distance d("t) between the stereographic of IS and i. H the points on the sphere are denoted by (ZI,3:"Z.), (z;,:/:;,z;), we have first (ZI - za' + (z, - zJ' + (z. - Z~)I ~ 2 - 2(z1:l:; + x.z; + Z1:l:'). From (35) and (36) we obtain eIter a short computation z,Z; + z,(z,; ++1Z)1(::l1:; + I') - (z - 2)(:1 - 2') + (1aI' - 1)(lz'I' - 1) .. (1 + W)(1 + Iii") . ~ (1 + lal1')+(1 I+z!'l)a('lI')+-la'21Ia) - " • • Aa a result we find thet • (28) d(z,.') .. ...., For II - GO the corresponding formula is d(z, GO) ~ VI 2 + "Izli.. EXERCISES J. Show that z and :I to diametrically opposite points on the RiP-mann sphere if and only: if Ii.' = -1. I. A cube has its v!lrlices on the sphere S and its edges parallel to the coordinats axes. Find the stercographic projections of the vertices. J. . problem for a regUlar tetrahedron in general position. ... Let Z, Z' denote the stereographic projectiollB of z,:I, and let N be the north pole. Show that the triangles NZZ' and Nze are similar, and use this to derive (28). s. Find the radius of the spherical image of the circle in the plane whoes center is a and radius R. , '. . , 2 PLEX FUNCTI s 1. INTRODUCTION TO THE CONCEPT OF ANALYTIC FUNCTION . The theory of functions of a complex variable aims at extending calculus to the complex domain, Both differentiation and inte- gration acquire new depth and signifiMIIll8; at the sam" time the range of applicability becomes radically Indeed, ouly the analytic Dr holomorphic functions MIl be freely differentiated and integrated. They are the Dnly true "functions" in the sense of the French "Thoorie des fonctions" or the German "Funktionentheorie.'J Neverthelees, we sball use the term "functiDn" in its modem meaning. Therefore, when stepping up to oomplex nwnbers we haw to eonsider four different kinds of functions: real functions of a real variable, reAl functions .nf a complex variable, complex functions of a real variable, and complex functions of a complex variable. As a practical matter we that the letters. and ID ,ball always denote complex variables; thus, to indicate a complex function of a complex variable we use the notation ID = !(z). t The notation 11 - J(z) will be Jlsed in a neutral manner with the UIlderstMding that z and 1/ can be either rea! or complex. When _ want to indicate that a variable is definitely restricted to real values, _ shall usually denote it by t. By these we * .. t Modem .tudOllla &Ie "ell .ware that f It"Ddo for the runntjpn aDd J{a) ··for. -'01. fn..,OD. &.e,_, "",')eta are taAitioDally minded and , wntJnue1O,~·.,·,·t"tba"DoteOiiJf..)." :;- ' ..••• • t. ~r; .. ... -"'(,-"---",',-.. .., .: ..' . .. , ~ . . -,,:,',:-.: ;:,; _.';-:.;": .. '~';::~;·~'..~~f;;i-'-i-'";.:.;,:: 0; :' , : .:";::'.:: -:. :.:, 21 .. :- . .:.' . ','., ",-,,"- ZI COIIPUX AIIALYSI. + do not wish to cancel the earlier convention whereby .. notation z = x iy automatically implies that x and 11 are real. . It is CBsentia! that the law by which a function is defined be formulated in clear and unambiguoup terms. In other WOrdB, aD functioDB mut be weU defined and consequently, until further notice, Mngl£-oalued. t It is 1Wl necessary that a function be defined for aD values of the independent variable. For the moment we shall deliberately under- emph&Bive the role of point set theory. Therefore we make merely an informal agreement that every function be defined on an open 1Iflt, by which we mean that if I(a) is defined, then f{x) is defined for all " suffi- ciently cloBe to a. The formal treatment of point set topology is deferred until the next chapter. 1.1. Limits and Continuity. The following basic definition will be adopted: TM Jundion f(,,) itt 84id to have the limit A IJII " tmda to a, (1) .....lim I(x) = A, iJ' . only if the loUOOIing is I!:m: Far tJVeTY • > 0 there exittls a number II > 0 with the property tIwl 1/(:) - AI < ,for all values of x such tIwllx - al < II and:r >" a. . . . !'-.,. ' . "'l'!rla definition makes decisive liRe of the absolute value. 8ince the notion'of absolute value has a meaning for complex as well 88 for real we can UBe the same definition regardless of whether the variable the function J(,,) are real or complex. ..........'. . an alternative simpler notation we sometimes write: f(x) --> A for . ' ''l'bere are some familiar variants of the definition which correspoud . . eere where a or A is infinite. In the real. case we can distinguish +'" the limits and - "', but in the complex case there is only ~infinite limit. We truet the reader to formulate COllect definitions tci'Cover all the possibilities. '.. '" 'The well.known results concerning the limit of a'sl1m, .. product, and a ,qilotient continue to hold in the complex Indeed, the proofs depend only on the properties of the absolute value expreesed by labl - lal . Ibl and la + bl :!Ii lal + Ibl. t We eb·D IIOmetim e& IISD the plecm'.mc term .in;. ,.sd/flAditJlt& to un_line that the f1"'efiou has cmly one yalue for: elM vr'u of the ~ .. .. . : -..... : .,: .. COMPLEX FUNCTIONS Condition (1) is evidently eqwvalent to (2) .l.i.m... i(x) = A. From (1) and (2) we obtain .li-m4 Re I(x) = Re A (3) ......lim 1m I(x) = 1m A. (1) is a con""quence of (3). The function I(x) is said to be continuous at a if and only if .....lim I(x) = I(a). A conlin"""" ,unction, without further qualification, is one which is continuous at all points where it is defined. + The sum/(x) g(x) and the product/(x)g(x) oftwo continuousfune- tions are continuous; the quo\ient I(x)/g(x) is de1ined and continuous at a if and only if I/(a) 'J"f o. If I(x) is continuous, so arc· Be I(x), 1m I{x), and 11(x)l. , The derivative of a function is defined as a particular limit and can be '; considered regardless of whether the variables are real or complex. The forn1a.l definition.is " , , ' (4) I'(a) = lim I(x) -/(a). ..... z-Q The usual rules for forming the derivative of a sum, a product, or a ". quotient are all valid. The derivative of a composite function is deter- mined by the chain rule. There is nevertheless a fundamental difference between the of a ',real and a complex independent variable. To illustrate our point, let /fII) be a r«Jl function of a complex variable whose derivative exists at ," Then I'(a) is on one side real, for it is the limit of the quotients + I(a 1) -/(a) Ii. , Ii. tends to zero through real values. On the other side it is also the of Uw quotients I(a + ih) - lea) = .. iii. 118 such purely imaginary. Therefore f(G) must be zero. Thus a of & oomiMex variableeitber hIIII the derivative zero, or eLte does DOt 1IliiIt.' , "~"'" .. ' ...... .. ,.'. , , , ,- •. , .,,:..: ':' -': .,:- -' '_ ' -' :.;_" ._"r. _.. . ,,',_ ": .,. .. . . ..-..;.::_~.:, ;'._~,' - , '" . - - ,~ . ' - , - ' , .",' ~,':, ;,;.'',~" ,-~ .. - c. ,. ',.. '.'-';:''0 .,;".;.;:":"'..H."' ".•' '._ '.:'"- - " ., ...·-,k-,":...., . ;"",'~' . " . ....... 24 COMPLEX ANALYSIS of a complex function of & real variable CM be reduced to the real + II we write z(t) = :I:(t) iy(t) we find indeed + ret) = :1:'(1) iv'(t), . and the existenee of s'(I) is equivalent to the simuItaruloU8 existenee of :e'(0 and TI(t). The complex notation hIlS nevertheless certain formal advantages which it would be unwise to give up. In contrast, the existence of the derivative of a complex function of a complex variable has far-reaching consequences for the structural proper- ties of the function. The investigation of these consequences is the central theme in complex-fllnction theory. 1.2. Ancdytfe Functio...... The class of aMlgticfunctioot is fom.ed by the complex fllnctions of a complex variable which a derivative wherever the function is defined. The term holDlltorphic fumlirm is nsed with identical meaning. For the purpose of this preliminary investiga- tion the reader may think primarily of functioU8 which are defined in the whole plane. The 811m and the product of two analytic functions are again analytic. The same is true of the quotient f(z)/g(%) of two analytic functions, pro- vided that I/(z) does not vanil!h. In the general calle it is to exclude the points at which g(z) = O. Strictly speaking, this very typi- cal case will t.hus not be included in our considerations, but it will be clear that the results remain valid except for obvious modifications. The definition of the derivative can be rewritten in the form fez) + _ lim fez 11) - fez) • ~o h All a first consequence fez) is continuous. Indeed, from + + f(1l 11) - fez) - h· (f(z h) - f(z»/II we obtain + lim ~o (J. (z h) - f(z)) = 0 . fez) ... o. + If we write fez) = v(s) w(z) it follows, moreover, that v(z) and II(Z) are both continuous. The limit of the difference quotient must be the same regardless of the way in which h approaches lIero. If we choose re.a\ values for h, then the imaginary part 'I is kept constant, and the derivative becomes .. partiaJ derivative with respect to:e. We have thus COMPLEX FUNCTIONS Similarly, if we substitute purely imaginary values ik for h, we obtain fez} _ lim I(z + i~) ~o ik -/(z) "" - i ~ ay ~ - i ~ ay + ~. ay It follows that I(z) must satisfy the partial difterential equation (5) ~ iIz - -ia~y whioh resolves into the real equations • au av (6) -iIz= -aJ y are the Caw:hy-Riemmr.n differential equations which must be satisfied by the real and imaginary part of any analytic function. t We remark that the existence of the four partial derivatives in (6) is implied by the existence of fez). Using (6) we can write down four form81Jy difterent expressions for fez); the simplest is fez} - ~az +i~a.z For the quantity If(z)I' we have, for instance, If(z)I' = aaxu •+ aauu' = aaxu' + aaxv' = aauxaavy The last expression shoWl! that If(z)I' is the Jacobian of u and v with respect to z and II. We shall prove later that the derivative of an analytic function is itself analytic. By this fact u and v will have continuous partial deriva- tives of all orders, and in particnl.... the mixed derivatives will be equal. Using this information we obtain from (6) a'v a'v AU=8z.+ay.-O a'u a", Av=az·+ayt=O. A function u which satisfies LGp/"::'. '-' :,'~'~'.'-' '- .- .....-..,,"' ~ .-' "•.'," '.. ,.~~, tibn of u. Actually, v is detem';ned only Up to an additive coDStant, 80 that the Il8e of the definite article, although traditional, is not quite aceu- rate. In the same sense, u is the conjugate h8Jmonic function of - •. o This is not the place to iliscuss the weakest conditions of regularity which CILll he imposed on harmonic functions. We wish to prove, how- + ever, that the function 1£ ill determined hy a pair of conjugate har- monic functions is always analytic, and for this purpose we make the explicit assumption that 1£ and v have continuousfil"ilfnlrder partial derivatives. It is proved in calculus, under exactly these regularity con- ditions, that we can write az a + + u(x. 1.,1/ k) - u(x,1/) = au I. + au Y k + 6, ii + + . v(x h,y k) - .(z,1/) = II +~ k + to, where the remainders "" " tend to zero more - rapidly than I. + ik in the + + + lienee that 6,/(1. ik) -+ 0 ILlld ../(1. iTe) -+ 0 for I. ik ..... O. With + the notation fez) = u(x,1/) iv(z,y) we obtain by virtue of the rela- tions (6) + + J(z I. ilc) - fez} = ~ iJz + i ~ ax (h + ik) + I. + iI, and hence , + + + + fun f(. I. ilc) - f(z) = ~ i ~. .H~ ,0 h. ik ilz ax .e conclude that f(l) is analytic. -,.-- If u(x,1/) and v(x,1/) have rontmOOWl jiTBt-order partial derivatives wh.ich j,aliwf1/ the Cauchy-Riemann differential equation" then J(z) = u(z) + w(z) _.. tmallltic with rontinU0U8 derivative /'(.), and COIWerMIII. The conjugate of a harlllonic function can be found by integtation, and in simple caseo the computation can be made explicit. For inst;a.nr,e, U = ",' - y' is harmonic and au/ax = 2x, au/ay = - 21/. The conju- 'pte function must therefore satisfy i-aN", ~ 21/, aay. = 2",. - - + From the first equation v = 2:I:y 0 the equation pe,) = 0 has at least one root. This is the ~ed /nDdamentaI theorem of algebra which we shall prove later. If pea,) ... 0, it is shown in elementary algebra that pe,) = (z - a,)p,(z) where P,(z) is a polynomial of deglee 1l - 1. Repetition of this finally leads to a complete factorization (8) P(z) = a,.(z - ",){z - al) • . . (z - ...) , t For formal reaSDns, if the constant Oiarel&rded M a polynomi&l, itadell'ee is. equal to - ... COMPLEX FUNCTIONS 21 where the (I., aI, • . • , .... are not distinct. From the fac- toriz"tion we conclude that P(z) does not Vlmish for any value of z different from a" a., ... ,..... Moreover, the factorisation is uniquely determined except for the order of the factors. If exactly h of the (lj coincide, their common value is called a zero of P(z) of the order h. We fiud tbt the 8Um of the orders of the zeros of a pcIynom;a] is equal to its degJee. More simply, if each zero is counted as ma.oy times as its order indicates, a polynomial of degJee n has exactly n zeros. The order of a zero a can also be determined by consideration of the 8ucceeeive derivatives of P(z) for z = a. Suppose that a is a zero of order h. Then we can write P(z) = (z - a)·P.(z) with P.(a) ¢ O. SUIl- ceseive derintion yields Pea) = pI(a) = . . . - pel-li{a) = 0 while pel)(a) ¢ O. In other words, the order of a zero equals the order of the first nonvanishing derivative. A of order 1 is called a simple zero and is characterised by the conditions Pea) = 0, pI(a) ¢ O. As an application we shall prove the fonowing theorem, known as L'l.laJI' ~ Theorem 1. If allll~08 oj a poll/flOmiol P(z) lie in a hnlJ plane, !hell all zeroB oj the derivative pI (z) lie in the same hnlJ plane. From (8) we obtain (9) + . . . + P'{z) _ 1 -,---1--=._. P(II) - z - a, z- .... Suppose tbt the half plane H is defined as the part of the plane where 1m (z - a)/b < 0 (see Chap. 1, Sec. 2.3). If ... is in H and z is not, we have then But the imegin&Iy parts of reciprocal numbers have opposite sign. Therefore, under the same 1m bez - ...)-1 < O. If this is true for all k we conclude from r - 1m bPP('(sII} < L" b 1m Z-Cl,t 0, I-I . and coDBequently P' (_) ¢ O. In a sharper fonnwation the theorem tells WI that the smaUest convex polygon tbt contains the aeroe of P(.) also contains the zeros of PI(Z). COIIPLEX ANALYSIS 1.4. RatiolMJl Functions. We turn to the ""_'Al of a rational function P(z) (10) R(z) = Q(z)' given lIB the quotient of two polynomials. We and this is ell89n- tial, that P(o) and Q(z) have no common factors and hence no common zeros. R(z) will be given the value QD at the zeros of Q(z). It must therefore be considered as a function with values in the extended plane, and as such it is continuous. The zeros of Q(z) are called poles of R(z), and the order of a pole is,by definition equal to the order of the corre- sponding lero of Q(z). The derivative (11) R ' (• ) _ - P'(z)Q(o) - Q'(z)P(z) Q(.). • , exists only when Q(z) ~ O. However, as a'rational Iunction defined by the right-hand member of (11), R'(z) has the same poles as R(z), the order of each pole being increased' by One. In case Q(z) has multiple zeros, it should be noticed that the expression (ll) does not appear in reduced form. Greater unity is aOOieved if we let the variable z 88 well 88 the values R(z) rang 11 R(z) has a zero of order III - 11 at QD. if III < 11 the point at .. is a pole of order,1I - tn, and if III = 11 R(oo) = a./b. ~ 0,00. , , -, . ''."".'