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Elementary theory of analytic functions of one or several complex variables
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Elementary theory of analytic functions of one or several complex variables
Henri CartanCategories:
Año:
1963
Editorial:
Addison Wesley Longman Publishing Co
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english
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227
ISBN:
0201009013 9780201009019
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COLLECTION ENSEIGNEMENT DES SCIENCES HERMANN ADIWES I NTERNAT I O NAL N SERIES M ATHEMATICS A. J. Lohwater, Consulting Editor HENRI CARTAN University of Paris Elementary theory of analytic functions of one or several complex variables EDITIONS SCIENTIFIQUES HERMANN, PARIS ADDISONWESLEY PUBLISHING COMP ANY, INC. Reading, Massachusetts  Palo Alto  London This is translated from T HEORIE ELEMENTAIRE DES FONCTIONS ANALYTIQUES D'UNE OU PLUSIEURS VARIABLES COMPLEXES Hermann, Paris © 1963 Hermann, Paris TABLE OF CONTENTS CHAPTER I. POWER SERIES IN ONE VARIABLE Formal power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. Convergent power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I6 1. 3. Logarithmic and exponential functions . . . . . . . . . . . . . . . . . . 28 4. Analytic functions of one variable . . . . . . . . . . . . . . . . . . . . . . . 36 Exercises 43 . . . . . . . . .. . . . . . . .. . . . . . . . . . .. ..... . . ... . . .. . . . CHAPTER II. HoLOMORPHIC FUNCTIONS; CAUCHY'S INTEGRAL I. Curvilinear integrals; primitive of a closed form . . . . . . . . . . . . 49 2. Holomorphic functions; fundamental theorems. . . . . . . . . . . . . 66 Exercises 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER III. TAYLOR AND LAURENT EXPANSIONS I. Cauchy's inequalities; Liouville's theorem . . . . . . . . . . . . . . . . 2. Mean value property and the maximum modulus principle . . 79 8I 3. Schwarz' lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4. Laurent's expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5. Introduction of the point at infinity. Residue theorem . . . . . . . 89 6. Evaluation ofintegrals by the method ofresidues . . . . . . . . . . . 99 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I08 CHAPTER IV. ANA; LYTIC FUNCTIONS OF SEVERAL VARIABLES; HARMONIC FUNCTIONS 1. Power series in several variables . . . . . . . . . . . . . . . . . . . . . . . . . I I8 2. Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I2 I 3. Harmonic functions of two real variables . . . . . . . . . . . . . . . . . . 122 4. Poisson's formula; Dirichlet's problem . . . . . . . . . . . . . . . . . . . 5. Holomorphic functions of several complex variables Exercises . I27 . . . . . . . I 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5 CARTAN TABLE OF CONTENTS CHAPTER V. CONVERGENCE OF SEQ.UENCES OF HOLOMORPHIC OR MERO· MORPHIC FUNCTIONS; SERIES, INFINITE PRODUCTS; NORMAL FAMILIES 1. Topology of the space e(D) ............................. 142 2. Series of meromorphic functions ......................... 148 3 Infinite products of holomorphic functions .. . . . . . . . . . . . . . . . 4. Compact subsets of�(D) Exercises CHAPTER VI. I. I 57 .............................. 162 .............................................. 168 HoLOMORPHIC TRANSFORMATIONS General theory; examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 72 2. Conformal representation; automorphisms of the plane, the Riemann sphere, the open disc.......... ... . ............. 1 78 3. Fundamental theorem of conformal representation .......... 184 4. Concept of complex manifold; integration of differential forms. I 88 5. Riemann surfaces ............ ........ ................. 196 Exercises 207 CHAPTER VII. HOLOMORPHIC SYSTEMS OF DIFFERENTIAL EQ.UATIONS 1. Existence and uniqueness theorem ....................... 210 2. Dependence on parameters and on initial conditions ........ 216 3. Higher order differential equations .. . ................... 218 Exercises 2Ig SOME ................. . ... ........................ . NUMERICAL OR Q.UANTITATIVE ANSWERS TERMINOLOGICAL INDEX NOTATIONAL INDEX • • . . • . . . . • • • . • • • • • • • • . • • • . • • • . . . . . . . . . • • • • • . • . • • . . • • . . • • • • • • • • • . • . • • • • • • . . . • • . • • • . . • • • . • • . • . . . . • • • • • • 222 223 228 PREFACE The present volume contains the substance, with some additions, of a course of lectures given at the Faculty of Science in Paris for the require ments of the licence d'enseignement during the academic sessions 19571958, 19581959 and 19591960. It is basically concerned with the theory of analytic functions of a complex variable. The case of analytic functions of several real or complex variables is, however, touched on in chapter IV if only to give an insight into the harmonic functions of two real variables as analytic functions and to permit the treatment in chapter vu of the existence theorem for the solutions of differential systems in cases where the data is analytic. The subject matter of this book covers that part of the" Mathematics II " certificate syllabus given to analytic functions. This same subject matter was already included in the " Differential and integral calculus " certifi cate of the old licence. As the syllabuses of certificates for the licence are not fixed in detail, the teacher usually enjoys a considerable degree of freedom in the subject matter of his course. choosing This freedom is mainly limited by tradition and, in the case of analytic functions of a complex variable, the tradition in France is fairly well established. It will therefore perhaps be useful to indicate here to what extent I have departed from this tradi tion. In the first place I decided to begin by offering not Cauchy's point of view (differentiable functions and Cauchy's integral) but the Weierstrass point of view, i.e. the theory of convergent power series (chapter 1). This is itself preceded by a brief account of formal operations on power series, i.e. what is called nowadays the theory of formal series. I have also made something of an innovation by devoting two paragraphs of chapter to VI a systematic though very elementary exposition of the theory of abstract complex manifolds of one complex dimension. What is referred to here as a complex manifold is simply what used to be called a Riemann surface and is often still given that name; for our part, we decided to keep the term Riemann surface for the double datum of a complex mani fold and a holomorphic mapping of this manifold into the complex plane 7 PREFACE (or, more generally, into another complex manifold). In this way a distinc tion is made between the two ideas with a clarity unattainable with orthodox terminology. With a subject as well established as the theory of analytic functions of a complex variable, which has been in the past the subject of so many treatises and still is in all countries, there could be no question of laying claim to originality. If the present treatise differs in any way from its forerunners in France, it does so perhaps because it conforms to a recent practice which is becoming increasingly prevalent: a mathema tical text must contain precise statements of propositions or theorems statements which are adequate in themselves and to which reference can be made at all times. With a very few exceptions which are clearly indicated, complete proofs are given of all the statements in the text. The somewhat ticklish problems of plane topology in relation to Cauchy's integral and the discussion of manyvalued functions are approached quite openly in chapter 11. Here again it was thought that a few precise statements were preferable to vague intuitions and hazy ideas. On these problems of plane topology, I drew my inspiration from the excellent book by L. Ahlfors (Complex Analysis), without however conforming completely with the points of view he develops. The basic concepts of general Topology are assumed to be familiar to the reader and are employed frequently in the present work; in fact this course is addressed to students of' Mathematics II ' who are expected to have already studied the'Mathe matics I ' syllabus. I express my hearty thanks to Monsieur Reiji Takahashi, who are from experience gained in directing the practical work of students, has consen ted to supplement the various chapters of this book with exersices and problems. It is hoped that the reader will thus be in a position to make sure that he has understood and ar ;imilated the theoretical ideas set out in the text. HENRI Die (Drome), August 41h, 1960 8 CARTAN CHAPTER I Power Series in One Variable 1. Formal Power Series I. ALGEBRA OF POLYNOMIALS Let K be a commutative field. We consider the formal polynomials in one symbol (or ' indeterminate moment we do not give a value to ' ) X with coefficients in K (for the X). The laws of addition of two poly nomials and of multiplication of a polynomial by a ' scalar ' makes the set K[X] of polynomials into a vector space over K with the infinite base r,X, ...,X", ... Each polynomial is a finite linear combination of the in K and we write it � a .X ", where n�O number of the coefficients coefficients. an X• with coefficients it is understood that only a finite are nonzero in the infinite sequence of these The multiplication table XP.Xq = XPH defines a multiplication in K[X]; the product ( I. I ) c. = � apbq. p+q=n This multiplication is commutative and associative. It is bilinear in the sense that (r. 2) 9 POWER SERIES IN ONE VARIABLE for all polynomials ment an = (denoted P, P1, P2, Q and all scalars 1. It admits as by 1 ) the polynomial � a nX n such that a0 o for n > o. n�O unit ele = I and We express all these properties by saying that K[X], provided with its vector space structure and its multiplication, is a commu tative algebra with a unit element over the field K; it is, in particular, a commutative ring with a unit element. 2. THE ALGEBRA OF FORMAL SERIES � anXn, where this tim e n�O Wf! no longer require/ that Qnly a finite number of the coefficients an are A formal power series in X is a formal expression · nonzero. We define the sum of two formal series by where Cn =an+ bn, and the product of a formal series with a scalar by The set K[[X]] of formal series then forms a vector space over K. The neutral element of the addition is denoted by o; it is the formal series with all its coefficients zero. The product of two formal series is defined by the formula ( 1. 1 ) , which still has a meaning because the sum on the right hand side is over a finite number of terms. The multiplication is still commutative, and bilinear with respect to the vector structure. algebra over the field K with a unit element (denoted by the series � anXn n�O such that a0 = I and a n = associative Thus K[[X]] is an ) 1 , which is o for n > o. The algebra K[X] is identified with a subalgebra of K[[X]], the subalgebra of formal series whose coefficients are all zero except for a finite number of them. 3· THE Denote ORDER OF A FORMAL SERIES � n�9 a nX n by S(X), or, more briefly, by S. The order w(S) of this . series is an integer which is' only defined when S =I= o; it is the smallest n such that a n =I= o. We say that a formal series S has order )> k if it is o o or if w(S) )> k. By abus de langage, we write w(S) )> k even when S = although w(S) is not defined in this case. IO I.1.3 FORMAL POWER SERIES Note. We can make the convention that w(o) = + that w(S) :;;;,,. that an = oo. The S such k (for a given integer k) are simply the series Li anXn such n�O o for n < k. They form a vector subspace of K[[X]]. Definition. A family (S;(X))ie1> where I denotes a set of indices, is said to be summable if, for any integer k, w(S;) :;;;,,. k for. all but a fin�te number of the indices i. By definition, the sum of a summable family of formal series S;(X) = Li an,iXn n�O is the series S(X )= Li anXn, n�O where, for each n, This makes sense because, for fixed an= Li an,i· n, all i but a finite number of. the an, 1 are zero by hypothesis. The operation of addition of formal series which form summable families generalizes the finite addition of the vector structure ofK[[X]]. The generalized addition is commutative and associative in a sense which the reader should specify. The formal notation Li anXn can then be justified by what follows. Let n�O a monomial of degree p be a formal series Li anXn such that an = o for n;::.o n =I= p and let apXP denote such a monomial. The family of monomials (anXn)ne:'! (N being the set of integers :;;;,,. o) is obviously summable, and its sum is simply the formal series Note. Li a,.X". n�O The product of two formal series is merely the sum of the summable family formed by all the products oi a monomial of the first series by one of the second. PROPOSITION 3. 