..,... COMPLEX fUNCTIONS 31 We can now count the total number of zeros and poles in the extended plane. The count shows that the number of zeros, including thoee at 00, is equal to the gIeater of the numbers m and n. The number of poles is the same. This common number of zeros and poles is called the ~ of the rational funetion. If a is any constant, the function H(,,) -' Q has the 8&me poles as R(o), and consequently the same order. The zeros of R(o} - Q are roots of the equation R(o} - a, and if the roots are counted as many times 88 the order of the zero indicates, we can state the following result: A rational function. R(o) of order p 11M p zer/HI and p pole8, and every equation. R(z} = a has e:MCtIy p roota. A rational function of order 1 is a linear fraction 8(z) - ; t~ with a8 - fh ;F O. Such fractions, or linear will be studied at length in Chap. 3, Bee. 3. For the moment we note merely that the equation." = 8(z) has exactly one root, and we find indeed z = 8-1(w) = aw -'\'V I-+fJa • The transformations 8 and 8-1 are inverse to each other. The linear transformation z + a is called a parallel tr~, and l/z is an ~simI. The fOlmer has a fixed point at CD, the latter inter- changes 0 and 00. Every rational function has a representation by parti6l ff'tJdiorul. In order to derive this representation we first that R(z) has a pole at DO. We carry out the division of P(z) by Q(z) until the of the remainder is at mOllt equal to that of the denominator. The result can be written in the form (12) + R{z) = G(z} H(z) where G(z) is a polynomial without constant teno, and H{.) is finite at DO. The of G{:.) is the order of the pole at GO, and the polynomial G(z) is called the ri~ part of R{z) at ... Let the distinct finite poles of R(z) be denoted by fJ., fJ.. • • • , fJ•• +} The function H fJi is a rational function of r with a pole at r = "'. By une of the decomposition (12) we can write • "',,,-,:. ' • • .,• '; • - ,"-,..,,--," - -.·,''tj , '.,", ,"--' ,"- , ". ' - . ' " ,. - ,- ,'- ,.". ~-. . -,- •• ', 32 COMPLEX ANALYSIS or with a change of variable R(o) = Gj 1 z - fl; +H; Here G; 1 • - fl; is a polynomial in s ~ fJ; without constant term, called the singular part of R(z) at flj. The function Hi 1 z - /l; is finite for z = flj. C(lnsider now the expression (13) I• . R(z) - G(z) - Gj ;-1 ---'1--". Z - fJ; This is a rational function which cannot have other poles than fl., fl" . . . , fl. and ... At z = fJ; we find that the two terms which become infinite have a difference Hj z ~ fJ; with .."finite limit, and the same is true at ... Therefore (13) has neither any finite poles nor a pole at ... A rational function without pole. must reduce to a constant. ""d if this constant is absorbed in G(z) we obtain (14) L R(.) = G(z) + • Gj 1 z - fJ •• ,. - 1 ' This representation is well known from the calculus where it is use 0 there exists an n. such that Ia.. - AI < • for n ~ no. A sequence with a finite limit is said to be ~, and any sequence which does not con- verge is divergent. If lim. , . ... = .. , the aequence may be said to diverg4 II) infinity. Only in rare C88eS can the convergence be proved by exhibiting the limit, so it is extremely important to make use of a method that JK>lmits proof of tbe existence of a limit even when it cannot be determined explicitly. The test that serves tbi. purpose hears the name of Cauchy. A seqilence . will be called or a Cav.chy ~, if it satisfies the follow- ing condition: given any I > 0 there exists an n. such that I... - a..1 < • whenever n ~ n. and m ~ nO. The test reads: A "'tIIcnu i. Cl)f/.vergent i/ and I)f/.ly i/ it il a Cauday.eq_. The neceeaity is immediate. If a.. --+ A we can find n. such that < ./2 for n iii!: n.. For tn,n ~ n. it fonows by the triangle that Ia.. - a..1 :i Ia.. - AI + Ia.. - Al < •. The 8ufficiooey is closely connected with the definition of real num- .\I:eIs, 8!ldOIll! way in which real numbers can be introduced is indced to c!)ndition.However, wewiah to lise bou~e!lrqonOto"'~uenc:eof real num. f -~ ' , ." . r-... - . -.-."<;.' .... '.,- . .- . -:.-. ... ..". -'\.!'.' - -. ,:.,..."...-.•. .. ......;., >" ••_,_;.,,',, . _,~"-.-.,. '''''"_-/:'-_.''"''.;.-,~: :-,..c_,_,,'"' • .- .,'.. " .<, ":-.--.;.,....__,'.: .' -' ':",'_.-""''.,:'0"-"::",~.'-:"-"1\<"'""-,,~:',-;, , .' -';,.,'-,,' _,..- '-_a,.,'-,.',, -,~,._,•" '~ ~',__., ..;. -_,,,-..-.... -,,-h,~.'".",','- .,- .. -m..-'-..-'-~"''-.',-,'"'-~,'-., , ..•.•" .>. ":):"n•"•",.'" , . .'., 'A--"".-",-. '·...k.~,".-'',,,.~-...'..."..-~'..';;.•.,;."y.~-,_.;.-'.;.--.-,-_--.',..,''...-,',, -_...' -,', •. ._,.~. 34 COIIPLEX ANALYSIS The real and imaginary parts of a Cauchy sequence are again Cauchy aequence., and if they converge, SO does the original aequence. .. For this reason we need to prove the sufficiency only for rel!l sequences. We use the opportunity to recall the notions of limes IlUperior and lime8 inferior. Given a real sequence \a.lr we shall set a. = ma.x \a., ... , ...1, that is, a. is the greatest of the numbers a" .... ,.... The sequence la.lr is + nondecn-smng; hence it has a limit A, which is finite or equal to 00. The nnmber A. is known as the leasl upper bouM or IlUpre",um (l.u.b. or IlUp) of the numbers ...; indeed, it is the least Dumber which is ~ all a:". Construct in the same way the least upper bound A. of the sequence \... ,: obtained hom the original sequenee by deleting a" • • • , ..... It is clear that \A.I is a nonino easing sequence, and we denote its limit + by A. It may be finite, GO, Or - GO. In any case we Mite ...... A=limsup .... It is easy to characterUe the limes superior by its properties. If A is + ., finite and I > 0 there exists an n. such that A.. < A and it follows + that "'.:;; A... < A & for .. $; .... In the opposite direction, if "'. :;; A - • for n ~ "0, then A.. :;; A - ., which is impossible. In other words, there are a.rbitrarily large n for which > CI_ A -.. If + A = CD there are a.rbitrarily large ..., and A = - CD if and only if ". tends to - CD. In all cases there caonot be more than one number A with these properties. The liMes inferi()r cao be defined in the Mme ma·nner with inequalities reversed. It is quite clear that the limes inferior and limes superior will be equal if and only if the sequenceconverges to a finite limit or diverges + to GO or to - GO. The notations are frequently simplified to lita and The reader should prove the following relati9ns: + + + li'"m- a. lim fl. ;li lim (... (1.) :> lim ... lim (1. + + + lim a. lim fl. ;li ffiii (... fl.) ~ lim ... lim fl•. Now we return to the sufficiency of Cauchy's condition. From + • I... - a..1 < • we obtain 1...1< 1....1 for n $; n., and it follows that A = lim ". and a = lim ... are both finite. If a ;of A choose (A .;.. a) 0= 3 and deu,m.ine a corresponmng no. By definition of a and A there exists +. an a. < a and an a. > A - 0 with m,n i1:; no. It follows that A - a - (A - ....) + (.... - a.) + (a. - < a) 3.,eontrarytothech.oi.~ of.. Henee a - A, and the sequence converges. . ." .- .... ", '" -'... "., CO.PLEX FUNCTIONS 35 2.2. Series. A very simple appli~tion of Cauchy's condition permits \18 to deduce the convergence of one sequence from that of another. If it is true that lb. - b.1 :; 1"- - ""I for all pairs of subscripts, the sequence Ib.1 may be termed a c 0 there exists an n. such that la.. + II" 11 + . . . + a..+,1 < • for all n ~ n. and p ~ O. For p = 0 we lind in particular that 111.1 < I. Hence the gen- eral term of a convergent series tends to aero. This condition is , but of course not 1I11fficient. If a finite number of the terms of the aeries (15) are omitted, the new series converges or diverges together with (15). In the C9lJe of conver- + gence, let R. he the Slim of the series which begillB with the term a"." Then the SUm of the whole series is S = a.. R•. The series (15) can be compared with the series (16) 11111 + la.1 + . . . + 111.1 + . . . fomltlli by the absolute values of the terms. The sequence of partial sums of (15) is a contraction of the sequence con esponding to (16), for III. + a..+1 + . . . + a..+o\ :; 111.\ + 1a..+,1 + . . . + 11.1" ,,1· There- fore, convergence of (16) implies that the original series (15) is convergent. A series with the property that the series forllled by the abeolute values of the terms converges is said to be oollOlut611l Clll'Wtrgent. ",eJIee. 1.J. Uniform eo.... Consider a sequence of functions f.(",), aDiii!fined on the _e set E. If the sequenile of values If.("'» con- ~ for eveJj z that belcmgs to E, then the limit f(z) is egeiu a function ,> ' dn·B';';" By'ileliDition, if c> 0 and '" . . 'toE· ·exiots aDllti sucb that 1/.(z) -/(z)1 < • for fa ~ 1It, but llti is allowed to depend on z. ,. . .... ...... .. ... .., , ": . ,.).... " ", . ',.'.,."..,...........,.,.',; .' ' ' ' ,,.. .'..". ' :' , COMPLEX ANALYSIS For instance, it is true that .. lim 1+-1 x=x • + for all x, but in order to have 1(1 l/n)x - xl = lxl/n < s for n l?; n. it is necessary that n. > lxi/eo Such an no exists for every fixed x, but the requirement cannot be met s; multaneously for all x. . We Bay in this situation that the sequence converglls pointwise, but not uniformly. In Positive fonnulation: TIuJ sequence (f.(x) I convergea .....iJolfnly to J(x) on IIuJ aet E iJ to every , > 0 ~re exiaIB an n. BUCk thai. If.(x) - f(x)1 < • for aU n l?; no and aU x in E. The most important consequence of uniform convergllnce is the following: TIuJ limit Iunaion f1j a uniformly c_rgenl sequence f1j conIinuOUI funclion& ia itself conIinuoua. . Suppose that the functions f.(x) are continuous and tend uniformly to f(x) on the set E. For any • > 0 we are able to find an n such that If.(x) - l(x}1 < e/3 for all zin E. Letxo be a point in E. Becausef.(x} is continuous at Xo we can find Ii > 0 such that II.(x) - f.(x.) I < _/3 for all x in E with Ix - x.1 < a. Under the same condition on x it follows that + IJ(x) - l(xo)1 ~ I/(x) - [.(:c)1 I/.(x) -J.(x.)1 + If.(x.) - l(zo)1 < " and we have proved that f(z) is continuous at :c•• In the theory of analytic functions we shall find uniform convergence much more important than pointwise converg1lnce. However, in most eases it will be found that the convergence is uniform only on a part of • the set on which the functions are originally defined. " Cauchy's necessary and sufficient condition has a counterpart for uniform convelgllnce. We assert: The sequence (f.(x) I c07lllergea unilorllllyon E if and only if to every a > 0 ~e exial8 an no BUCk thall/.(z) - f.(x}1 < afar all m,n l?; n. and aU zinE. The neceMity is again trivial. For the sufficiency we remark that the limit function fez) exists by the ordinary form of Cauchy's test. In the ineqnality If.(z) - I.(x) I < & we can keep n fixed and let m tend to GO. It follows that 1/(:e) - f.(x) I ~ e for n l?; n. and -all z in E. Hence the convergence is uniform. For practical use the following test is the most applicable: If a sequence of functions 11.(:&) \ is a contraction of a convergllnt sequencc of constants (0,,\, then the sequencc (f.(:e)1 iSllniformly convergent. The hypothesis means that 1/..(x) - I.(x)\' ~ law. - 0,,1 on E, and the, con- . . .' -. . . d ' COMPLEX FUNCTIONS n clusion follows immediately by Cauchy's condition. In the case of series this criterion, in a somewhat weaker fonn, becomes particularly simple. We say that a series with variable terms j,(1') +ft(1') + . . . +1.(1') + . . . has the series with positive terms a,+a.+··· +a.+'" for a majortmt if it is true that 1/.(x)1 ~ M a. for some eonstant M and for all sufficiently large ..; converaely, the first series is a minoran! of the second. In these circumstances we have I/.(x) +1.+,(1') + ... +1.+,(x)1 ~ M(a. + a.+l + ... + "-+.). Therefore, if the majorant the minorant oonverges uniformly. This condition is frequently refened to as the WeierstI'dB8 M lest. It has the slight weakness that it applies only to series which are also absolutely convergent. The general principle of contraction is more eomplicated, but has a wider range of applicability. EXERCISES I. Prove that a convergent sequence is bounded. ...... 2. If lim z. ~ A, prove that .l.i.m......!. (z, + z. + . . . + z.) = A· 3, Show that the sum of an absolutely convergent seriee does not change if the terms are rearranged. 4. Disc"'" completely the convergence and uniform convergence of the sequence Inz-Ir. 50 Discuss the uniform convergence of the series for real values of x. c. If U = v, + u. + ... , V = v, + v. + ... are convergent oeries, prove that UV = u.v, + (u,v, + UoV.) + (v,v, + u,v, + v,v.) + ... provided that at least One of the oeries is abeolutely convergent.. (It is easy if both mes are abeolutely convergent. Try to arrange the proof so economically that the absolute convergence of the second series is not needed.) .-.. . -, - ";' .-" '. "' :'::';":,:- -;'-.'- ;- ....~.-.. ' . ,-,'.,:';,,_.;.:-.:,"-;.:~,:,"'.::,-'""'.;:,:':--',. :':':-:..!,~;..:,;'.,.::,-~,..j,~:':;:-"."":"""",-"~,.~,-',">","""."-''':''''.~~\-'"+-'''- '''~_''''''''':~'''-'''''-:'';'''.'-"-' COMPLEX ANALYSIS :4.4. P01fIer Series. A power seriu is of the fonll (17) a.+a.z+a,z' + ... + a.z" + ... where the coefficients a- alld the variable z are complex. A little more generally we may consider series z., which are power series with respect to the center but the difference is so slight that we need 1I0t do so in a formal manner. As an almost trivial example we consider the geomelric IlerU8 1+"+.'+,,, +1"+ • • • whose partial sums can be written in the form 1 + z + . . . + z"-' = 1l--1z". Since I" -> 0 for \z\ < 1 and \1"\ ~ 1 for \z\ ~ 1 we conclude that the geometric series converges to 1/(1 - z) for Izi < I, diverges for Izl ;;,. 1. It turns out that the behavior of the geometric series is typical. Indeed, we shall find that every power series converges inside a circle and diverges outside the same circle, except that it may happen that tltEl aeries converge. only for z = 0, Or that it converges for all values of z. More precisely, we shall prove the following theorem due to _.A. bel: - Theorem 2. For ellery power aeria (17) tAere eziMa a number R, 0 ~ R ;l! 00, called 1M rodiua oj , IlIith 1M JollOVJing properlie8: (i) .TM 8eMa ronv...gu absolutely Jor ellery Z fDith \z\ < R. IJ 0 ;:;; p < R 1M convergen<:e is uniJ•• HI Jor \z\ ;:;; p. (ii) IJ Izl > R 1M leall8 oj 1M senu are unbounded, and 1M IleN8 i& conaequently divtrgtllt. , (iii) In 1.1 < R 1M aum of 1M aeria is an analylic fufldioo. TM derivative can be obtained by lermwiBe differtlltiation, and 1M derived aeria haa 1M same radius oj CUTIIIergenee. " , The circle ItI .. R is called the circle oj _ ...ge'/lC6; nothing is claimed about the convergence on the circle. We shall show that the assertionein the theorem are true if R is chosen according to the formula (18) -- l/R = lim sup v\a-I. - .-• --C_."'. --.-.., COMPLEX FUNCTIONS 31 This is known as Hadamard'sfof'WIlJla for the radius of convergence. If 1.1 < R we can find p 80 that 1.1 < " < R. Then IIp> l/R, and by the definition of limes superior there exists an 110 such that 1a.1 1/" < 1/p, 1a.1 < IIp"forn ~ n•. Itfollowsthat!a"z"1 < (1'lIp)"forlargen,sotbat the power series (17) bas a convergent geometric series as a majorant, and is consequently convergent. To prove the uniform convergence for Izi ;:;; p < R we cboose a 1" with p < l < R and find 10,,"1 ~ (P/l)" for n.. n S: Since the majorant is convergent and has coutant term~ we conclude by Weierstrass's M test that the power series is uniformly convergent. If I_I > R we choose I' 80 that R < " < 1.1. Since III' < llR there are arbitrarily large n such that 1a.11I· > 1/p, 1a.1 > l/pa• Thus la"zal > (1.1/1')' for infinitely many n, and the terms are unbounded. r • The derived series 110,,,---1 has the same radius of convergence, 1 + because Vn --+ 1. Proof: Set Vn = I 3,. Then ba > 0, and by use + + of the binomial theorem n = (1 6.)" > 1 ! n(n - 1)8~. This gives i! < 2/n, and hence i, --+ O. For I_I < R we shall wtite • r + • /(.) = aoZ- = B.(.) R.(.) • where .-. () + + . 8•• =a. al' .. + a._•I·1ZR,.(') = • 4 ~ aoZ•• and also r • !t(z) = na"zo-I = lim B~(Z), 1 ...... We have to show that 1'(.) = ft(-). Cl'nsider the identity (19) » B.(.) - Sa("'~ _ 8~('.) + (.~(".) - ft(•• Z - 20 + RaC.) - R.(••) , .2 -.I. where we a&IIUIlle thah ". " and 1.1. 1••1 < I' < R. The last u,lin eat> be .. . ... . .. • 'o}t;-. Il10(.......' + .......'... + + et-'), . '. ,. ., . • :;""'": :.,', ';',";", >.c, .,, .. :.'-' " :, " ,:",' ---~"::'-;"~:' ,~:."~" l'i.l;;'i~ ', ,', ','" ','. ,'" -, ';;,.','":-': ,,:::,..~ '.': , ';, _ -:...x~.:' "." " , . : • . . .