1. The ring K[[X]] is an integral domain (this means that S =I= o and T =I= o imply ST =I= o ) . Proof Suppose that S(X) = Li apXP and T(X) =� bqXq are nonzero. p Let p = w(S) and q = w(T), let S(X)·T(X) =�CnXn; n II POWER SERIES IN ONE VARIABLE obviously c. = o for n <P + q and since ap # o, bq # o, we have that cPH = Since K is a field and apbq. # o, so ST is not zero. cP+,1 What is more, we have proved that (3. I ) Note. w(S T) · = w(S) + w(T) for S #o and T#o. One can consider formal series with coefficients in a commutative ring A with a unit element which is not necessarily a field K; the above proof then establishes that; if A is an integral domain, then so is A[[X]]. 4· SUBSTITUTION OF A FORMAL SERIES IN ANOTHER Consider two formal S( X ) series = S n�O T(Y) a.X", It is essential also to assume that b0 = = S p?:0 bpYP. o, in other words that w(T)> I. To each monomial a.X• associate the formal series a.(T(Y)) •, which has a meaning because the formal series in Y form an algebra. Since b0 o, the order of a.(T(Y))• is> n; thus the family of the a.(T(Y))• (as n takes = the values o, I, ... ) is summable, and we can consider the formal series S (4· I) n�O a.(T(Y))•, in which we regroup the powers ofY. This formal series in Y is said to be obtained by substitution ofT(Y) for X in S(X ); we denote it by S(T(Y)), or So T without specifying the indeterminate Y. The reader will verify the relations : (4. � (S1 + S2)o T S1o T + S2o T, ( (S1S2) o T (S1o T) (S2o T), 2) = = 1 o T = 1. But, note carefully that So (T1 + T2) is not, in general, equal to The relations (4· 2) express that, for given T ping S + which transforms the unit element Note. If we substitute o in S(X ) reduces to its ' constant term I2 (of order> 1 ), the map So T is a homomorphism of the ring K[[X]] in the ring K[[Y]] ' = I into I. S n�O a0• a. X •, we find that the formal series FORMAL POWER SERIES I. If we have a summable family of formal series Si and if w(T) I ·4' > 1, then the family S; o T is summable and ( � S,) o T = � (Si o T), which generalizes the first of the relations (4· Si (X) � .an, = n?0 2). For, let ;X•; we have � i S;(X) � (� a•,i) = i n�O Xn, whence while �Si o T = � C�o an,i(T(Y))•) . To prove the equality of the right hand sides of (4. 4) and (4. observe that the coefficient of a given power only a finite number of the coefficients a., i YP 5), we in each of them involves and we apply the associativity law of (finite) addition in the field K. PROPOSITION 4. 1. The relation (S o (4. 6) holds whenever w(T) > Proof. 1, T) o U w(U) > = So (To U) (associativity of substitution). I In the case when Sis a monomial, Both sides of (4. 6) are defined. they are equal because (4. T•o 7) which follows by induction on n U (To U) • = from the second relation in (4· 2). The general case of (4· 6) follows by considering the series S as the (infinite) sum of its monomials a.X"; So T and, from (4· = by definition, � a.T•, n;?0 3), (So T) o U = � a.(T" o U), n�O POWER SERIES IN ONE VARIABLE which, by (4. 7), is equal to � n�O an(T 0 U) n s = 0 (T 0 U). This completes the proof. 5· ALGEBRAIC INVERSE OF A FORMAL SERIES In the ring K[[Y]J, the identity (5. I ) ( I  Y) ( I + Y + can easily be verified. PROPOSITION 5. r. • · • + yn + Hence the series For S(X) = � I • • • ) = I  Y has an inverse in K[[Y]]. nXn to have an inverse element for the multi· a n plication ef K[[XJJ, it is necessary and sufficient that a0 =I= o, i.e. that S(o) =I= o. Proof. The condition is necessary because, if T(X) = � bnXn and if S(X)T(X) = 1, then a0b0 1 and so a0 =I= o. Conversely, suppose that a0 =I= o; we shall 1 show that (a0)' S(X) S1(X) has an inverse T1(X), whence it follows that (a0)1T1(X) is the inverse of S(X). Now = = S1(X) = I  U(X) with w(U) > 1, and we can substitute U(X) for Y in the relation (5. 1 ) , from which it follows that 1  U(X) has an inverse. The proposition is proved. Note. By considering the algebra of polynomials K[X] imbedded in the algebra of formal series K[[X]], it will be seen that any polynomial Q(X) such that Q(o) =I= o has an inverse in the ring K[[X]]; this ring then contains all the quotients P(X)/Q(X), where P and Qare polynomials and where Q(o) =I= o. 6. FORMAL DERIVATIVE OF A SERIES Let S(X) the formula = � anXn; by definition, the derived series S'(X) 1s given by n (6. I ) It can also be written I4 S'(X) = ;� .or d� S. � nanXnt. n�O The derivative of a (finite or infinite) 1.1 .7 FORMAL POWER SERIES sum is equal to the sum of its derivatives. mapping of K[[X]] into itself. The mapping S? S' is a linear Moreover, the derivative of the product of two formal series is given by the formula d dX (ST) (6. 2) dS dT dX T + S dX0 = For, it is sufficient to verify this formula in the particular case when and If T are monomials, and it is clearly true then. S(o) =fa o, let T be the inverse of S (c.f. n°. 5). gives ( )= d I dX s (6. 3) s<n>(X)= n! an + (6. 2) I dS s2 dx· Higher derivatives of a formal series S(X) = � anXn , its derivative of order n Hence, The formula S are defined by induction. If is terms of order > I. s<n>(o) = n! an, (6. 4) S<n>(o) means the minate X in S<n>(X). where result of substituting the series o for the indeter 7. COMPOSITIONAL INVERSE SERIES The series I(X) defined by I(X) = Xis a neutral element for the composition of formal series : SoI= S = I o S. PROPOSITION 7. I. Given a formal series S, a necessary and sufficient condition for there to exist a formal series T such that T(o) = o, (7. 1 ) SoT =I is that S(o) = (7. 2) o, S'(o) =? o. In this case, T is unique, and ToS = I: in other words T is the inverse of S for the law of composition o • Proof. Let S(X) = � anXn, T(Y) = � bnYn. n�O n�I S(T(Y)) = Y, If POWER SERIES IN ONE VARIABLE then equating the first two terms gives (7. a0 4) Hence the conditions (7. = o, are necessary. 2) Suppose that they are satisfied; we write down the condition that the coefficient of yn is zero in the left hand side of (7. This coefficient is 3). the same as the coefficient of yn in which gives the relation (7. 5) where Pn is a known polynomial with nonnegative integral coefficients and is linear in a2, mines b1; then, for (7. 5). ,a.. Since a1 # o, the second equation (7. 4) deter � 2, h. can be calculated by induction on n from ••• n Thus we have the existence and uniqueness of the formal series T(Y). The series thus obtained satisfies T(o) o and T'(o) # o, = and so the result that we have just proved for S can be applied to T, giving a formal series S1 such that This implies that S1 = I o S1 = (S o T) o S1 = S o (T o Hence S1 is none other than S and, indeed, T S1) o S = = S o I = S. I, which completes the proof. Remark. Since S(T(Y)) = Y and T(S(X)) = X, we can say that the 'formal transformations' Y = X S(X), = T(Y) are inverse to one another; thus we call T the ' inverse formal series ' of the series S. Proposition 7. I is an 'implicit function theorem' for formal functions. 2. Convergent power series I. THE COMPLEX FIELD From now on, the field K will be either R or C, where R denotes the field of real numbers and C the field of complex numbers. Recall that a complex number z = x + 9' ( x andy real) is represented by a point on the plane R2 whose coordinates are 16 x and y. If we associate CONVERGENT z with each complex number we define an automorphism z POWER SERIES = x > z + ry z z; is z ' conjugate ' = x  ry, of the field C, since zz' = zz'. z + z' = z + z', The conjugate of its I.2.2 in other words, the transformation z �z is znvo lutive, i.e. is equal to its inverse transformation. The norm, absolute value, or modulus by lzl of a complex number z is defined iZI = (z·z)112• It has the following properties : jz + z'/ � lzl + lz'I, The norm lzl lzz'I = lzl.lz'I, is always;;;;,:.. o and is zero only when z= o. This norm enables us to define a distance in the field C : the distance between is lz  z'I, z z' R2• and which is precisely the euclidean distance in the plane The space C is a complete space for this distance function, which means Zn e C that the Cauchy criterion is valid : for a sequence of points to have a limit, it is necessary and sufficient that lim zm  Znl = m� l n.;.. O. oo oo The Cauchy criterion gives the following wellknown theorem : if a series �Un n of complex numbers is such that �lunl < + n oo, then the series converges (we say that the series is absolutely convergent ). Moreover, We shall always identify R with a subfield of C, i. e. the subfield formed by the z such that z = z. The norm induces a norm on R, which is merely the absolute value of the real number. norm of the field C (or R) R is complete. The plays an essential role in what follows. We define = _!__ (z + z) Re(z) and 2 Im(z) = � (zz) 2Z the ' real part ' and the ' imaginary coefficient ' of 2. REVISION OF THE THEORY OF CONVERGENCE OF z e C. SERIES OF FUNCTIONS (For a more complete account of this theory, the reader is referred to Cours de Mathematiques I of J. Dixmier : Cours de l'A.C.E.S., Topologie, chapter VI, § 9.) POWER SERIES IN ONE VARIABLE Consider functions defined on a set E taking real, or complex, values (or one could consider the more general case when the functions take values in a complete normed vector space; cf. u, we write llull =sup a:EE lac. cit.). /u(x)/, which is a number > o, or may be infinite. for any scalar :A, when For each function Evidently, llull < + oo : in other words, llull is a norm on the u such that llull < + oo. of functions Un is normally convergent if the series of vector space of functions We say that a series }:llunll n norms }:11u.!1< + n is a convergent series of positive terms, in other words, if oo. This implies that, for each xeE, the series convergent, and so the series }: un(x) }:/u.(x)/ n is is absolutely convergent; moreover, " ifv(x) is the sum of this last series, u.11  n}: =:O p lim !Iv p�oo = o. p }: Un converge uniformly n=O Thus, a normally convergent series is uniformly E, the series whose general term is Un is said The latter relation expresses that the partial sums v as P convergent. to tends to infinitiy. If A is a subset of to converge normally for xeA u� = Un I A is normally convergent. each lu.(x) I series }: <n n if the series of functions (restriction of A) This is the same as saying that we can bound on A above by a constant is convergent. Un to En> o in such a way that the Recall that the limit of a uniformly convergent continuous functions (on a topological space E) is continuous. the sum of a normally convergent series of continuous functions is sequence of In particular, continuous. An important consequence of this is : PROPOSITION I. 2. Suppose that, for each n, lim x�:ro un(x) exists and takes the value an. Then, if the series }:Un is normally convergent, the series }:an is convergent and ( }:a.= lim }:u.(x) n X�Xo R ) (changing the order of the summation and the limiting process). All these results extend to multiple series and, more generally, to sum mable families of functions (cf. the abovementioned course by Dixmier). I8 CONVERGENT POWER SERIES 3· RADIUS OF CONVERGENCE OF A POWER SERIES All the power series to be considered will have coefficients in either the field R, or the field C. Note however that what follows remains valid in the more general case when coefficients are in any field with a complete, nondiscrete, valuation, that is, a field K with a mapping I of K into the set of real numbers ;;:. o such that x'> /x �Ix+ YI<( [xi+ [y[, [xy[ [x[.[y[, ? ([xi o) � ( x o), exists some x =I= o with /xi =I= 1. = = and such that there Let S(X) = � anXn n�O = be a formal series with coefficients in R or C. We propose to substitute an element z of the field for the indeterminate X and thus to obtain a 'value' S(z) of the series, which will be an element of the field; but this substitution is not possible unless the series is convergent. � anzn n�O In fact, we shall limit ourselves to the case when it is absolutely convergent. To be precise, we introduce a real variable r;;:. o and consider the series of positive (or zero) terms called the associated series of S(X). > o, which may be infinity. Its sum is a welldefined number The set of r ;;:. o for which is clearly an interval of the half line R+, and this interval is nonempty since the series converges for r = o. The interval can either be open or closed on the right, it can be finite or infinite, or it can reduce at the single point o. In all cases, let p be the least upper bound of the interval, so p is a number ;;:. o, finite, infinite, or zero; it is called the of the formal power series called the disc of convergence is empty if p = o. � anXn. n�O The set of z radius of convergence such that fzl < p is of the power series; it is an open set and it It is an ordinary disc when the field of coefficients is the complex field C. PROPOSITION 3· I. For atry r< p, the series � anzn converges normally for Jzl <( r. n�O ticular, the series converges absolutely for each z such that I z I < p; a) In par· rg POWER SERIES IN ONE VARIABLE the series 2i anzn diverges for