-,-,..' •. '.i:.;:. <:~-',';.F.....:";."~;.'.;-'.,. ,).~:•.. :',.::"';.~-; , , ' ' , ,-'-,'.-:.:- l(!:-);.....). ",,:,:;, :. ... ,,\-,.,.•.'!.vl"(' ~,."-,':r~:'':,,'".I"~""""1..~j: .. CO ...LEll ANALYSIS and we conclude that The expression On the right is the remainder term in a convergent series. Hen"" we can find no such that ~•.(8) - R.(z.) 0 such that 0 < II - %.1 < 0 implies I 8.(Z) Z - 8.(Z.) Zo - &•' (z".j < • 3- a . When all these inequalities are combined it follows by (19) that I(z) ..,. I(zo) _ /t(Zo) <. z - %0 when 0 < Iz - z.1 < o. We have proved that {(zo) exists and equals it~OJ. Since the reaeoning can be repeated we have in reality proved much more: A power series with positive radius of convergence has derivatives of all orders, and they are given explicitly by I(z) = ao + a,z + a,z' + . ; . f(z) = a, + 2a.z + 3a,z' + ... . . . . + + + ... f"(%) = 2,.. 6a,z 12aot' . . . .. . .. . . . .. . .. . . . . .. . . .. .. . . . . . . . + + + . . . f Ci) (z) = k '.a. (k+1)1 I! a,+12 (k+2)! 21 a~tZ' -T In particular, if we look at the last line we that a. = jC"(O)/Ic!, and the power series becomes J(z) = J(O) + + {(O)z {,(OJ Zl + ... + J(O) (0) ZO + ... 2 ! 11! " Thie is the familiar Taylor-Maclaurin development, but we have proved it only under the assumption that/(z) has a power development. We do know that the is uniquely determined, if it exists, but the main part is still namely that every analytic function has a Taylor development. • COMPLEX FUNCTIONIJ 41 EXEReI SES L Expand (1 - z)-, m a positive integer, in powel'8 of z. 2. Expand ~::13 in powel'8 of • - 1. What is the radius of convergence? .. Find the radius of convergence of the following power series: L L:r, L L L "Pz', n!zo, q"%"(lql < 1), %"' .. If %a.z" has radius of convergence R, what is the radius of con- vergence of %/J,.!t.? of ~~Z"? S. If f(z) = %a.%", what is %n'II,,"'? .. If %110%" and %b..' have radii of convergence R 1 and R., show that the radius of convergence of %a.bo%" is at least R.R,. 7. If Iim.__ 111.1/111.+11 = R, prove that %a.z" has radius of COn- vergence R. - .. For what values of • is • convergent? • t. Same question for J.5. Abel'. Limit Theorem. There is a secoild theorem of Abel's which refere to the case where a power series conwrges at a point of the circle of convergence. We lose no generality by 888Uming that R = 1 and tbat the convergence takes place at z = 1. - !- Theorem 3. If}; a. convergea, IAen f(ts) = 0.J" tenda tD J(1) aa z o 0 IIppr0acAe8 1 in """" II UNJfI tMt 11 - zl/(1 - 1_1> remllim bounded. HemIIrk. Geometrically, the condition means that" stays in an angle < lSO° with vertex 1, symmetrically to the part (- 00 ,1) of the real axis. It is euatomary to sa,y that the approach place in a SUlk IJfI9k. - Proof. We may assume,~t.I a. ":" 0, for thisean bea.ttNned by adding """',. - -. - . • • .. . ...... .. '" .. .. ".. '-'" , .. - . . , .'. "':,' . .. , ,_.' ... ,:,' , '...-.....,.", ;..,--~',.-,''. ._.:_,• •'.,.>. ,_-"- "-,'--,:'. ,:,"-_';". ,_- .' -' .''. " " ...'.:..-,". ' , , '-".,',.,"_~-,, .,,'.',~",' " :.,..~,., . , _"~··__·,:."-"'·A':':.""_"'_; ___ ' ~-_"'_~"··~, '"">·'-_ ',.:.-_·,"~.,"-:.,·.'"•""~.~,,",'." ', ' ,''_"l- Q~","...',,''..:.~".,".c.".:".',.,·. .....- , ,-' .._.t-..~~ 42 COIIPLEX ANALYSIS + + ... + a constant to a.. We write Sa = a. a, tIa and make use of the identity (summation by parts) 8a = (Z) = 8.(1 4. - + aiZ + + z) 8,(, ... + a.r = - Zl) + ... + 8. (81 - 80)' + 8._,(,-' - + r) ... + + s.r (8. - s__ . ) z · + + ... + + = (1 - ,) (80 8iZ 8._1"-') S,$'•. But 8.," -> 0, so we obtain the representation . L • I(z) = (1 - z) Sal'. o . We are assuming that \1 - z\ ;:;; K(l - It\), say, and that 8. -> O. Choose m so large that \s,l < 8 for 11 !5;. m. The remainder of the series 2:8,$', from 11 = m on, is then dominated by the geometric series •L..• \z\" = _\z\"/(l - \zl) < _/(1 - \z\}. It follows that \/(z) \ ;:;; \1 - z\1 ..L-1 8~i I+ KE. o The first tenn on the right cail be made arbitrarily small by choosing z sufficiently close to 1, and we conclude thatf(z) -> 0 when z ..... 1 subject to the stated restriction. 3. THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS The person who approaches calculus exclusively from the point of view of real numbers will not expect any relationship between the exponential • function e" and the trigonometric functions cos z and sin z. Indeed, these functions seem to be derived from completely different sources and with djil'erent purposes in mind. He will notice, no doubt, a similarity between the Taylor developments of these functions, and if willing to use imaginary + arguments he will be able to derive Euler's lonnula e;Z = cos Z i sin z as a formal identity. But it took the genius of a Gauss to analyze its full depth. ' With the preparation given in the preceding section it will be easy to Jefine e', cos z nnd sin z for complex z, and to derive the relations between these functions. At the same time we can define the' logarithm as the inverse function of the eXpOnential, a.nd .the logarithm leads in tum to the correct definition of the argument of a complex number, and hence to the nongeometrie definition of angle. S.l. TIuJ Exponential_ We may begin by defining the ~ial function as the solution of the differential equation COIIPLEX FUNCTIONS (20) 1'(&) = 1(,,) with the initial value 1(0) = 1. We solve it by setting I(z) = a. + a,z + . . . + a"Z" + . . . I'(z) = a, + 2asz + ... + + 1Ia"z...., ... If (20) is to be satisfied, we must have a.-I = 1ICI., and the initial condition gives IJo = 1. It foTIows by induction that a. = 1/1I!. The solution is denoted by e' or exp II, depending on purely typo- graphical cODsiderations. We must show of CoUl"l!e that the series (21) e' = 1 + IZi + Zl 2! + ... + srt ni + ... converges. It does 80 in the whole plane, for Vni..... 00 (proof by the reader). It is a consequence of the differential equation that e' satisfies the addition tIoearem (22) ee I i = e- . ~. + Indeed, we find that D(e" e-') = e' . e-' e' • (-e-') - O. Hence eo . e' 'is eo constant. The value of the constant iafound by setting .. = o. + We conclude that e" e-' = e<, and (22) follows for Z = fI, C = a b. Remark. We have used the fact thatj(z) is constant ilf'(2) is identically zero. This is eertainly 80 il!is defined in the whole plane. Foril! = u+ W we obtam. -aa"u, = -clu = iJy -iIIJ = iJz i-iIJIJy - 0, and the real version of the theorem shows that! is constant on every horiJJOntal and every vertical line. As a particul&r case of the addition theorem eo· r ' = 1. This showa that e' ill r&eVeJ zero. For real :z: the aeries development (21) ahows that e" > 1 for:z: > 0, and since eo and .,. are reciprocals, 0 < e" < 1 for:z: < O. The fact that the aeries bas real coefficients shows that exp il is the complex conjugate of eJql z. Hence 1_"1" = e'" r" - 1, and le' = 6', or e' = 1. It follows that c = i .. with real ..; we prefer to say that .. is a period of e'-. We ohaIi show that there are periods, and that they are all integral multiples of a positive period "'" Of the many ways to prove the existence of a period we choose the following: From D sin II = cos II ~ 1 and sin 0 = 0 we obtain sin II < 1/ for y > 0, either by integration or by use of the mean-value theorem. In the same way D cos 1/ = - sin II > -II and cos 0 = 1 gives cos 1/ > 1 11'/2, which in tum leads to sin 1/ > y - 1/'/6 and fiDally to cos 1/ < 1 - va + y'/2 1/'/24. This inequality show. that cos < 0, and therefore va there is a II. between 0 and with cos II. = O. Because + COS'1/. sin'lI. = 1 we have sin II. = ±1, that is, ..... = ± i, and hence ."" - 1. We have sho wn that 411. is a period. Actually, it is the smallest positive period. To see this, take 0 < II < II.. Then sin II > y(1 - y'/6) > 11/2 > 0, which shows that cos 1/ is + strictly decreasing. BecaUse sin 11 is positive and COS'1/ sin'1/ ~ 1 it follows that sin II is strictly increasing, and hence sin II < sin 1/. = 1. The double inequality 0 < sin II < 1 guarantees that ei • is neither ± 1 nor ±i. Therefore e"· .., 1, and 4y. is indeed the smallest positive period. We denote it by .... Consider now an arbitrary period "'. There exists an integer n such + that n",. ~ '" < (n 1)"". If .. were not equal to ""'0, then '" - n",. would be a positive period < .... Since this is not possible, every period must be an integral multiple of "'" The 81IUIl/es/ plMitive period of fi' is denoted bll 2..-. In the course of the proof we have shown that en/S = i, These equations demonstrate the intimate relationship between the nUrn- bers e and 11'. When 1/increases from 0 to 2..-, the point 111 = e'· describee the unit circle Iwl = 1 in the positive sellll6, namely from 1 over i to -1 and back over -i to 1. For every wwith Iwl = 1 there is one and only one 1/ from the half-open interval 0 ~ II < 2..- such that w = e'v. All this follows readily from the elltablisbed fact that cos II is strictly decreasing in the "first .i. quadrant," that is, between 0 and 11'/2. Froman algebraic point of view the mapping w= establishes a M'IOIIIOI'JIM8"~ .between the additiw group of real numbers and the muitipUClMive"aroupof,«lOinp!eJ; Rumbas with ab80lutevalue 1. The ~ IIf·tM;· . ..~~. fo"i\V~ by all integral mul~ 211&. ,', ,. ,-,- .... . . ", ..: '... ~.. . ' " . : : " :..: ' . ,' ..);~~. " ~,:' ,"~"':~I.'..,)..:.:~.~::.\,'.. '. i .. . ': "": " '.....' , ' .' . .. , . ',.,,:.. , '." , :, ,'._ ... COIIPI,.I!X ANALYSIS 3.4. The Logarithm. Together with the exponential function we must slso study its inverse function, the logarithm. By definition, z - log w is a root of the equation 6' = w. First of all, since 8' is always ,-!O, the number 0 MIl no logarithm. For w ... 0 the equation.- - w is equivalent to (24) .,. = Iwl, . .;0 = 1It/lwl. The first equation has a unique solution % = log Iwl, the r.allogarithm of the positive number liDl. The right-barul member of the second equation (24) is a complex number of absolute value 1. Therefore, &8 we have just seen, it has one and only one solution in the interval 0 ;;; 'II < 2-r. Inaddi- tion, it is satisfied by all 'II that differ from this solution by an integral multiple of 2-r. We see that every comple:e number other than 0 MIl i.nJiniW.'II man'll whiM di.ffer from each other btl multip!e8 of 2-ri. The imasjnary part of log w is also cal.led the argument of w, arg w, and it is interpreted geometrically 88 tbe angl6, measured in rsdiallB, between the positive real axis and the balf line from 0 through the point ID. Accord- ing to this definition the argument has infinitely many values whicb differ by multiples of 2-r, arul log ID = log Iwl + i arg lit. With a change of notation, if Izl = rand arg z = 8, then II - re". This notation is 80 convenient that it is used constantly, even wben the expo- nential function is not otherwise involved. By convention the logarithm of a positive number shall always mean the real logarithm, unIese the contrary is stated. The symbol a', where a and b are arbitrary complex numbers except for the condition a ... 0, is always interpreted as an equivalent of exp {b log a). If a is restricted to positive numbers, log a shall be real, and a> has a single value. Otherwi.., e·..... log a is the complex logaritbm, and a> has in general infinitely many values which differ by factors There will be a single value if and only if b is an integern, and then a" can be interpreted as a power of aora-'. Ifbis a rational number with the redueed form pig, then a> has exactly g values and can be represented as W. The addition theorem of the exponential function ,clear.ly implies • + log (%,%.) = log B, log z, + arg (z,z.) = arg %, arg z" , but only in the sense that both sides represent the same infinite !let of complex numbers. If we want to compa.te a value on the left with a value on the right, then we can merely assert that they differ by a multiple of 2-ri (or 2-r). (Compare with the remarks in Chap. I, Sec. 2.1.) ", .,\ ",:, -'.'," '"." '-'~''''-':-. , COMPLEX FUNCTIONS .7 Finally we discuss the the equation cosine which is obtained by solving + COB ~ .. ~ (e" rI') = w. 6" This is a quadratic equation in with the roots ~.. =- to ± VWt - 1, and :e =arc cos w - -ilog (w ± ";w' - 1). We can also write these values in the form + arc cos w = ± i log (w v'w' - 1), + for w v'w' - 1 and w - Vw' - 1 are reciprocal nnmbers. The infinitely many values of arc cos w retIed the evenneBII and periodicity of eos :e. The sine is most eflSi1y defined by arc • SID UI = f2r " - arc cos w. It is worth emphasizing that in the theory of complex analytic func- tions all elementary transeendental functions can thus be through t;* and its log z. In other words. there is one elementary transcendental function. ..XIRCIIII L For real 1/, show that every remainder in the series for COS 1/ and sin 1/ has the 88me sign as the leading term (this generalizes the inequali- va. ties used in the periodicity proof, Sec. 3.3). 2. Prove, for instance, that 3 < fr < 2 i, : i IL Find the value of e' for z = - ri, rio .. For what values oiz is t;* equal to 2, -1, i, - i/2, -1 - i, 1 + 2i? .. Find the real and imaginary parts of exp (e')• .. Det6t1line all values of 2', i', (-1)". "I. Determine the real and parts of %". L ExpreBII arc tan w in terms of the logarithm. ,to Show bow to define tbe "angles,f in a triangle, bearing in mind that they $hould lie between 0 and fr. With thiB definition, prove that the 811m of the angles i, fr. Jt. Show tba\ ,he lOOts of the binomial .. - aarethever- recuJar tiUI of a "'< . ," po!;y&Oll (equal eidee and ",. " ,,',"."...-., " . ""'",:,":..,~"'".,'' . ',':. . . '.' " ",'"-h",;';,~,;'':,'"::;~'-i'-"'~";:.:' ,i:,'''.",.,',>'-',:·.,. . .l'.:.~.'Y.;.~:"., " .::':.,;.'. "..:.'" :~.. '. - ,- .',i-·:".-.:-·,":·./,~i,:·,,-.:.:-:~':-, '~,,,;":,.' 3 ALYTIC FUNCTIONS AS APPINGS A function tD - fez) may be viewed 88 a mapping which repre- sents a point z by its image tD. The purpot!e of this chapter is to study, in a preliminary way, the special properties of defined by analytic functions. In order to carry out this program it is desirable to dewlop the underlying concepts with sufficient generality, for otherwise we would soon be forced to introduce a great number of ad hoe definitions whose mutual relationship would be far from clear. Since present-day students are exposed to abstraction and gen- erality at quite an early stage, no apologies are needed. It is perhaps more appropriate to sound a warning that gJeatest possi- ble generality should not become a purpose. In the firat section we develop the fundamentals of point set topology and metric spaces. There is no need to go very far, for our main concern is with the properties that are essential for the atudyof analytic functions. If the student feela that he is already thoroughly familiar with this material, he ahould read it only for terminology. The author believes that proficiency in the study of analytic functions requires a mixture of geometric feeling and computa- tioDAlakill. The second and third aections, only loosely connected with the firat, are expressly designed to develop geometric feeling by way of detailed study of elementary mappings. At the 88me time we try to stress ri&or in geometric thinking, to the point where the geometric imap b. ecomes th.e suide but not the found.... tion of reaa9J'jDg~ • .. - - " ' .'---_",.' - •..,. . . . ",'_ .-,-_••,•'•."••.,,' . -•" . . -.."", _,-_-,_"w, --"" "'~_" .-...... .. •-.. . . . :.,.- ' " -,., .,' .' . . • . • • • . . • . . _.;:._' .~_ • • •_• c_ . . • • _.· _~ .......•.•.•.•. ~'.:,.'..-;,.~', •• CO..PLEX ANALYSIS :L ELEMENTARY POINT SET TOPOLOGY The brancb of mathematics which goes under the name of t01W/ogy is con· cerned with all questions directly or indirectly related to continuity. The term is traditionally used in a very wide sense and without strict limits. Topological considerations are extremely important for the foundation of tbe study of analytic functions, and the first systematic study of topology was motivated by this need. The logical foundations of set theory belong to another discipline. Our approach will be quite naive, in keeping with the fact that all our applications will be to very familiar objects. In this limited framework no logical paradoxes can occur. 