lzl > n�O when lz/ p.) b) (We say nothing about the case p. = Proof. Proposition 3. I follows from Let r and r0 be real numbers such tha t exists a finite number M > o such that o < r < r0• ABEL'S LEMMA. for any integer n > If there o, then the series 2i anzn converges normally for [zl < r. n�O For, /anzn[ <; /an/rn <; M(r/r0)n, En and M�r/r0)n = is the general term of a convergent series  a geometric series with common ratio We now prove statement a) of proposition 3.1: if r < p, r/r0 < r. r0 such choose 2i lan/(r0)n converges, its general term is bounded n""o above by a fixed number M, and Abel's lemma ensures the normal convergence of 2i anzn for /zl <; r . Statement b) remains to be proved: n""O if Jz/ > p, we can make /anzn[ arbitrarily large by chasing the integer n suitably because, otherwise, Abels' lemma would give an r' with p < r' < /z/ such that the series 2i lan/r'n4 were convergent and this n�O that r < r0 < p; since would contradict the definition of p. Formula for the radius of convergence (Hadamard) : we shall prove the formula (3. r) lim sup Jan!lfn. r/p = n;. oo Recall, first of all, the definition of the upper limit of a sequence of real numbers Un: lim sup n::oo To prove (3. r), Un = ( lim sup p..:;,..oo n�p Un ) • we use a classical criterion of consequence: if sequence of nonnegative numbers such that lim sup 2ivn < + oo; moreover, if they are such n }: Vn + oo (this is " Cauchy's rule " and n 2ivn with a geometric series). n Here we put Vn /an/r n and find that = n_,oo that lim sup n�oo n�oo 20 (vn)l/n = ( is a then then follows by comparing the series = lim sup Vn (vn)1fn <I, (vn)1Jn > r, r lim sup n..:;,..ao Jaq/1/n) • CONVERGENT POWER SERIES Zi la.Ir " and so the series for I /r < lim sup la.I''"· n.;..oo Some examples.  4• AND (3· I ). Zi n ! z• has zero radius of convergence; n�O � z•, � n�O n:>O _!__ z•, n � � z· n>O has radius ofconvergence n MULTIPLICATION '' OF CONVERGENT POWER kl = I. SERIES. Let A(X>) and B(X) be two formal power series whose radii Let PROPOSITION 4. I. of convergence are )> p. S(X) A(X) + B(X) = be their s um and product. and P(X) A(X) . B(X) = Then : a) the ser ies S(X) and P(X) have radius of convergence ;;> b) for lzl < p; p, we have ( 4.I ) A(X) and diverges It can be shown that they behave differently when ADDITION S(z) Proof. la.I''", has infinite radius of convergence; .�on.  each of the series equal to I . This proves The series � z" �  the series converges for I /r >lira sup = A(z) + B(z), P(z) · = A(z)B(z). Let = 2i a.X•, B(X ) n�O = Zi b.X•, n?0 S(X) = � c.X•, n?O P(X ) � d.X• , = n�O and let '{n We have = la.I + jb.I, le.! < '(n, ld,.I < o,.. If r < p, the series converge, thus It follows that the series � lc.jr• n�O and 2i Id.Ir" n�O � la,. Ir" n�O and � lb.Ir" n�O converge and therefore that any r< pis less than or equal to the radius of convergence of each of the series S(X) and P (X ) . The two relations (4· I ) Thus both radii of convergence are ;;> p. remain to be proved. The first is obvious, and 21 CARTAN 2 POWER SERIES IN ONE VARIABLE the second is obtained by multiplying convergent series; to be precise, we recall this classical result : Let � Un and � Un be two absolutely convergent series. n�O n�O PROPOSITION 4. 2. Wn = If � UpVn P> o::s;;p�n then the series � w,. is absolutely convergent and its sum is equal to the product (P�.�O Up) (q��OVq ) • Write a.p = � Ju.I, �q n�p moreover, if = 1 �q\ v.I; we have � m ;> 2n, lupj.\vq\, where for n; thus, this sum is is less than a sum of terms the integers p and q is > which tends to zero as n tends to infinity. each term, at least one of less than a.0�. It follows that � � the product of the infinite sums £.1 u. and £.1 v . n�O • n?0 � + k�m 1 wk + �0a.. + h tends to 5· SUBSTITUTION OF A CONVERGENT POWER SERIES IN ANOTHER For two given formal power series the formal power series PROPOSITION 5.1. p(S) and p(T) are =!= =!= o. So T S and T with T(o) in paragraph Suppose T( X) o, = 1, = o, we have defined no. 4. � b.X•. If the radii of convergence ·�t So T is also such that � \b.\r• < p(S); the radius of then the radius of convergence of U To be precise, there exists an r> o = ·�t convergence of U is ;>. r, and, for atry z such that \z\ < r, we have \ T(z) i < p(S) and (5. I) 22 S(T(z)) = U(z). CONVERGENT 1.2.5 POWER SERIES � anXn. For sufficiently small r > o, � lbn/r• is finite n�t n;:::.o since the radius of convergence of T is ¥= o. Thus, � I bn irn t is finite n;::>t for sufficiently small r > o, and, consequently, Proo}� Put S(X) = tends to o when r tends to o. There exists, then, an r > o such that � \bnlrn < p(S) as required. It follows that R�i � cnXn, we is finite. However, this is a series � y . rn, and, if we put U (x) n�O n�O clearly obtain !cnl <:;: "(n· Thus � lcnlrn is finite and the radius of convern;::>O gence of U is > r. Relation (5. 1 ) remains to be proved. Put Sn(X) � akXk and let O<S";k<S;n Sn o T Un. For lzl <:;: r, we have = = = U.(z) = S.(T(z)), since the mapping T � T(z) is a ring homomorphism and Sn is a polyno mial. Since the series S converges at the point T(z), we have S(T(z)) = lim Sn(T(z)) . n On the other hand, the coefficients of U Un by those of = (SS.) o T are bounded a series whose sum tends to o as n+ + oo. It follows that, for lzl <; r, U(z)  U.(z) tends to o as n � + oo. Finally, we have U(z) = lim Un(Z) n�oo = lim S.(T(z)) n...;.oo = S(T(z)) for lzl < r, which establishes relation (5. I ) and completes the proof. Interpretation of relation (5. I ) : suppose r satisfies the conditions of propo sition 5. I. Denote the function z � T(z) by T, defined for l zl <:;: r, and similarly denote the functions defined by the series S and U by S and (J respectively. The relation (5. I ) expresses that, for lzl <; r, the composite function S o T is defined and is equal to (J. Thus the relation U S o T between formal series implies the relation U S o T if the radii of convergence of S and T are ¥= o and if we restrict ourselves to sufficiently small values of the variable z. = = 23 POWER SERIES IN ONE VARIABLE 6. ALGEBRAIC INVERSE OF A CONVERGENT POWER SERIES We know (§ 1, proposition 5. 1 ) that, if there exists a unique formal series to T(X) S(X) such � anXn with a0 =F o, n�O that S(X)T(X) is equal = 1. PROPOSITION 6. I. If the radius of convergence of S is =F of convergence of the series T such that ST 1 is also =F o. o, then the radius = Proof. S(X) a0 T(X) is Multiplying by a suitable constant reduces the propos1t10n to the special case when The inverse series 1 + � yn; = = 1  U(X) so that U(o) = o. moreover, the radius of convergence of the latter is equal to n>O 6. I and so =F o; proposition 7. Put S(X) I. obtained by substituting U(X) for Yin the series I then follows from proposition 5. 