1.1. Sets ..nd Elements. In our langnage a lUll will be a collection of identifiable objects, its ekments. The reader is familiar with the notation '" e X which expresses that", is an element of X (as a rule we denote sets by capital letters and elements by smallletteJ:s). Two sets are equal if and only if they have the some elements. X is a subset of Y if every ele- ment of X is also an element of Y, and this relationship is indicated by X C Yor Y J X (we do not exclude the possibility that X = Y). The empty set is denoted by !If. A set can be referred to as a space, and an element as a poiR!. Subsets of a given space are usually called point sets. This lends a geometric lIavor to the language, but should not be taken too literally. For instance, we shall have occasion t., consider spaces whose elements a.... functions; in that case a "point" is a function. The imerlNJCtioo of two sets X and Y, denoted by X () Y, is formed by all points which are elements of both X and Y. The unioo X U Y con· sists of all point.s which are elements of either X or Y, including those which are elements of both. One can of course form' and union of arbitrary collections of sets, whether finite or infinite in number. The compl£menl of a set X consists of all points which are not in X; it will be denoted by ~X. We note that,the complement depends on the totality of points under consideration. For instance, a ""t of leal numbers has one complement with respect to the real line and another with respect to the complex plane. More generally, if X C Y we can consider the relative complement Y ~ X which oon$ists of all points that are in Y but not in X (we find it clearer to lise this notation only when X C Y)• . It is helpful to keep in mind the diatnDUtive lowl XU (Y ("\ Z) = (X V Y) {"\ (X U Z) X () (Y U Z) = (X ("\ Y) U (X () Z) . ._...' - ANALYTIC FUNCTIONt At MAPPING' " and the De Mtwgan laws ~(X V Y) = ~X fi-Y -(X fi y) = -X V -Yo These are purely logirAl identities, and they have obvious generalizations to arbitrary collections of sets. 1.:1. Me",", Spaces. For all considerations of limits and continuity it is essential to give a precise meaning to the terms "sufficiently near" and "arbitrarily near." In the spaces Rand C of real and complex numbers, respectively, such nearness can be expressed by a quantitative condition Iz - yl <.. For iostance, to say that. a set X contains all :t 1lU1/lIMntiti near to y meaDS that there exists an $ > 0 suoh· that z E X whenever Iz :"'111 < II. Similarly, X c01llains poifItIJ a,bil,ariltlma, to tI if to every • > 0 there exists an z E X such that Iz - 111 < •. What we need to describe neam""" in quantitative terms is obviously a d~ d(z,y) between any two points. We say that a set S is a marie 3pQU if there is defined, for every pair Z E S, 11 E 8, II nC)Dnegative real num- ber d{Z,II) in such a way that the followiog conditions are fulfilled: 1. d(z,y) = 0 if and only if x = y. 2. dCy,z) = d{z,y). + 3. d{z,.) ~ d(x,y) dCy,')' The last condition is the triangle inequality. For instance, R and C are metric spaces with d(x,y) = 13: - til. The n-dimensional euclidean space Re is the set of real ...tuples x = (x" • . . ,xe) ! • with a distance defined by d(3:,y)' = (Xi - y.)'. We rcr&! tbat we 1 have defined a distanee in the extended complex plMe by (8ee Chap. I, Sec. 2.~); Bioce this represents the euclidean dista.nee between the stereoglapbic images on the Riemann sphere, the triangle inequ.ljty is obviously f"lIi1Ied. An example of a function space is given by C[a,bl, the set of all continuous functions defined on the ioterval a ;:ii z ~ h. It beOOmes a metric space if we define dist··"ee by dC/,g) = max If(z) - g(x) I· In '. of distance, we introduce the following termjnology: For any. > 0 and~YII e8,.thB!I8~ B(V,4}ota,U x e~with II(z,l/) < 'is r&!ed "', . ," ., ,.J. " , "~: __ ' : . • _. _",,", .' _.. , ", _. 52 CO"~LI!X ANALYSIS the baa with center y and radius 8. It is also refeIled to as the oS-neighbor- hood of y. The general definition of neighborhood is as follows: A Bet N C 8 ia called II neighborhood of YES 'f it contain8 • In other words, a neighborhood of y is a set which contains all points sufficiently nAIIT to y. We use the notion of neighborhood to define open, set: Definition 2. A sel is open if it ia II neighborhood of each of iU element8. The definition is interpreted to mean that the empty eet is open (the condition is heeanee the set has no elements).. The following is an innnediate consequence of the triangle inequality: Every baa ia an IJPell Bet. Indeed, if Z E B(y,Ii), then 6' = a - d(y,z) > O. The triangle in- equality shows that B(z,f,) C B(y,a), for d(x,z) < 8' gives d(x,y) < 8' + d(y,z) '" o. Hence B(y,a) is a neighborhood of z, and since Z was any point in B(y,f) we conclude that B(y,a) is an open set. For gJ eater em- phasis a ball is sometimes referred to as ILIl open ball. to distinguish it from the cklsed baa fonned by all XES with d(x,y) :i! 8. In the complex plane B(z.,f) is an OpM diak with center z. and radius 8; it consists of all complex numbers z which satisfy the strict inequality Iz - z.1 < o. We have ju$ proved that it is an open set, and the reader is urged to interpret the proof in geometric tefillS. The complement of an open set is said to be ckI81ld. In any metric space the empty set and the whole space are at the same time open and closed, and there may be other sets with the same property. The following properties of open and closed setH are fundamental: The intersecliun of II finite number of open 8e18 i$ open. The uniun of any collectiun of open seiIt i8 open. The union of II finite number of cl08ed sels is clolted. The inter8ectiun of any collection of closed 8818 is closed. The proofe are so obvious that they can be left to the reader. It should be noted that the last two statements follow from the first two by use of the De Morgan laws. There are many tenus in common usage which are directly related to the idea of open sets. A complete list would be more confusing than helpful, and we shall limit ourselves to the following: i!'lel'ior, cklBurll, ANA~YTIC FUNCTIONS AS MAPPINGS b&ufldary, ez!eriM. (i) The interior of a set X is the largest open set contained in X. It exists, for it may be characterized as the union of all open sets ex. It can also be described as the set of all points of which X is a neighborhood. We denote it by Int X. (ii) The closure of X is the smallest closed set which contains X, or the intersection of all closed sets "JX. A point belongs to the closure of X if and only if all its neighborhoods intel/loot X. Tbe clOlllll'e is usually denoted by X-, infrequently by Cl X. (iii) The boundary of X is the closure minus the interior. A point belongs to the boundary if and only if all its neighborhoods intersect both X and ~X. Notation: Bd X or ax. (iv) The exterior of X is the interior of ~X. It is also the oomph,. ment of ~e closure. As such it can be denoted by ~X-.. e e Observe that Int X X X- and that X is open if Int X = X, e e e closed if X- = X. Also, X YimplieslntX lnt Y,X- Y-. For added convenience we shall alSl) introduce the notions of iaolated point and auumulatioo poi"'. We say that :I: ~ X is an isolated point of X if :t has a neighborhood whose intersection with X reduces to the point x. An accumulation point is a point of X- which is not an isolated point. It is clear that z is an accumulation point of X if and ouly if every neighborhood of x contains infinitely many points from X. EXERCISES L If S is a metric space with distance function <1(:1:,1/), show that 8 + with the distance function a(z,1/) = d(:I:,1/)/[1 d(x,1/)] is also a metric space. The latter space is bounded in the sense that all distances lie under a fixed bound. 2. Suppose that there are given two distance functioll8 d(z,1/) and tlt(x,1/) on the same space S. They are said to be equivalent if they deter· mine the same open sets: Show that d and il, are equivalent if to every e > 0 there exists a & > 0 such that d(z,y) < & implies d,(z,1/) < " and vice versa. Verify that trus condition is fulfilled in the preceding exercise. .. Show by strict application of the definition that the closure of Iz - z.\ < aislz - z~ ~ a. .. If X is the set of complex nnmbers whose real and parts are rational, what is Int X, X.,., aX? s. It is sometimes typographically simpler to write X' for ~X. With this notation, how is X'-' related to X? Show that X-'-'-'-' = X-'-'. .. A set is said to be discrete if all its points are isolated. Show that a ""tin discrete. . ".,- .-. R -.- -,o' r C . is c'o-u" nta.ble. 7. Shol" that the. accmriulation points of . , any set form a closed set. . . <...;,.: ,'.-."--.' ,-" - -,",-"~" • . :'.'~ - ',-:- COMPLEX ANALY81S 1.1. Conneeledneo. If E is any nonempty subset of a metric space S ;Fe may consider E as a metric space in its own right with the same dis- tance function d(:Z:,lI) all on all of S. Neighborhoods and open sets on E are defined as on any metric space, but an open seton E need not be open when regarded as a subset of S. To avoid confusion neighborhoods and open sets onE are often to as relative neighborhoods and relatively open sets. As an example, if we regard the closed interval 0 ;:it :z: ;$ 1 as a subspace of R, then the semiclosed interval O·:::i! :z: < 1 is relatively open, but not open in R. Henceforth, when we say that a subset E has some specific topological property, we shall always mean that it has this prop- erty as a 8Ilbspace, and its 8ilbspace topology is called the relative topology. Intuitively speaking, a space is connected if it consists of a Bingle piece. This is meaningness unless we define the statement in terms of The easiest way is to give a negative characterisation: 8 i8 not if tllMe uiat3 II pa,tition 8 = A V B into Opel~ aOOset3 A lind B. It is understood that A and B are di8joint and nfnIemPIy. The connected- neN of a space is often used in the following manner: Suppose that we are able to construct two complementary open subsets A and B of 8; if S is conn'Mlted, we may conclude that either A or B is empty. A subset E C S is said to be connected if it is connected in the rela- tive topology. At the risk of being pedantic we repeat: Definition 3. A aOOHt of II 'IMtric. $p4C6 i8 conneded if it cannot be r6Fe- 86fIted (JI the union of tfDO disjoint ,ellltively open 8eta none of fDhillll. i8 6"'pl'/l. If E is open,· a subset of E is relatively open if and only if it ie-open.. Similarly, if E is closed, relatively closed means the same as closed. We can therefore state: An open ad i8 connected if it cannot be decomposed into lIDO opm ad8, and II closed 8e1 i. connected if it C/lnnot be decompow ink) tlDO c/ow Beta. Again, none of the sets is allowed to be empty. Trivi&! examples of connected sets are the empty set and any set that consists of a single point. In the case of the real line it is possible to name all connected sets.· The most important result is that the whole line is connected, lind this is indeed one of the fundamental properties of the real-number system. An i1lterval is defined by an ineqllality of one of the four types: + .. a < ~ < b, a;:it ~ < b, a < ~ ~ b, II. ~ :z: ;:it b. t For· a - - GO or b - this includes the semi-infinite interv&!s and the whole line. , t We denote open iRte.vals by (0,6) and cloBed intervals by [...61. Another common practice i•.to denote open intervals by Ia,b[ and 8emic1.....l intervals by ]a,bl or 1..,6[, It is alwa undentood that" < 6. . ANALYTIC FUNCTIONS AS MAPPINGS 55 Theorem 1. The nonemply connuted ~ oj the real line or, the intervaU. We reproduce one of the classic'" proofs, based on the fact that any monotone sequence has a finite or infinite limit. Sup~ that the rett.lline R is represented 88 the union R ~ A V B of two disjoint closed sets. I f neither is empty we rAD find a' E A and b, E B; we may !l8SUme that a' < bt. We bisect the interval (o"b,) and note that one of the two halves has its left end point in A and its right end point in B. We denots this interv'" by (a2,b.) and continue the process indefinitely. In this way we obtain a sequence of nested inte'."'s (a.,b.) with a. E A, b. E B. The sequences (a.1 and (b.1 have a common limit c. Since A and B are closed c would have to be a common point of A and B. This contradiction shows that either A or B is empty, and hence R'is connected. With minor modifications the same proof applies to any interval. Before proving the converse we make an important remark. Let E be an arbitrary subset of R and call a a lmDer bound of E if co ~ It for all :u E. Consider the set A of all lower bounds. It is evident that the complement of A is open. As to A itself it is easily seen that A is open whenever it does not contain any largest number. Because the line i. connected, A· and its complement cannot both be open unless one of them is empty. There are thus three possibilities: either A is empty, A con- tains a largest number, or A is the whole line. The largest number a of A, if it exists, is called the greatest IOt.Der bound of E; it is commonly denoted as g.l.b. x or inf x for x E E. If A is empty, we ""glee to set + a ~ - 00, and if A is the whole line we set a ~ 00 • With this con- vention every set of real numbers has a uniquely determined greatest lower bound; it is clear that a ~·+oo if and only if the set E is empty. The 1to8! upper bound, denoted 118 tu.b. x or sup x for x E E, is defined in a corresponding manner.t Returning to the proof, we assume that E is a connected set with the greatest lower bound a and the least upper bound b. All points of E lie between II and II, limits included. Suppose that a point ~ from the open interval (11,11) did not belong to E. Then the open sets defined by:t: < ~ and:t: > f cover E, and because E is connected, one of them must fail to meet E. Suppose, for instance, that no point of E lies to the left of t· Then ~ would he a lower bound, in contradiction with the fact that II is the greatest lower bound. The opposite assumption would lead to a similar contndietiol), and we cooclude that ~ must belong to E. It follows that E is an open, cJ.08ed, or ... interval with. the end points II and b; the are aDd fi . , . . . __ II .;,' ..;:~., ~. ",.,'- to be1nCluded. --,.- II COMPLEX ANALYSIS In the course of the proof we have introduced the notions of gteatest lower bound and least upper bound. If the set is closed and if the bounds are finite, they must helong to the set, in which cere they are called the minimum and the maximum. In order to he sure that the bounds are finite we must know that the set is not empty and that there is some finite lower bound and some finite upper bound. In other words, the eet must lie in a finite interval; such a set is said to be botmded. We have proved: Theorem 2. Any cloaed and bounded _ p t y 3d of real numbers htJg a minimum and a mazimum. The structure of connected sets in the plane is not nearly 80 simple as in the MOO of the line, but the foUowing charact.eJUation of open connected sete contains essentially all the information we shall need. Theor..m 3. A - p l y open 36/ in tile plane ill conmcted if and only if any tVlO of ita points can be joined by a polygon which lies in tile 8Ot. The notion of a joining polygon is so simple that we need not give a formal definition. We prove first that the condition is neceBBarY. Let A he an open con- nected oot, and choose a point a fA. We denote by A, the subset of A whose pointe can be joined to a by polygons in A, and by A. the subset whOile pointe <,annot he 80 joined. Let us prove that A, and A. are both open. First, if a, fA, there e. jsis a neighborhood \z - a,\ < a contained in A. AU points in this neighborhood can be joined to a, by a line seg- ment, and from there to a by a polygon. Hence the whole neighborhood is contained in A " and A, is open. Secondly, if a. fAt, let \z - a.