1. DIFFERENTIATION OF A CONVERGENT POWER SERIES PROPOSITION 7. I. Let S(X) � anXn be a formal power series and let = n�O S'(X) = � nanXn1 n�O be its derived series (c£ § 1, no. 6). Then the series S and S' have the same radius of convergence. Moreover, if this radius of convergence p is =F o, we have, for I.el< p, (7. I ) S'(z) where h tends to o _  _!11 1.1 S(z + h)  S(z) ' h without taking the value Preliminary remark. If lzi < p, values of h (in fact, for Jhj < o. lz + h i < p for sufficiently small JzJ); thus S(z + h) is defined. On relation ( 7. 1 ) that h tends to o through then p the other hand, it is understood in  nonzero real values if the field of coefficients is the field R, or by non zero complex values if the field of coefficients is the field C. of the field R, relation derivative equal to S'(z); (7. 1 ) expresses that the function proved directly. 24 S(z) S(z) is has (7. 1 ) derivative with respect to the complex S'(.<:) obviously continuous for JzJ < p, which can also be In both cases, the existence of a derived function implies that the function __.,.. in the case of the complex field C, relation shows that we also have the notion of variable z. In the case z CONVERGENT POWER SERIES Proof of proposition 7. Let r. ix.= la.I and let ( ) � ix.r• < r � nix.r•1 ·�t and, consequently, r r< < r' < p; then p. Conversely, if since r' < r < + oo, < p, choose an r' such that n11' (ix.r ) .n ( 1'r ) •1. I na.r whence n�o ' p and p be the radii of conver ' nix.r•1 If r < p , the series ·�0 � gence of the series S and S' respectively. converges, and so 1.2.7 '• p, there exists a finite , M > o such that ix.r'• 1; n ( ;, ) •l, n� n (_!_,) l < M for all n, nix.r•1 < and, since the series < converges, the series � nix.r•1 also <p and !zl <p n�I ' ' converges; thus r p . We have then that any number < p is ' ' any number < p is p , from which it follows that p p . (7· 1 ) Relation and an r such n�I T < = remains to be proved. lzl < r < p that o =I= ( 7 2) · Choose a fixed and suppose that z with lhl < r l zl in what follows. Then S(z + h) is defined, and we have S (z + (7. 3) S(z)  S, (z) h) · where we have put u.(z, h) = a. l (z Since !zl and iz h + h)•1 + + hi are + With this choice of n0, the finite sum when h = o; Thus I S(z we + (7. 3) � u.(z, h), ,:.. n�I + · · · + z•1  nz•1 j. Ju.(z, h)I< wix.r•1; E that h)  S (z)  ' S (z) h have 'proved the and, since > o, there exists an integer � u.(z,h) n�no , I n�n0 � u.(z,h) I< E/2 h (7. 2) is a polynomial in h which it follows that smaller than a suitably chosen 'Ti· we deduce from h) • 2 oo; thus, given n>no vanishes + we have < r, r < p, we have � nix.r• 1 < ·�1 n0 such that z(z = Finally, if satisfies when and l< I n�n0 � u.(z,h) I+ � 2nix.r•1 < n>n0 relation (7. 1 ) . Jhl is Jhl < 'Tj, •· POWER SERIES IN ONE VARIABLE It can be shown that the convergence of Note. S'(z) is uniform with respect to z for !zl < r (r less than the radius of convergence 8. p). S(z + h)  S(z) h towards being a fixed number strictly CALCULATION OF THE COEFFICIENTS OF A POWER SERIES Let S(x) so that p =I= o, The function S(z) be a formal power series whose radius of convergence S(z) _Li anzn is the sum of the series for lzJ < p. has for derivative the function n�O S'(z) proposition 7. I to the series to obtain its derived function sum of the power series also p. S' = _Li nanzn1• n�O ,Li n(n  I)anzn2, whose n�O We can again apply S"(z), the radius of convergence is This process can be carried on indefinitely, and by induction we see that the function tive of order n S(z) is infinitely differentiable for lzl < p; its deriva is where Tn is a series of order ;:> I, in other words Tn (o) = o. From this, we have (8. 1) an = I S< nl(o) . n! This fundamental formula shows, in particular, that, if the function S(z) is known in some neighbourhood of o (however small), the coefficients an of the power series S are completely determined. Consequently, given a function/(z) defined for all sufficiently small l zl, there cannot exist more than oneformal power series S(X) ,Li anXn whose radius of convergence is =I= o, n�O and such that f(z) = .Li anzn for lzJ sufficiently small. n�O = 9· COMPOSITIONAL INVERSE SERIES OF A CONVERGENT POWER SERIES. Refer to § I, proposition 7. 1. PROPOSITION 9.1. Let S be a power series such that S(o) and let T be its inverse series, that is the series such that T(o) = o, If the radius of convergence of S is =I= o, = o and S'(o) = o, SoT=I. then the radius of convergence of T is =I= o. The reader can accept this proposition without proof because a proof (which does not use power series theory) will be given later (chap. IV,§ 5, proposition 6. I ) . CONVERGENT POWER 1.2.9 SERIES Here, however, a direct proof using power series theory is given to satisfy the reader with an inquisitive mind. It uses the idea of ' majorant series ' (cf. chap. vu). Let us keep to the notations of the proof of proposition 7.1 in§ I and let us consider relations (7. 5) of§ I which enable us to calculate the unknown coefficients bn of the required series T(X) . Along with the series S(X) , we consider a 'majorant' series, that is a series S(X) = A1X � n�2 A.X• with coefficients A. > o such that I a. I < A. for all n; moreover we assume that A1 la1j. Applying § 1 proposition 7. I to the series S, gives a series = T (Y) such that S(T(Y) ) = = � n�t B.Y• Y; its coefficients B. are given by the relations (g. 1) which are analogs of (7. 5) of§ We obtain from them by induction on n r. (g. 2) lb.I< B•. It follows that the radius of convergence of the series T is not less than that of the series T. We shall prove proposition g. 1 by showing that the radius of conver gence of T is > o. To this end, we choose the series S as follows: let r > o be a number strictly less than the radius of convergence of the series S (by hypothesis, this radius of convergence is =I= o) ; the general tenn of the series � la.Ir" is then bounded above by """' by a finite number M > o and, if we put (g. 3) for n )> 2, A.