[ < • be a neighborhood contained in A. If a point in this neighborhood could be joined to a by a polygon, then a. could be joined to this point by a line segment, and from there to a. This is contrary to the definition of A., and we conclude that A. is open. Since A was connected either A, or A. must be empty. But A, contains the point a; hence A. is empty, and all pointe can be joined to a. Finally, any two points in A can he joined by way of a, and we have proved that the condition is neceaeary. For we remark that it is even poaeible to join any t.to points by a polygon whose aides are parallel to the coordinate axes. The proof is the same. • .In order to prove the sufficiencY we MOUrne that A has a representa- tion A = A. V Alas the union of two disjoint open sete. ChOOllea, fA" as E A. and suppose that these pointe can be joined by a jIOlygon in A. . , .' , ANALYTIC FUNCTIONS AS .APPINGS II One of the sides of the polygon must then join a point in A, to a point in A" and for this re'son it is ,mffieient to consider the case where a, and a. are joined by a line segment. This segment has a pa.rametric representa- tion z = a, + I(a, - a,) where t nms through the interval 0 :;; t :;; 1. The subsets of the interval 0 < t < 1 which conespond to points in At and A., respectively, are evidently open, disjoint, and nonvoid. This contradicts the conneetednel!8 of the interval, and we have proved that the condition of the theorem is sufficient. The theorem generalizes easily to Ro and Co. Definition 4. A fI01Iem¢lI connected open set i8 called a regiqn. By Theorem 3 the whole plane, an open disk I.. - al < p, and a half plane are regions. The same is true of any 6-neighborhood in R". A region is the-more dimensional analogue of an open interval. The closure of a region is called a cloaed region. It should be observed that di1ferent regions may have the same closure. It happens frequently that we have to analyze the structure of sets which are defined very implicitly, for instance in the course of a proof. In such cases the first step is to decompoee the set into its maximal connected comptmenl3. As the name indicates, a component of a set is a connected subset which is not contained in any larger connected subset. If E is the given set, consider a point 0 E E and let C(a) denote the union of aU connected subsets of E that contain a. Then C(o) is sure to contain a, for the set consisting of the single point a is connected. If we .can show that C(a) is connected, then it is a maximal connected set, in other words a component. It would follow, moreover, that any two components are either disjoint or identical, which is preeisely what we want to prove. Indeed, if e E C(o) (\ C(b), then C(o) C C(e) by the definition of C(e) aod the conneotedness of C(o). Hence a" C(e) , and by the same reasoning C(e) C C(a), so that in fact C(/I) = C(e). Similarly C(b) = C(e), and consequently C(a) = C(b). We call C(/I) the component of a. Suppose that C(a) were not conneeted. Then we could find relatively open sets A, B '" JJ Buch that C(a) - A V B, A (\ B = S. We may 8SS'llnp. that II" A while B contains a point b. Since b EC(/I) there is a connected set E. C B wbinb contains II aod b. The representation B, .. (E. (\ A) V (8. f"\ B) W(Mlid be a decomposition into relat.iveIy open subsets, and B.n A, beE.f"\ B Dei~ would be emptY'. 'tbis is a and we oonclude that .... ,-- ,. . . :. ',. ""--'" .... ,.," '., . . . . . . . . .'...,... ,:, . ',,:'._., .."--' ",,","-,'- :,.,_.. '_'::.:'.'_'_"":"._. ___ ,._H , .. ..; .. ,.,\,._ .. ,,~, ... C;O...LElI ANAL YS .. Theorem 5. In R" the components of any opIln Mare opIln. This is a CODBequence of the fact that the ~neighborhoods in R· are connected. Consider a ! C(a) C E. If E is open it contains B(a,8) and bec&l1!!R B(a,8) is connected B(a,&) C C(a). Hence C(a) is open. A little more gene~aIly the is true for any space S which is looally connected. By this we mean that any neighborhood of a point a contains a connected neighborhood of a. The proof is left to the reader. In the case of R" we can conclude, furthermore, that the number of components is countable. To see this we observe that every open set must contain a point with rational coordinates. The set of points with rational coordinates is countable, and may thWl be exprewted as a sequence (p.l. For each component C(a), determine the smallest 1c such that p. E O(a). To difierent component.s correspond difierent 1c. We con- clude that the componentS are in one-to-one correspondence with a subset of the natural numbers, and consequently the set of components is COlmtable. For instance, ~ opIln 3Ubaet uJ R ia a countable "nion oj dia,ioint interval,. Again, it is possible to analyze the proof and thereby arrive at a more general result. We shall say that a set E is denae in S if E- = 8, and we sball say that a metric space is separabk if there exists a countable s..bset which is dense in S. We are led to the following result: In a locally 8epaf'abk spaC6 we'/I opIln &Ill is a CQUfIIabk union oj di8joint regions. EXEaCISES L If xes, show that the relatively open (closed) subsets of X are precisely those sets that can be expressed as the intersection of X with an open (closed) snbset of S. 2.. Show t bat the union of two regions is a region if and only if they have a common point. S. Prove that the closure of a connected set is connected. 4.. Let A be the set of points (x,1/) E R" with x = 0, 1111 ~ I, and let B he the set with x> 0, 1/ = Bin l/x. Is A V B connected? s. Let Ebe the set of points (x,/I) E Rlsuch that 0 ~ x ~. 1 and either y = 0 or y = l/n for some positive ~teger n. What are the com- ponents of E? Are they all dosed? Are they relatively open? Verify that E is not locally connected. '" Prove that the components of a dosed set are closed (use Ex. 3). 7. A set i8 said to be diM:rele if all its Points are isolated. Sbo" that a set in a separable metric space is countable. ANAL YTIC FUNCTIONS AS IIAPPINGS 511 1.4. Compactne... The notions of convergent sequences and Cauchy sequences are obviously meaningful in any metric space. Indeed, we would say that x...... x if d(x.,x) ..... 0, and we would say that {x. I is a Cauchy sequence if d(x.,x..) ..... 0 as n and m tend to 00. It is clear that every convergent sequence is a Cauchy sequence. For Band C we have proved the converse, namely that every Cauchy sequence is convergent (Chap. 2, See. 2.1), and it is not hard to see that this property carries over to any B'. In view of its importance the property deserves a special name. Ddinition 5. A metric space i8 said wbe comp/.u 'f wery Cauchy sequence iB convergent. A subset i8 complete if it is complete when regarded as a subspace. The reader will'find no difficulty in proving that a compleU BUb.., of a mell ie space iB closed, and that a tWsed BUb.oet of a compleU 8fHJI'Il iB compkle. We 8haJI now introduce the stronger concept of compacl'IIU'. It is stronger than completeness in the sense that every compact space Of set is complete, but not conversely, As a matter of fact it will turn out that the compact subsets of Band C are the closed bounded sets. In view of this result it would be possible to dispense with the notion of compactness, at least for the purposes of this book, but this would be unwise, for it would mean shutting our eye8 to the most striking property of bounded and closed sets of real or complex numbers. The outcome would he that we would have to repeat essentially the 8ame proof in many different connections. There are several equivalent characterizations of compactness, and it is a matter of taste which one to choose as definition. Whatever we do the uninitiated reader will feel somewhat hewildered, for he will not be able to di.seern the purpose of the definition, This is not surprising, for it took a whole generation of mathematicians to sgrnsensus of present opinion is that it is hest to focus the attention on the different ways in which a given set can he covered by open sets. Let us say that a collection of open sets is an open COVer1'11f/ of a set X if X is contained in the union of the open sets. A IlUbcoveri'llf/ is a 8ubcolIection with the same property, and a finite covering is one that consists of a finite number of sets. The definition of compactness reads: Definition 6. A Bel X iB compacl 'if and only if every Opll/l COIJer1'11f/ of X conl4im a finite BUbcovering. . '.- • ..... '.- . ..J. ',' .. ' ..... :- .. .. ' , ' . . - .-"., .', ,.: : .. , ,-.' ' . . ':. ',,' -, .. ".'.~."). '•-.·,1.>-, . .. : ",'- ::;'-'. " ,','" IG COMPLEX ANALYSIS and the covering is by open sets of S. .But if U is an open set in S, then U (\ X is an open subset of X (a relatively open set), and conversely every open subset of X can be expressed in this fonn (Sec. 1.3, Ex. 1). For this reason it makes no difference whether we formulate the definition for a full SpllOO or for a subset. The property in the definition is frequently referred to 88 the HeifUlBorel properly. Its importance lies in the fact that many proofs become particularly simple when formulated in terms of open coverings. We prove first that every compact SpllOO is complete. Suppose that X is compact, and let Ix.) be a Cauchy sequence in X. If y is not the limit of Ix.l there exists an , > 0 such that d(x.,y) > 2, for infinitely many ... Determine ... suchthatd(x..,x.) < dorm,n ~ no. Wechoose a fixed n ~ n.forwhichd(x.,y) > 2.. Thcnd(x..,y) ~ d(x.,y) - d(x..,x.) > • for all m ~ no. It follows that the .-neighborhoOd H(y,e) contains only finitely many x. (better: contains x. only for finitely many n). Consider now the collection of aU open sets U which contain only finitely many x.. If Ix.l is not convergent, it follows by the preceding reasoning that this collection is an open covering of X. Therefore it muet contain a finite 8ubcovering, formed by U ... ,UN. But that is clearly impossible, for since each U, contains" only finitely many x. it would follow that the given sequence is finite. Secondly, a compact set is necessarily bounded (a metric space is bounded if all distances lie under a finite bound). To see this, choose a point Xo and consider all balls B(x.,r}. They fonll an open covering of X, and if X is compact, it contains It finite subcovering; in other words, X C B(xo,r,) U ... U B(x.,r..), which means the same as XC H(xo,r) + with r = max (r" , .•, r..). For any X,y E X it follows that d(x,y) ;:;; d(x,x.) d(y,x.) < 2r, and we have proved· that X is bounded. But boundedn".,.. is not all we can prove. It is convenient to define a stronger property called total boundedne88: .. D,.6nitlon 7. A sel X is totally bounded iI, I'" every e > 0, X can be OOfI61'ed by finitely """,y ball.! 01 radim e. This is certainly true of any compact set. For the collection of all balls of radius e is an open covering, and the compactness implies that we can Belect finitely many that cover X. We observe that a. totslly bounded set is necessarily bounded, for i(X C B(xl, e) U ... U B(x.., e), + then any two poin~ of X have a distance <2. max d(x"xj). (The preceding proof that any compact set is bounded becomes redundant.) We have already proved one part of the following theorem: Theorem 6. A BIll ia cumpacl if and tmly if it ia complete and IotIJIly botmded. "N"LYTIC FUNCTIONS " • •"I'PINOS To prove the other part, 888lU1le that the metric space S is complete and totally bounded. Suppose that there exists an open covering which does not contain any finite subcovering. Write t. = 2-. We know that S can be covered by finitely many B(x,o,). If each had a finite subeovering, the same would be true of S; hence there exists a B(x".,) which does not admit a finite subeovering. Because B(x.,E,) is itself totally bounded we can find an x. E B(x,,!!) such that B(x"e.) has no finite subcovering.t It is clear how to continue the construction: we obtain a sequence x. with the property that B(x., E.) has no finite sub- covering and X.+1 E B(x., ••). The second property implies d(x.,x....,) < •• and hence d(x.,x.+p ) < o. + .....' + ... + .""0-1 < 2-·+1. It follows that x. is a Cauchy sequence. It converges to a limit 1/, and this 1/ belongs to one of the open sets U in the given covering. Because U is open, it contains a ball B(I/,8). Choose n 80 large that d(x..,I/) < 8/2 and •• < &/2. Then B(z., ••) C Bty,&), for d(x,x.) < e. implies d(x,l/) ;:;; d(x,z.) + d(x.,I/) < 3. Therefore B(x.,e.) admits a finite subeovering, namely by the single set U. This is a contradiction, and we conclude that S has the Heine-Borel property. Coroll ery. A IlUb.eI of R or C is compact if and onll/ if it is closed and bounMa. We have already mentioned this particular consequence. In One direction the conclusion is immediate: We know that a compact set is bounded and complete; but R and C are complete, and complete subsets of a complete space are closed. For the opposite conclusion we need to show that every bounded set in R or C is totally bounded. Let I1B take the case of C. If X is bounded it is contained in a disk, and hence in a square. The I!quare can be subdivided into a finite number of I!quares with arbitrarily small side, aud the squares can in turn be covered by disks with arbitrarily small radius. This proves that X is totally bounded, except for a small point that should not be glossed over. When Definition x e s 7 is applied to a subset it is slightly IUDbiguous, for it is not clear whether the ...neighborhoods should be with respect to X or with respect to S; that is, it is not clear whether we reqnire their centers to lie on X. It happens that this is of no avail. In fact, suppose that we have covered X by ...neighborhoods whoee centers do not nece:marily lie on X. If such a neighborhood does not meet X it is superfluous, and can be dropped. If it does a point from X, then we caD replace it by a 2a-Deighborhood around that point, and ...e obtain a finite covering by 2..neighborhoods with centers on X. F!lrthi8 reason the ambiguity is only apparent, and our proof that boundedqbeet/l , ," of. C . .are totally o o. u n d ' e d is valid. t Here we eM n,hl tbe INt that any subset of a totally bonodtd set is totally bounded. , The ieael .. ~ PO" Ihia. ... . , -.- '""," '-." ',',"'''I',- ':'-'.,i_.-''.;".";, " "" . " :,:~ ",' ,. ·"·'t·",,"., -,",", : '. ""' ~'.'-" ,~.;:."_':..-,..: ' . ," ,'-,' , ," ' . ''-. . :-', - '''.: . : '-,.,,,",.,-.,'., . ,.' -;, . ,.- .. ~;, . ''-- - .. . .:,:"" ,..:. " ' - .-, - ,. ~~ 12 COMPLEX ANALYSIS There is a third characterization of oompact sets. It deals with the notion of limit paint (sometimes called c/usler value): We say that 11 is a limit point of the sequence Ix.l if there exists a subsequence (x.. l that converges to 11. A limit point is almost the Bame as an accumulation point of the set formed by the points x., except that a sequence permits repeti- tions of the ~ame point. If 11 is a limit point, every neighborhood of 11 contains infinitely many x.. The converse is also true. Indeed, suppose that n -+ O. If every B(y, £.) oontains infinitely many x. we can choose subscripts no, by induction, in such a way that x •• E B(y, e.) and n",., > no. It is clear that 1z..1 oonverges to y. Theorem 7. A flIIJtrU! &pIJCe Ur co,npad if IJtId only if every infinite aeq~ Iuu a limit point. This theorem is usually referred to as The original formulation WIU! that every bounded sequence of oomplex numbers has a convergent subsequence. It came to be recognized 8B an important theorem precisely becal'''e of the role it plays in the theory of analytic functions. The first part of the proof is a repetition of an earlier argument. If 11 is not a limit point of Ix.1 it has a neighborhood which contains only finitely mAny x. (abbreviated version of the conect phrase). If there were nO limit points the open sets containing only finitely many x. would form an open covering. In the compact ease we could select a finite subcover- ing, and it would follow that the sequence is finite. The previous time we used this reasoning was to prove that a compact space is complete. We showed in that every sequence has a limit point, and then we observed that a Oauchy sequence With a limit point is necessarily con- vergent. For strict economy of thought it would thus have been better to prove Theorem 7 before Theorem 6, but we prefened to emphasize the importance of total boundedness as early as possible. It remains to prove the converse. In the first place it is clear that the Bolzano-WeieistnL"8 property implies colDpletene88. Indeed, we just pointed out that a Cauchy sequence with a limit point must be convergent. Suppose now that the space is not totally bounded. Then there exists an • > 0 such that the space cannot be covered·by finitely many ""neighbor- hoods. We construct a sequence (x.1 ""follows: x. is arbitrary, and when ,x. X" ••• have been selected we cho,Qse X.+. 80 that it does Iiot lie in B(x.,e) V . . . V B(x.,e). This is always pollSible because these neigh- borh(lods do not cQver the whole space. But it is clear that (x.I has no convergent subsequence, for d(x..,x.) > • for all m and fl. We conclude that the Bolzano-Weierstrass property implies total boundedness. In view of Theorem 6 that is what we had to prove. .. . . .._-.,~ '."- . """LYTIC FUIICTIOII. " • •"""IIIGB 13 The reader should re8eet on the fact that we have exhibited three ch8l'acteruations of compactne:-'" whose logical equiv&lence is not at all trivial. It should be clear that results of this kind are particularly v&luable for the PUIpOse of presenting proofs as concisely as possible. . EXERCISES 1. Give an alternate proof of the fact that every bounded sequence of complex numbers has a convergent subsequence (for instance by use of the limes inferior). 2. Show that the Heine-Borel property can &leo be expretl8ed in the following manner: Every collection of closed sets with an empty intersection contains a finite 8ubcollection with empty intenlection. .. Use compactness to prove that a closed bounded set of real nllmben! has a maximum. ... If E, ::> E. ::> E.:J • . . is a decreasing sequence of nonempty • compact sets, then the interse.tion r\ E. is not I'mpty (Cantor's lemma). I Show by exl'mple that this nlled not be true if the sets are merely closed. 50 Let S be the set of all sequences % = (%.I of real num bers such that only a finite number ofthe %. are ... O. Defined(%,y) = max 1%. - y.l. Ia the space complete? Show that the ~neigbhorhoods are not totally bounded. 1.ti. Continrcou.o Functro.... We shall consider functions/which are defined on a metric apace S and have values in anotber metric space S'. Functions are also referred to as mappinga: we say that / ma}l8 B into B', and we write/:B .... B'. Naturally, we shall be mainly concerned with real or complex-valued functions; occasicoually the latter are allowed to take values in the extended complex plane, ordinary distance being replaced by distAnce on the Riemann spbere. The space S is the dl>lll4in of.the function. We are of free to consider functions / whose domain is only a subset of B, in which ease the domain is regarded as a subspace. In moat it is safe to slur over the distiDction: a function on B and its restriction to a subset are usually denoted by tbe same symbol. If XC Bthe set of all values/(z) for %f B is called the itrUlf18 of X under I, and it is denoted by I(X). The inverse itrUlf18rl{X') oC X' C B' conoistl!of all %e Bsuch that/(%) eX'. Observe that/U-,{X'» C X', and rIU(X» ::> X. The definition of a continuous function needs praetically no modifica.- ~on: / is continooWl at a if to every & > 0 there exists a > 0 such that ~(z,a) < a implies d'U(z),/(a» <.. We are mainly concelD~ with rnnetiollS thllt'arecontiDuoua at all points in the 'domain of definition. '. ...... '--, ..... "'._" -,.' " ", . .-'.,,"- " • . ..:",·"·,,',·'.~'·,,~-.i·,-" " ' . , ' . , ;,,-·.,!:·,'-~'-":"'-:1'_,·' . . ' . -' '•__.I-.,"--!:' . -..'f..:.,,-1"-":".'". , ,, ' , . , - . '..... . , '..".-. . .-.: . :... : - . _'. ," .••. ':- , ","'.' , '--,:,."r,...:. "::i ::~,,;:~~~> .(;~::-~,:..~:_:".,:.:):~~'.:.._~-:.:..,;..(';:.<'' y < I} which is neither open nor closed. In tills eXll.lllple I(R) fails to be closed heeaWle R is not compact. In fact. the following is true: TheorelD 8. Untkr a comintuma fJUlpping tIuJ image 01 every compact set ., and consequently clOMl. Suppose that I is defined and continuous on the compact set X. Consider a covering of I(x) by open sets U. The inverse images I-'(U) are open and form a covering of X. Because X is compact we "an select a finite sUbcovering: X C I-'(U,) U ... U I-I(U.). It follows that f(X) CU, V • . • V U.., and we have proved that I(x) is compact. Corollary. A contin1WllB real-valued function on a compad set hall a marl- mum and a minimum. . . .. The image is a closed bounded subset of R. The existence of a maximum and a minimum follows by Theorem 2. TheoreJD 9. Under a continooUB mapping the imnge 01 anll connected 8el ia connected. We may assume that I is defined and continuous on the whole space 8, and that f(8) is ell of 8'. Suppose that 8' = A V B where A a.nd B are open and disjoint. Then S = f-I(A) U f-'(B) is a of S as a union of disjoint open sets. If 8 is connected either I-I(A) = 0 or l-I (B) = 0, and hence A=-O or B.~ O. We conclude that 8' is connected. A typical application is the assertion that a real-valued function which is continuous and never zero on·a connected set is either always positive or always negative. In fact, the image is connected, and hence an interval. But an interval which cont.ains positive and negative .011111- ANALYTIC fUNCTIONS A. MAPPINGS bers also contains zero. A mapping f:8 ..... S' is said to be one to one if I(x) = fey) only for x = Yi it is said to be onto if f(8) = 8'.t A mapping with both these properties has an inverse j'-', defined on 8'i it eatisfiesf-'(f(x» .. xand f(j'-'(x'» - x'. In this situation, if f and f-1 are both continuous we say that f is a topoWgiool mapping or a 1wmeOlllorphi8m. A property of a set which is shared by all topological images is called a topological properly. For instance, we have proved that compactneJ!S and connecrednesa are topological properties (Theorems 8 and 9). In this connection it is per- haps u-eful to point out that the property of being an open subset is not topological. If X C 8 and Y C 8' and if X is homeomorphic to Y there is no resson why X and Y should he simultaneously open. It happens to be true if 8 .. 8' = R' (invarionce of the regirm), but this is a deep theorem that We shall not need. The notion of unilornt crmtinud1l will be in constant use. Quite generally, a condition is said to hold uniformly with respect to a parameter if it can he expwllJed by inequalities which do not involve the parameter. Accordingly, a function f is said to be uniforlll11l continuous on X if, to every • > 0, there exists a 3 > 0 such that d'(/(xl),J(z.» < • for all pairs (Xl,X.) with d(x,,x.) < 3. The emphasis is on the fact that ais not allowed to depend on x,. Theorem 10. On 0 ro.npact BBl every ronlinuotU function is uniforml1l contimroUB. The proof is typical of the way the Heine-Borel property can he used. Suppose thatf is continuous on II> compact set X. For every y • X there is II> baIl B(y,p) such that d'(f(x)J(1I» < ./2 for x • B(1I,p) i here p may depend on 11. Consider the covering of X by. the smal.ler balls B(1I,P/2). There exists a finite subcovering: X C B(1IbP,/2) V ... V B(1I.,p../2). Let 6 be the smallest of the numbers Pl/2, .•• , p../2, and suppose that d(x"z,) < I. There is a y. with d(Xl,II') < p./2, and we obtain d(x',II') < p./2 + 3 ;:iii 3.. Hence d'(/(z,).!(1I.» < ./2 and d'(f(x,),f(1I.» < ./2 80 that d'(f(xl),f(xi) < 1&8 desired. On sets which are not compact 80me continuous functions are uni- formly continuous and otbers are not. For instance, the function z is uniformly continuous on the whole complex plane, but the func\ion .' is not. t TheM IiDauiatico.ll7 c1um.,. ~ c." be ...pleced b7 .Jfi.. (for on. to one) and Iwjsc'" (lor onto). A-.;.q·Wn& with both paoperi.tM ia..ned. wjn'":" H COlfPLEX ANALYSIS EXlRella 1. Construct a topological mapping of the open diu Izi < 1 onto the whole pl&ne. 2. Prove that a subset of the real line which is topologically equiva- lent to an open interval is an open interval. (Consider the effect of removing a point.) S. Prove that every continuous one-to-one mapping of a compact space is topological. (Show that closed sets are mapped on closed sets.) 4. Let X and Y be compact sets in a complete metric space. Prove that there exist x E X, 1/ E Y such that d(x,y) is a minimum. s. Which of the foUowing functions are uniformly ,continuous on the whole real line: sin x, x ain x, x sin (x'), Ixlt sin x? 1.6. Topofogialf SPOC6IJ. It is not necessary, and not always con- venient, to express nearness in terUlsof distance. The observant reader will have noticed that most results in the preceding sections were fOrInulated in terms of open sets. True enough, we used distances to define open sets, but there is really no strong reason to do this. If we decide to consider the open sets as the primary obieets we must postulate axioms that they have to satisfy. The following axioms lead to the commonly accepted definition of a topologirol space: Definition 8.,' A topologirol space ill a set T togotlter with a colled:ian of its Btlb8et8, called open sel8. The JoUlllJ!ing coMtlilmB have 10 be fuljUled: (i) The emply set {lJ and !he whole apace Tare open w. (ii) The inUrsectWn of Imy lIDO upM Btll. ill an open set. " (iii) The union of on arbitrary co/led.ion oJ open set. ill an open set. We recognize at once that this terminology is consistent with our earlier definition of an open subset of a metric space. Indeed, properties (ii) and (iii) were strongly emphasized, and (i) is trivial. Closed sets are the complements of open sets, ....d it is immediately clear how to define interior, closure, boundary, and 80 on. Neigbbor- hoods could be avoided, but they are rather convenient: N is aneighbor- hood of x if there exists an open set U such that x E U and U C N. Connectedness was defined purely by means of open sets. ' Hence the definition carries over to topological spaces, and the theoreJfu! remain true. The Heine-Borel property is alai> one that deals only with open sets. Therefore it,makes perfect sense to speak of a compact topological space. However, Theorem 6 become. meaningless, and Theorem 7 becomes false: As a matter of fact, the first serious difficulty we encounter iz, with ANALYTIC fUNCTIONS AI MAPPINGS .7 say z convergent sequences. The definition is clear: we that %. -> if every neighborhood of % contains all but a finite number of the %.. But if %. -> % and %. -> Y we are not able to prove that % = y. This awkward situation is remedied by introducing a new axiom which characteri~e8 the topological space as a HausdtJrfl opace: Definition 9. A topological opace ill called a HautJorfl space if any two distinct points are contmned in di8;joint open seta. In other words, if % "" Y we require the existence of open sets U, V such that % f U, 1/ f V and U f\ ·V = /If. In the presence of this condition it is obvious that the limit of a convergent sequence is unique. We shaD never in this book have occasion to consider a space that is not a HaU8dorfr space. This is not the place to give examples of topologies that cannot be derived from a distance function. Such ell'ampies would necessarily be very complicated and would not fUrther the purposes of this book. The point is that it may be unnatural to introduce a distance in situations when one is not really need.ed. The reasOn for including this section has been to alert the reader that distances are dispensable. 2. CON FORMALITY We now return to our original setting where all functions and variables are restricted to real or complex numbers. The role of metric spaces will seem disproportionately small: all we actually need are some simple applications of connectedn""" and compactness. The whole section is mainly descriptive. It centers on the geometric consequences of the existence of a derivative. %.1. ,4,."" and ClDaed Cur...... The equation of an are "f in the plane is most conveniently given in parametric form % = %(t), y = yet) where t + runs through an interval .. :l!! t :;; P and :ret), 1/(t) are continuous func- tions. We can also use the complex notation. - !let) = :r(1) iy(t) which has several advantages. It is also customary to identify the are "f with the continuous mapping of [..,P]. When followiug this custom it is preferable to denote the mapping by. = "f(t). . Considered as a point set an are is the imagll of a closed finite interval under a continuous mapping. As such it is compact and connected. How- ever; an are is not merely a set of points, but very Lssentially also a 8UC- cession of points, ordered by inczening values of the parameter. Jf a n I on ::Ii d/J'._, eMing -. then ". function t = a ( ,-" . " , ( T ) ~ .-- , . maps ·the' ,III' ." - an me siuno,tcee-. nesaiol n...i:c:ifli-p.o.'.m:. :It,istJa', on s" to ~ ' .. :i a(t) . ._. . ._. ," ,", ,,-' .....-......,". -, -;'_":, '_' ", • ,j:j• '-',"-'-. , .•:- .. -.- - .,,' : "-," -.. -.. __. __ . _'_: :' -"-- '-•. "-. ... --,~,>.,~~", " " , . , . . - ·.',,'.,.-~._..:.w.f,.~.l'",·>.'".,-~.".f.'"'_:,_•'. ~--.,-.~-.,. '•, _,"--_n,..,."."'..~,'_~_,I' "' £. '.."..'... -"~,':~'..,....,...:,.....,"F._.,~•'•,.......:...."-..'..-'.'..'.,.~'"-~I •.".-..',~.~-"--c·"-"';'':-"''I"n,,-:';. . . ','' II COIIPLEX ANALYIIS We say that the first equation arises from the second by a change 01 parame- ter. The change is rever8ihle if and only if (1)(,,) is strictly increasing. For + instance, the equation 11 = t' it", 0 ~ t ;;; 1 arises by a reversible of parameter from the equation z = t + 11.', 0 ;;; t :iii 1. A change of pa.rametric interval (a,{J) can always be brought about by a l i _ + of parameter, which is one of the fonn t = 111' b, a > O. Logically, the simplest course is to consider two &l'CS as difierent as soon as they are given by diilerent equations, of whether one equation may arise from the other by a change of parameter. In follow- ing this course, as we will, it is important to show that cerf.ain properties of arcs are invariant under a change of parameter. For instance, the initial IUld tel ",ina! point of an arc remain the same after a cbal;lge of parameter. + If the derivative t(t) - Z/(!) iy'(t) exists and is ~O, the arc "I has a tangent whose direction is determined by arg t(l). We shall say that the arc is differentiable if t(l) exists and is continuous (the term con- tinuously differentiable is too 1lDwieldy); if, in addition, z'(t) ~ 0 the arc iR said to be regular. An arc is piecetl1ise differentiable or ~ regular if the same conditions hold except for a finite number of values t; at these points z(t) shall still be continuous with left and right derivatives which are equal to the left and right limits of z' (t) and, in the case of a piecewise regular arc, ¢O. The diilerentiable or regular character of an arc is invariant under the change of parameter t = 'P(T) provided that 'P'(T) is continuous and, for regularity, ¢O. When this is the we speak of a difierentiable or regular change of parameter. An arc is mnple, or a Jordan arc, if z(t,) = z(/.) only for"''''' t.. An arc is a closed CUf'!I6 if the end points coincide: z(a) = z(ft). For closed curves a shift of the parameter is defined as follows: If the origin•.! equa- tion is z = z(t), a ~ t ;;; /J, we choose a point t. from the interval (a,(:I) and define a new closed curve whose equation is 11 = ,et) for I. ~ t ~ fJ and + 2 = z(t - fJ + a) fol' (J :; t ;;; t. {J - a. The purpose of the shift is to get rid of the distinguished position of the initial point. The correct definitions of a differentiable or regular cloiIed curve and of a Bimple clo8ed CUnJS (or Jordan cunJs) are obvious. The opposite arc of 2 = z(I), a ;;; t ~ /J, is the arc z ~ z( -t), - II ;:ii t ;;; -a. Opposite arcs are sometimes denoted by "I and -"I, sometimes by 'Y and "1-1, depending on the connection. A constant function z(t) defines a point CW'II6. ' + A circle C, originally defined as a locus 12 - "I = r, can be considered as a closed curve with the equation z = a rsu, 0 :; t :; 2r. We will lise this standard parametrization whenever a circle is introduced. This convention saves us from writing down the eqll8tion each time it is ANALYTIC FUNCTIONS AS MAPPINGS .. needed; also, and this is it. most important purpose, it serve. as a definite rule to distinguish between C and - C. 2.2. Analytic Functions In Regions. When we consider the derivative f'(e) _ lim !