= Mfr" we obtain the coefficients of a majorant series of S; its sum S(x) is equal to S(x) = A1x  M x2/r2 x/r  r for lxl < r. We seek, then, a function T(y) defined for sufficiently small values of y which is zero for y o and which satisfies the equation S(T(y)) y identically; T(y) must satisfy the quadratic equation = (g. 4) = (A1/r + M/r2) 'f2  (A1 + yr)T + y which has for solution (which vanishes wheny = = o, o) When I y I is sufficiently small, the surd is of the form A1 v'I+u , with I u I < I, and so T(y) can be expanded as a power series in y, which converges for sufficiently small I y I· Thus the radius of convergence of this series is =I= o, as required. POWER SERIES IN ONE V ARlABLE 3. Logarithmic and Exponential Functions I. EXPONENTIAL FUNCTION We have already remarked (§ 2, no. infinite radius of convergence. For 3) that the formal series � �X· has n z complex, we define ·�0 • that is, the sum of an absolutely convergent series. This function has derivative d (e') = e' ( I. I ) dz by proposition 7. 1 of§ 2. On the other hand, applying proposition 4. 2 of § 2 to two series with general terms I I Vn = lzIn, n. n Un= lz, n. gives w. = I � ..:::. ' 'zPz'np o�p�np.(n p). = I I n. (z + z')•. Consequently ( 1. 2) (the fundamental functional property of the exponential function). In particular, e'.e'= (1. 3) Putting z= x 1, e' so =I= o + iy (with x andy real) gives so we need only study the two functions variables. We have d (1. 4) 2. for all ;:;. dx (e"') = d  e"', REAL EXPONENTIAL FUNCTION e"' dy and . (e'Y) = = 1 x2 + x + + 2 where x and y are real z..e'Y. e"' We have seen that e"' =I= o: what is more, e"' the expansion e"' eir, · · · = (e"'12)2 > o. Moreover, shows that e"'> I +x when x>o. LOGARITHMIC AND EXPONENTIAL FUNCTIONS r.3.2 Thus =+ lim e'° ��+oo  x for substituting x oo; leads to lim e'° = o. We deduce that the function e'° of the real variable from o to + oo. The transformation t = e'° x increases strictly has therefore a inverse trans formation defined for t > o; it is denoted by x = log t. This function is also strictly monotonic increasing and increases from oo to+ oo. The functional relation of e'° is written log (tt') and, in particular, log I = = log t+ log t', o. On the other hand, the theorem about the derivative of an inverse function gives d (2. 2) (log t) dt = I/t. Let us replace t by I+ u (u> I); log (I+ u) is the primitive of which vanishes for u = o; 1I+ u moreover we have the following power series expansion I  I+ U =I u+ u2+ ... + ( I)nlun1 + ... whose radius of convergence is equal to I. From proposition 7. I of § 2, it follows that the series of the primitive has the same radius of convergence I and that its sum has derivative __; I+ U (2. 3) log (I+ u 2 u ) = u + 2 whence, for lul < I, un ... + ( I)n1_+ ... n (in fact this expansion is also correct when u = 1 ). Now put (2. 4) S(X) = � n;::>I � Xn, n • and examine the composed series U 5. I of§ 2, for  1 < u <+ 1, V(u) = =So T. We have from proposition S(T(u) ) ; 29 POWER however, T( u) =log V(u) SERIES ( I + u), = IN ONE S(x) e1•g(l+u)_ This shows that the formal series VARIABLE =e" I= I, so (1 + u)  I= u. U is merely I because of the uniqueness of the power series expansion of a function (cf. series S 3· THE IMAGINARY EXPONENTIAL The series expansion of thus ei1. ei1 to by relation I § 2, no. REAL ) 8). Thus the and T are inverse. ei1 FUNCTION shows that ei1 (y is the complex conjugate of is the square of the modulus of (1. 3). eiY eit; ei1; but this product is equal Thus We note that, in the Argand plane representation of the complex field C, the point ei1 is on the unit circle, from the origin o is equal to form a group U I. that is the locus of points whose distance The complex numbers u such that /uj = I under multiplication and the functional property : the mapping y  ei1 is a homomorphism of the additive in the multiplicative group U. This homomorphism will be studied expresses the following group R more closely. The homomorphism y  ei1 maps R onto U, and its' kernel' (subgroup of the y such that eit I, the neutral element of U) is composed of all the integral multiples of a certain real number > o. By definition, this number will be denoted by 2'7t. THEOREM. = Proof. Let us introduce real and imaginary parts of e;Y; we put, by defini tion, ei1 =cosy+ siny, which defines two real functions cos y and sin y, such that cos2 y + sin2 y = 1. These functions can be expanded as power series whose radii of convergence are infinite : I (3• I ) I COS y I = I  y 2+ sin y =y 2 I y3 + __ �! ( 1 ) ny2 I' + n ' (zn)!  I )n 2n +I + .... ... + ( (2n + I ) ! y · · · · · · LOGARITHMIC AND EXPONENTIAL FUNCTIONS We shall study the way in which these two functions vary. Observe that separating the real and imaginary parts in the second equation gives � When y � (cosy) =  siny, ( I. 4) (siny) = cosy. o, cosy is equal to I ; since cosy is a continuous function, there exists a y0> o such that cos y> o for o �y �Yo· Hence sin y, whose = derivative is cos y, is a strictly increasing function in the interval Put sin y0 =a> o. [o,y0]. We shall show that cos y vanishes for a certain Suppose in fact that cosy> o for y0 �y �y1; value ofy which is> o. we have (3. 2) cosy1  cosy0 =  ly, sinydy. Yo However, sin y ;> a, because sin y is an increasing function in the interval [y0, y1] where its derivative is > o, thus ly, sinydy;> a(y1 y0). Yo By substituting this ( . 2) 3 m and noting that cos y1 > o, we find that Y1 yo I < cOSYo· a [ y0, Yo + This proves that cos y vanishes in the interval ; cos YoJ Write _::... for the smallest value ofy which is > o and for which cosy = o 2 (this is a definition it). of the number In the interval [ o, :} decreases strictly from I the mapping y is a bijective mapping of the compact interval [ u o,  eiY to o, and sin y increases strictly from o to cosy : J onto the set of points and v are both ;> o. The mapping y of the unit circle u2 + v2 For _::... 2 that and eiY <Y <it, of the unit circle whose coordinates compact  = eiY is I space, we deduce : a homeomorphism ef we have ordinate [ in the positive quadrant u eiY = iei(yf), is;> o, and takes o, :J ;> o, whence we takes each complex value of modulus whose ; thus By a theorem of topology about continuous, bijective, mappings of a LEMMA. (u, v) I v onto the sector ;> o. easily deduce I whose abscissa is <: o each value precisely once. Analogous results can be deduced for the intervals [ it, 32it J and [ 37t2 , 27t} POWER SERIES IN ONE VARIABLE for o �y < 27t, i er precisely once, whereas 2i"' Thus, e takes = each complex value of modulus Therefore the function 1. i  er of period 27t, and the mapping y maps R on U. i e r 1 is periodic This completes the proof of the theorem. 