(z + h) - !(!) ..... h of a complex-valued function, defined on a set A in the complex plane, it is of course understood that Z E A and that the limit is with respect to values + h such that z h EA. The existence of the derivative will therefore have a different meaning depending on whether. is an interior point or a boundary point of A. The way to avoid this is to insist that all analytic functions be defined on open sets. We give a formal statement of the definition: Definition 10. A complez,..valued !utICtion f(.), deji.1It!d on an open Bel n, is 8fJid to be analy/i£ in n if it 00s a t1.erioolive at each point of O. Sometimes one says more explicitly that /(z) is complex analytic. A commonly used synonym is holmnorphic. It is important to stress that the open set n is part of the definition. As a rule one should avoid speaking of an analytic function f(z) without referring to a specific open set n on which it is defined, but the rule can be broken if it is clear from the context what the set is. Observe that f must first of all be a fundion, and hence ringk-flalued. If 0' is an open subset of n, and if f(.) is analytic in Il, then the restriction of f to II' is analytic in 0'; it is customary to denote the restriction by tbe 88me letterf. In particular, since the components of an open set are open, it is no loss of generality to consider only the case where a is connected, that is to say a reg1•on. For greater flexibility of the l8ngll.g~ it is desirable to introduce the following complement to Definition 10: Definition ll. A function f(.) i. analytic on an arbitrary point let A if it is 1M reslridion to A of a function which is analytic in _ open 86t conlaining A. The last definition is merely an to use a convenient te....j- nology. This i8 a ca"" in which the set Q need not be explicitly men- tioned, for the specific choice of Q is usually immaterial as long as it contain. A. Another instance in which the mention of a can be 8Uppreased i.e the phrase: "Let I{z) be at z.." It mean8 that .. f.mction f(z) is defined and hae .. of 2<> . <- ,.--"--- COIIPLEX ANALYSIS Although our definition requires all analytic functions to be single- valued. it is possible to consider such multiple-valued functions 88 VZ. log z, or e.rc cos z, provided that they e.re restricted to a definite region in which it is possible to select a single-valued and analytic branch of the function. For instance, we may choose for n the complement of the negative vz real axis .0 ~ 0; this set is indeed open and connected. In 0 one and only one of the values of has a positive real part. With this choice to ~ VZ becomes a single-valued function in 0; let us prove that it is continuous. Choose two points z,• .., E 0 and denote the corresponding + values of w by w• ., 11. iv,. w, = u, + iv, with u,. u, > O. Then 1.0. -- .0.1 = Iw1 - to;1 = Iw, - to,l . Iw. + w.1 and Iw. + to.1 !l:; u, + u. > 11,. Hence Iw, - w. I < .1-.'-0-.'---,---.0-.'1" It. VZ and it follows that w = is continuous at .0,. Once the continuity is established the analyticity follows by deriV&tion of the inverse function • = w'. Indeed, with the notations used in calculus . O. consider the set A in the w-plane which is defined by the inequalities Iw - w,1 iii:; ", Ivl ;!!!! T, lu .,.. u,l ~ log 2. This set is closed and bounded, and for sufficiently small. it is not empty. The continu- OUS function Ie" - e"> I has consequently a minimum p on A (Theorem 8, Corollary). This minimum is positive, for A does not contain any point + w, n . 2wi. Choose 5 ,,;. min (P,~"). and aBBume that IZI - ...1= Ie"> - e"'1 < i. - ANALYTIC FUNCTION. AS MAPPINGS 71 Then w. cannot lie in A, for this would make Ie... - e"\ ~ p ~ B. + Neither is it possible that Us < fl, - log 2 or u. > u, log 2; in the former case we would obtain \e'" - 6".\ ~ eV' - 6" > jeW, ~ 4, and in the latter Ie'" - e"'\ ~ eO, - e"' > eO, > B. Hence w. mWJt lie in the disk ltD - tD.\ < c, and we have proved that w is a continuous function of z. From the continuity we conclude as above that the derivative exists and equals l/z. The infinitely m8.llY values of arc cos z are the same 88 the values of + i log (z y z· - 1). In this ease we restrict z to the complement (I' of the half lines z ~ -1, 11 = 0 and z ;;:; 1, 11 - O. Since 1 - 2' is never + real and ~ 0 in 0', we ean define y1 - z· 88 in the first example and then set y,,' - 1 ~ iyl - %t. Moreover, % YZ' - 1 is never real in 0', for z + Y zt - 1 and z - y z' - 1 are reeiprooals and hence real only if z and yz' - 1 are both real; this happens only when z lies on the excluded parts of the real axis. Because 0' is connected, it follows that all values of. z + Y zt - i in 0' are on the same side of the leal axis, and since i is such a value they are all in the upper half plane. We can therefore define an analytic branch of log (z + yz' - 1) whose imaginary part lies between oand.... In this way we obtain a: single-valued analytic function arc COBS = ilog (.. + yz· - 1) in (I' whoBe derivative is +. . + D arc cos % . -. z 1 1 yZ'-1 yz'Z -1 where yi - zi has a positive real part. There is nothing unique about the way in which the region and the single-valued branches have becn choBen in theBe examples. Therefore, each time we consider a function euch as log z the choice of the branch has to be specified. It is a fundamental fact that it is impo8lible to define a single-valued and analytic branch of log z in certain regions. This will be proved in the chapter on integration. All the results of Chap. II, Sec. 1.2 remain valid for functions which o.re analytic on. an open set. In partiClllar, the real and imaginary parts )f an analytic function in (I satisfy the Cauchy-Riemann equations -ia/xu= -aiIyv, Jonversely, if II and v satisfy theBe equations in. (I, and if the partial lerivativee are continuous, then u + iii lsan BD&Iytic (nnctijl/1 in O. An analytic f,",-ction in O. .. . . ... it.educee to • e6_n~ l"M! : - -. .,', '. . . • _ . ,-.' _.:,-.:;"'0;;;' _.;:,~ , '":;:. .·:',:' ""_;""';-.:,·:,"'~';h.-.""~-"-:':~'~_."~".',:':;':.:";":~'"-"'·.;<·;"":-:",':.;l-.'"::';·~<~~'.\;.o~_:i":,:'•,.·;;·-"..-.:,,:.,. ..•.;.. ·:,.~:" 72 COMPLEX ANALYSIS the following theorem we shall list BOrne simple conditions which have this consequence: Theorem ll. An analytic fUndWn in a region 0 whose dMivative van- w.a identicaUII IIlmt re8t1C1l to a eonatam. The 3IJm6 Ut truei! eilker the real pa1'~, the imaginary part, the flWdulm, or the argument Ut constant. The vanishing of the derivative implies that au/in, au/ay, ltv/in, ltv/ayare all zero. It follows that u and v are constant on any line seg- ment in Il which is parallel tc one of the coordinate axes. In Sec. 1.3 we remarked, in connection with Theorem 3, that any two pohits in a region can be joined within the region by a polygon whose sides are parallel 10 + the axes. We conclude that u ill is constant. If u or II is constant, 1'(z) -- au ax - ~. aauy-_a-ayll + J .alt- v - x_ 0' and hence I(z) must be constant. If u· + 112 is constant, we obtain uaa-3u+ : v-ilJt= 3v: O and uaa-uy + v ltv ay = -uiJa-xv +11aa3u-: = O. • These equations pCI'II1it the conclusion au/ax = ltvja:r: = 0 uDless the + + determinant u' v' vanishes. But if u' v' = 0 at a single point it is constantly zero and I(z) vanishes identically. Hence I(z) is in any case a constant. Finally, if arg I(z) is constant, we can set u = ku with constant k + (uDless II is identically zero). But u - kv is the real part of (1 ik)!, and we conclude again that I must reduce tc a constant. Note that for this theorem it is essential that 0 is a region. If not, we can oDly that 1(%) is constant on each component of O. EXERCISES + + Vi L Give a precise definition of .. mngle-valued branch of II VI - % in .. suitable region, and prove that it is analytic. 2. Same problem for log log z. I. Suppose that !(z) is analytic and satisfies the condition \/(z)' - 1\ < 1 in a region o. Show that either Re I(z) > 0 or Re I(z) < 0 throughout o. ANALYTIC FUNCTIONS AS MAPPINGS 2.3. Conformal Mapping. Suppose that an arc 'Y with the equation z = z(t), a :;; t ;;;; fl, is contained in a region n, and let I(z) be defined and continuous in ll. Then the equation w ~ w(t) = f(.(t)) defines an arc .,' in the w-plane which may he called the image of .,. Consider the case of an I(z) which is analytic in ll. If .'(t) exists, we find that ""(t) aiM exists and is determined by (1) w'(t) = 1'(.(I»z'(I). , We will investigate the meaning of this equation at a point Zo = z(I.) with z'(t.) ;!If- 0 andr(z.) ;!If- O. The first eonclusion is that w'(Ie) ;!If- O. Hence 'Y' has a tangent at w. = fC••), and its direction is determined by (2) + arg w'(t.) = argl'(z.) atg %'(1.). This relation that the angle between the directed tangents to 'Y at z. and to 'Y' at w. is equal to arg f(z.). It is hence independent of the curve .,. For this reason curves through ". which are tangent to each other are mapped onto curves with a eornmon tangent at w.. Moreover, two curves which form an angle at z. are mapped upon curves forming the same angle, in sense 88 well as in size. In view of this property the mapping by w = f(.) is said to be conformal at all points with fCz) ;!If- O. A related property of the mapping is derived by consideration of the modulus 1/'(4)1. We have lim If~z) - 1(4)1 = If(z.)I, --.. I· - ••1 and this means that any small line segment with one end point at ZD is, in the limit, contracted or expanded in the ratio If('o)l. In other words, the linear change of scale at 4, effected by the tranzfomlation w = f('), is independent of the direction. In general this change of scale will vary from point to point. Conversely, it is clear that both kinds of eonformaIity together imply the existence of f (ZD). It is less obvious that each kind will separately imply the same result, at least under additional regularity assumptions. To he more precise, let us asswne that the' partial derivatives iJflax and iJlllIy are continuous. Under this condition the derivative of 1I>(t) = f(lI(t» can he expressed in the form ! . "+., w'(l.) - X' (I.) 1I'(to) . .' . where the partial dari~ _fUkM at .0;' In 1enne of %'(to)"ftllB " . _ ->.' . , :'."" .. . ,- '-'''-:'~ -. ,,".: .. ;~ --' • ..'. .....•. ......-.. ....•.. ,....••_..c.'.:··.;':••••·· ....•.,.:,.'.. 74 COIIPL•• ANALYSIS be rewritten as If angles are preserved, arg [w'(to)/z'(to») must be independent of arg z'(to). The expression (3) must therefore have a colll!tant argument. AIl arg z'(to) is allowed to vary, the point represented by (3) describes a circle baving the radius ll(allib:) + i(fJllay)\· The argument eannot be constant on this eircle unlees its radius vanishes, and hence we must have (4) -aaxl= - t.aa-yl which is the complex form of the Cauchy-Riemann equations. Quite similarly, the condition that the ehange .of sceJe sheJl be the same in all directions implies that the expression (3) has a constant modulus. On a circle the modulUJl is constant only if the radius vanishes or if the center lies at the origin. In the fim case we obtain (4), and in the second case ,,_. -al . al ax ay :;::::::. The last equation expresses the facit that I(z) is analytic. A mapping by the conjugate of an analytie fnnction with a nonvanisbing derivative is said to be indirectly wnlllN'ilal. It evidently preserves the size but reverses the sense of angles. If the mapping of (l by '" = I(z) is topological, then tbe inverse fnnc- tion z - I-I(w) is also analytic. This follows easily if f'(z) '" 0, for then the derivative of the inverse function must be equal to III' (z) at the point z ..; I-'(w). We shall prove later that I'(z) can never vanish in the case of a topo\ogiceJ mapping by an analytic function. The knowledge thatf'(zo) '" 0 is sufficient to conclude that the ma~ ping is topological if it is restricted to a sufficiently small neighborhood of Zo. This follows by the theorem on implicit functions known from the oal- culus, for the Jacobian of the functions u = u(x,y), v .. v(x,y) at the point Zo i81/'(zo)\' and hence '" O. Later we shall present a simpler proof of this important theorem. .. But even if I'(z) '" 0 throughout the region 0, we cannot that the mapping of the whole region is necesse.rily topological. Toillust.mte ANALYTIC FUNCTIONS AS MAPPINGS l' what may happen we refer to Fig. 3-1. Here the mappings of the sub- nI' FIG.lt Doubly oovered region. regions (1, and Il, are one to one, but the images overlap. It is helpful to think of the image of the whole region &8 & trlUlsparent film which partly covers itself. This is the llimple IUld fruitful idea used by Riemann when he introduced the genera.1ized regions now known &8 Riemann IIUrlaus. . . 11.4. Length and Ar..... We have found that under a COnfOl'lllal mapping I(z) the length of an infinitesimal line segment at the point z is multiplied by the factor If'(,,) I. Because the distortion is the same in all directions, infinitesimal areas will clearly be multiplied by If'(o) I'. Let us put this on a rigorous basis. We know from calcul\18 that the length of a differentiable are 'Y with the equation. = o(t) = x(l) + iy(t), a ;:I t ~ b, is given by J' J' L('Y) = y'",'(O· + y'(t)1 de = 1.'(01 dt. • • The image curve 'Y' is determined by tD = w(t) = I(.(t» with the derivative w(O = f(.(I»"(I). Its length is th\18 I: L(..,') = 1f'(0(1)) 11.'(1) I tit. It is customary to use the ahorter notations (5) f f L('Y) = Idol, L(..,') = If'(z) lldol· T T Observe that in complex notation the calculus eymbol d3 for integration with respect to.arc length is replaced by ldo I. Now let E be a point set in the pilUle whose area /fdz A(E)= dll • . - '- ,- . .'-, ..•.,.,", . .' " - '.-". ",' -.;' "-" ':.:'.',., " '.' -" '.::.'-- ".\ ',.,~' COMPLEX ANALYSIS + ran be evaluated as a double Riemann integral. If f(.) - u(x,y) iv(z,y) is a bijective differentiable mapping, then by the rule for changing integration variables tbe area of the image E' = feE) is given by A(E') = fflu... - ..,.v.1 d% dy. " But if f(.) is a confol'lMI mapping of an open set containing E, then ..... - ..,.v. - 11'(.) I' by virtue of the Cauchy-Riemann equatiollll, and we obtain (6) ff A(E') = 11'(.)I' d% dy. B The (ol'mnl"" (5) and (6) bave important applications in the part of complex analyais that is frequently referred to 88 geometric function theory. I. LINEAR TRANSFORMATIONS Of all analytic functions the firslr o. For arbitrary complel< Ie ". 0 We can set Ie = 1.1:1 • k/lkl, and hence w = kz can be represented &8 the lCSalt of a hornothetic transformation followed by a rotation. The third transformation, to = l/z, is called an inversion. If c ". 0 we can write , COMPLEX ANALYSIS and this decomposition shows that the most general linear transform&ion is composed by a translation, an inversion, a rotation, and a homo- thetic tl'ansformation followed by another translation. If c == 0, the inversion falls out and the last tr8 muation is not needed. aXERCISES L Prove that the reflection z ..... i is not a linear transformation. 1. If z+2 T,z = z +3' T,z = z +z l' find T,T,z, T.T,z and T;:'TtZ, L Prove that the most general transforulation which leaves the origin fixed and preserves all distances is either a rotation or a rotation followed by reftexion in the real axis. ... Show that any linear transformation which transiOIlIll! the real axis into itself can be written with real coefficients. 8.2. The CrOSlt Ratio. Given three distinct points 1$0, z., z. in the extended plane, there exists a linear tra.nsforulation S which carries them into 1, 0, co in this order. If none of the points is co, 8 will be given by (9) Bz = :I - : :Is %1 - %1. z - %" %2 - z . If %"~' or 2. == co the transformation reduces to , Z - Za Z - Zt , Z2 - Zf, Z - z.. respectively. If T were another linear tl.LDsiormation with the same property, then ST-' would leave 1, 0, co invariant.. Direct calculation shows that this is true only for the identity trausf4)nnation, and we would have 8 - T. We conclude \hat S is uniquely deWlilined. Definition 12. The eros. roIw (z"z.,.,.,.,.) iB the im4(J' of z, umkr the linear traM/oltlilll'Wn which carriB8 z"z.,~. into 1, 0, co. The definition is meaningful only if Z"Z,,%, are distinct. A conven- tional value can be introduced 88 llOon 88 any three of the points are distinct, but this iaunimport9u t. ANALYTIC FUNCTlON8 108 MAPPINGS 71 The cross ratio is invariant under linear tran.'ormatiOIl8. In more precise formulation: Theoreln 12. 1/Z" Zt, Za, ii, are dulind point. in the alMidod plIJne and T any linear tTam/ormaiian, then (Tz"Tzt,Tz"Tz.) ~ (ZI,Z.,Z,,2.). The proof is immediate, for if. Bz = (z,zt,z,,z,), thenBT-' canies Tzt, Tz" Tz. into 1, 0, 00. By definition webave hence (Tzl, Tz., Tz" Tz.) = BT-'(Tz,) = 8Z1 = (z,,.f.,.,,,,,.). With the help of this property we can immediately write down the linear trMsfQrmation which carries three given P.:...~~:.:..,.•..,••,.. ,.~;. ,,',,'•.,.:.:.',, eo COMPLEX ANALYSIS divide by this coefficient and complete the square. After a simple computation we obtain ad-be = lie-fa which is the equation of a circle. The last resalt makes it clear th.e.t we should not, in the theory of linear transformations, distinguish between circles and straight lines. A further justification was found in the fact that bo~h correspond to circles on the Riemann sphere. Accordingly we sha.ll'e.gl ee to UBe the word circle in this wider sense.t , The following is an immediate corollary of Theorems 12 and 13: Tbeorem 14. A linear tranaformalian carrie. circle. iNo circle.. EXERCISES L Find the linear transformation which carries 0, i, - i into 1, -1, O. Z. Express the cross ratios cOnesponding to the 24 permutations of four points in tern.s of " = (Z.,%2,%,,%,). a. If the consecutive vertices %1, %t, Z" z. of a qlllulrilateral lie on a circle, prove t.bat \z, - z.\·\z, - %.\ = \It. - z,\ ·Iz. - z.\ + \z. - 1t.1 . lit. - z.1 and interpret the result geometrically. ... Show that four distinct points can be earried by a linear transformation, to positions I, -1, Te, -Te, where the value of k depends on the points. How many solutions are tbere, and how are they related? 3.3. Symmetry. The points z and j are symmetric with rupect to the real axis. A linear transformation with real coefficients carries the real axis into itself and z, Ii into points which- are again sym~tric. More generally, if a linear transformation T carries the real axis into a circle C, we shall say that the points 10 = Ta and 10* = Ti are symmetric with re8pect to C. This is a relation between 10, 10* and C which does not depend on T. For if S is another transformation which carries the real axis into C, then S-'T is a real transfornlation, and hence S-'w - S-'Tz and 8-'111* = S-'TJ are also conjugate. Symmetry can thus be defined in the following tenus: , t'I'hil eg'eement will be in foroe only when desliDC with linear t;ranefonnation ., ANALYTIC FUNCTIONS AS MAPPINGS II Definition 13. The points It and.* an said 10 be qmmetric wiQl rupect to the circle C thrMJ{/h z" Z" z, if mul onlll if (z*••••z.,z.) = (z.z.,z,.z.). The points on C. and only those. are symmetric to themselves. The mapping which carries z into •• is a one-to-one oonespondence and is called reflection with respect to C. Two redections will evidently result in a linear transformation. . We wish to investigate the geometric significance of symmetry. Sup- pose first that C is a straight line. Then we can choose " = co and the condition for symmetry becomes (10) Taking absolute values we obtain 1.* - ••1= I.r - 8.1. Here .. can be any finite point on C, and we conclude that. and .. are from all pointe on C. By (10) we have futther 1mz" · -., = -1m z- .. • 1:1-';. *1-Z, and hence 11 and z· are in dilferent half planes determined by C.t We leave to the reader to prove that C is the bisecting normal of the segment betWW.1. and , •. Consider now the case of a finite circle C of center a and radiu R. Sy"'sllllclmatic use of the invariance of the cross ratio allows us to conclude as foOows: (z,z.,z,,z.) = (z - a,z. - a,z, - a,z. - il) - i-a_, R..'., R' • :CI-=t':,-- .11-4 &'s-a z,-a = R" ,z,- 0_1- Gs.-a '-4 ' I - ._R'4+ a,zI•••••• • + This equation shows that the symmetric point ohis.* - R'/(! - a) a or that. and z· aatilsfy the relation (11) (,. - 4)(1 - a) = R·. The produet 1.* - 01 . I_ - al of the distances to the center is hence R'. Further, the ratio (II· - a}/(. - a) is positive. whioh that z and z· are on the same half line from o. There is a simple geometric coDlltruction for the 8ymmetrie point of " (Fig. 3-2). We note that the eymmetric point of (I is ... tUDlIiII . ..... .. .. .... _,_ _ _. . . . , ':": ,. ' ' ·r,.,·.',.·' ."",",',, .. ;,,::'", , " , ',:, '" '~',,:' . ,: ,,'" . . ....... ' ...'....,..._,' ...."'~'''''.',..,,,..:..' .. : .. .': '~' .•'., .•...:..-........~.:•.... '<...;.,.',.'.....:.....,:.:',:.;.'..>, .:. IZ COMPLEX ANALYSIS ZO c '" Fl•. W. ReHection in 8 circle. Theor.,.,. IS. (ThB symmetry priMiple.) If a linear Irana/ormation carries a circle C. into a circle C., then it trana/o""'8 any pair '" SY1n'IMtric poinIB with """Peel to C. into a pmr of symmetric poinIB with respecl to C•. Briefly, linear transfonnations preserve symmetry. If C, or C, is the real axis, the principle follows from the definition of BY mmetry. In the general the follows by use of an intermedis.te transfOn.,ation which carries C, into the real axis. There are two ways in which the principle of symmetry can be used. If the images of z and C under a certain linear transformation are known, then the principle a.1lows us to find the ime,ge of ~*. On the other hand, if the images of z and ~. are known, we conclude that the image of C must be a line of symmetry of these images. While this is not enough to determine the image of C, the infonnation we gain is nevertheless . valuable. The principle of symmetry is put to practical use in the problem of finding the linear transformations which carry a circle C into a circle C'. We can always determine the transformation by requiring that three w., points 21" 21., z. on C go over into three points WI, W, on C'; the trans- formation is then (w,w"w.,w.) = (z,z"z.,z.}. But the transformation is also determined if we prescribe that a point z. on C sball couespond to a point w. on C' and that a point z. not on C sha.1l be carried into a point w: w. not on C'. We know then that z: (the symmetric point ,of z. with respect to C) must correspond to (the symmetric point of Wt with rwpect to C'). Hence the trllnsformation will be obtained from the relation (w,w.,wo,w:) = (21,21,,210,21:). EXERCISES L Prove that every reflection carries circles into circles. • ANALYTIC FUNCTION' AS MAPPINGS 2. Reflect the imaginary axis, the line :I: = 1/, and the circle Izi = 1 in the circle Iz - 21 = l. I, Cany out the reflections in the preceding exercise by geometric construction. 4. Find the linear transform a.tion which ca.rries the circle Izi = 2 iuto + I. 11 - 1, the point - 2 into the origin, and the origin into i. L Find the most general linear transforma.tion of the circle 1.1 = R into itee\f. 8. Suppose that a linear transformation carries one pair of concentric circles into another pair of concentric circles. Prove that the ratioe of the radii must he the same. 7. Find a linear transformation which ca.rriea 1.1 - 1 and Iz - tl ~ {- into concentric circles. What is the ratio of the radii? L Same problem for Izi = 1 and z = 2. . 11.4. Oriented C...",.... Because S(z) is analytic and S'(.) = (aczd+-bde). ¢ 0 the mapping to .. S(.) is conformal fou ¢ -die and 00. ItIollows that a pair of intersecting circles are ma.pped on circles that include the same angle. In addition, the sense of an angle is preserved. From an intui- tive point of view this means that right and left are preserved, but a more precise formulation is desirable. An orientation of a circle C is determined by an ordered triple of points '.,Z.,ZI on C. With respect to this orientation a point" not on C i8 nid to lie to the right of C if 1m (.".".".) > 0 and to the left of C is 1m (Il,","."') < 0 (this checks with everyday .IIIA beea".... (i,l,O, 00) = i). It is eesential to show that there are only two different orientatioll8. By this we mean that the distinction left and right is the eame for all triples, while the meaning may be revelsod. Since the Ct088 ratio is invari- ant, it is sufficient to consider the ease where C is the real axis. Then cz + /Ill +b (z,...,.,,") = d coo be writtsn with real coefficients. and a simple calculation gives • 1m (s,z.,'.,") = Ia..d+-bdel' 1m •. We recognize tha~ the distinction between right and left is the same as . half plene. Which is which .:.;~.. .. ',..-' «-" - '' -",.-'"'. '-'" ,... ..- . .,' , :'. '.. -:< :.,-, ... - "':""",,' '. " .-:. ,: ",:> ~-'~'=.i:~:, ";':J1.:".':~+&i[~~k:~cii':i'i~0~,,,,~'~ -~~;j.>~.~- ~,, • -.,:,·).~·:,.';i,:~~~~t:,:J.;,i1.~~ii.-:J.:'::.;:;;";;.~~:z,';{,L~ ..COIIPLEX ANALYSIS A linear transformation S carries the oriented circle C into a circle which we orient through the triple SZl, &., Sz.. From the invariance of the crOM ratio it follows that the left and right of C will be mapped on the left and right of the image circle. If two circles are tang o.. 2. Prove ~hat a tangent to a circle is perpendicular to the radius through the point of contact (in this connection a tangent should be defined as a straight line with only one point in common with the circle). .. Verily that the inside of the circle Iz - al - R is formed by all points z with Iz - al < R. . 4. The angle between two oriented circles at a point of inteIsection is defined as the angle between the tangents at that point, equipped with the orientation. Prove by analytic r8ll8ODing, rather thaD geometric inspection, that the angles at the two points of intersection are opposite to each other. 3.5. Familiett of arc".. A great deal be done toward the visual-· ization of linear transformations by the introduction of certain families of circles which may be thought of as coordinate lines.. in a circular coordinate system. Consider a Iioear transformation of the form ID = k. I-a. z-b Here z = a = = to ID 0 and 1= b to ID GO. It follows that the straight lines through the origin of the tD-p1ane are-images of the ANALYTIC fUNCTIONS AS MAPPINGS circles through a and b. On the other hand, the concentric circles about the origin, IIDI = p, correspond to circles with the equation ~-G "z----"";:b = pllll· TheBe are the circle8 oj.ApolkmiWl with limit points a and b. By their equation they are the loci of points whoBe distances from a and b have B constant ratio. Denote by C, the cireles through A, b and by C. the cireles of Apol- Ioniua with these limit points. The configuration (Fig. 3-3) formed by all the circles C, and C. will be referred to 88 the circulaf- 11& or the 8teiner circle8 determined by a and b. It has many interest.ing properties of which we shalIli.t a few: 1. There is P.xa.,tly one C, and one C. through each point in the plane with the exception of the limit pointe. 2. Every C, meets every C. under right angles. 3. Reflection in a C1 transforms every C. into itself and every C1 into another C,. lUl1Iection in a C. transfomlS every C1 into itself and every C. into another C•. 4. The limit points are symmetric with respect to each C., but not with l'68pect to any other cirele. PIG. ..... Stain !I circlM. .-, , .' .'.. . ". ., . ,- .. . ..... . "'- "" . ', ' "- '.' ,- -: '''--' ,- • COMPLEX ANALYSIS These properties are all trivial when the limit points are 0 and GO I a, a. i.e., when the are lines through the origin and the concentric circles. Since the properties are invariant under linear transfonnations, they must continue to hold in the general "&se. lf a transfolmation UP = T. carries G, b into G', b' it can be written in the f6rm (12) • It iB clear that T transforms the circles C, and C. into circles C; and a~ with the Ii mit point. a', b'. The situation i8 particularly simple if G' = G, b' = b. Then a, b are said to be fiud points of T, and it i8 convenient to represent" and Tz in the· same plane. Under these circumstance. the whole circular net will be mapped upon itself. The value of k serves to identify the image a; a circles and c~. Indeed, with appropriate orientations 1 forms the an«le arg k with its image a;, and the quotient of the oonBtant ratios \. - al/lz - b\ on a; and a. iB Ik\. The zpeeial c&ses in which all C1 or all C. are mAPped upon themselves are particularly important. We have C: = C1 for all C1 if k > 0 (if k < 0 the circles are still the same, but the orientation i. rev,,,seJ). The transformation is then said to be hyperbolic. When k increases the a points T" z ~ G, b, will flow along the circles 1 toward b. The con- sideration of this flow provides a very elear picture of a hYPlrbolic transfonnation. , a. The ~.,., C; = oecurs when Ik\ = 1. Transformations with thi8 property are called elliplic. When srgk varies, the points Tz move along the eircles C.. The eorresponding flow circulates about G and b in difierent directions. The gtlnerallinear transformation with two fixed points is the product of a hyperbolic and an elliptic transformation with the same fixed points. The fixed. points of a linear transformation are found by solving the equation (13) In gtlneral this is a quadratic equation with two roots; if ., = 0 one of the fixed points i. GO. It may happen, however, that the roots coincide. A linear tr&psformation with ooinciding fixed points i8 said to be parabolic. The condition for this i. (a - ~)' = ~.,. If the equation (13) i8 found to have two distinct roots G and b. the transformation can be written in the f o m . . ANALYTIC FUNCTIONS AS IIApPING8 a7 v, -a = v-b k zz--ab. We can then use the Steip.er circles determined by II, b to discuss the nature of the transformation. It is important to note, however, that the method is by DO means restricted to this CIlSe. We can write any linear transformation in the form (12) with arbitrary II, b and use the two circular nets to great advantage. .For the discussion of parabolic transfOiDIatioDS it is desirable to intro- duce still another type of circular net. Consider the transformation ., w = z-a +c. It is evident that straight lines in the tD-plane conespond to eircles through a; moreover, pareJIellines correspond to mutually tangent circles. + w In particular, if v = u the lines u = constant and • - constant correspond to two families of mutually tangent circles which intersect at right angles (Fig. 3-4). This configuration can be considered 88 a degenerate set of Steiner circles. It is determined by the Point II and the tangent to one of the families of circles. We shall denote the images of the lines v = constant by V .. the circles of the other family by V•. Clearly, the line v ~ 1m c conesponds to the tangent of the circles 01; its direction is given by arg