4. MEASUREMENT OF ARGUMENT OF ANGLES. COMPLEX NUMBER A Let 27tZ denote the subgroup of the additive group the integral induces an The multiples isomorphism inverse of the number 27t. The R formed by mapping y i  er cp of the quotient group R/27tZ on the group U. 1 cp of U on R/27tZ associates with any isomorphism complex number u such that !u! 1, = a real number· which is defined up to addition of an integral multiple of 27t; this class of numbers is called the argument of u and is denoted by arg u. By an abuse of notation, arg u will also denote any one of the real numbers whose class modulo 27t is the argument of u; the function arg u is then an example of a manyvalued function, that is, it can take many values for a given value of the variable u. This function resolves the problem of ' measure of angles ' (each angle is identified with the corresponding point of U) : the ' measure of an angle ' is a real number which is only defined modulo 27t. We topologize the quotient group R/27tZ by putting on it the topology of the usual topology on the real line R : let p quotient be the canonical mapping of R on its quotient R/27tZ, a subset A of R/27tZ is said to be if its inverse image p 1 (A), which is a subset of by 27t, is an open set of R. It is easily verified that the topological space R/27tZ is Hausdorff (that is, open neighbourhoods). open R invariant under translation that two interval [0,27t], the natural mapping I  distinct compact; Moreover, it is points have disjoint for, if I is the closed R/27tZ takes the compact space I onto the Hausdorff space R/27tZ which is then compact by a classical theorem in topology. The homomorphism cp : R/27tZ  U is continuous and is a bijective mapping of the compact space R/27tZ onto the Hausdorff space U; hence cp is a homeomorphism General definition ef argument of R/27tZ on U. : for any complex number t of= o, define the argument oft by the formula arg t = arg (Fl)· The right hand side is defined already since the argument of o is not defined.) addition of integral multiples of 27t. (4· 32 1 ) t = t/!ti e U. (Note that As above, arg tis only defined up to We thus have ltlei•rgt. I.3.5 LOGARITHMIC AND EXPONENTIAL FUNCTIONS To solve the equation t" =a (where a =I= o is given) equation is equivalent to Application. Jtj = Jajlfn, the I arg t = arg a, n and has n complex solutions t because one obtains for arg t a real number defined up to addition of an integral multiple of 27r/n. 5· COMPLEX LOGARITHMS Given a complex number t, we seek all the complex numbers z such that =t. Such numbers exist only when t =I= o. In this case, relation shows that the z that we seek are the complex numbers ofthe form e' (4· I ) log !ti+ i arg t. (5. I ) We define log t =log (5. 2) It I + i arg t, which is a complex number defined only up to addition of an integral multiple of 27ri. From this definition, we have e10g1 =t. When t is real and > o, we again have the classical function log t if we allow only the value o for arg t. For any complex numbers t and t' both =I= o and for any values of log t, log t' and log tt' , we have (5· 3) log (tt') =log t + log t' (mod Branches of the logarithm. 27ri). So far we have not defined log in the proper sense ofthe word. t as a function Definition. We say that a continuous functionf(t) of the complex variable t, defined in a connected open set D of the plane C, not containing the point t = o, is a branch oflog t if, for all t e D, we have ef(t) =t (in other words, if f(t) is one of the possible values oflog t). We shall see later (chapter n, § I, no. 7) what conditions must be satisfied by the open set D for branch oflog t to exist in D. We shall now examine how it is possible to obtain all branches of log t ifone exists. PROPOSITION 5. I If there exists a branch f (t) of log t in the connected open set D, then any other branch is of the for m f(t) + 2k7ri (k an integer); conver sely, f ( t) + 2k7ri is a br anch of log t for arry integer k. 33 POWER SERIES f (t) Let us suppose the that IN ONE VARIABLE g(t) and are two branches of log t. The difference f(t)  g(t) h(t) = 2'1tZ_ is a continuous function in D which takes only integral values; since D is assumed connected, such a function is necessarily of points closed. te D h( t) such that constant. For, the set is equal to a given integer n is both open and Thus the set is empty or is equal D. is One defines similarly what must be understood by a branch of arg t That f (t) + 2k'lti The constant must of course k be an integer. is a branch of log t for any integer obvious. in a connected open set D which does not contain the origin. Moreover, any branch of arg Example. t defines one of log t and viceversa. (t) > o Let D be the open halfplane Re denotes the real part oft). value of arg t t in this For any (recall that Re (t) halfplane, there is a unique which is > _2:_ and < .2:. ; denote this value by Arg t. 2 We shall show that Arg t is a 2 continu ous functi on log It I + i Arg and that consequently t (t) > o. It will be called the prin cipal branch of log t. Since Arg t Arg (t JI t J) and since the mapping t + t JI t / is a continuous mapping of the halfplane Re ( t) > o on the set of u such that Jul 1 and Re (u) > o, it is sufficient to show that the is a branch of log t in the half plane Re = = mapping y u = = Arg u is continuous. However, this is the inverse mapping of ; [; the function u = eiY the compact interval [  ; , + ; ] ] fir as y ranges over the open interval  is a continuous bijective mapping of on the set of u Iu/ such that = 1 and Re ;. (u) ;> + o; this then is a homeomor phism and the inverse mapping is indeed continuous, which completes the proof. 6. SERIES EXPANSION OF THE COMPLEX LOGARITHM PROPOSITION 6. 1. The sum of the power series which converges for / u/ < Note first that if 34 f ul < 1, 1, i s equ al to the principal branch of log ( 1 + u). t= 1 + u remains inside an open disc contained LOGARITHMIC AND EXPONENTIAL FUNCTIONS m the half plane Re ( t) > o. Again we use the notations of relation ( 2. 4) and remember that the series S and T are inverse to one another; proposi tion 5. I of§ 2 shows that S(T(u)) = u for any complex number that f ul < I. In other words, is a branch of log (I + u). eT(u) = 1 such u + u; and consequently T(u) To show that this is the principal branch, it is sufficient to verify that it takes the same value as the principal branch for a particular value of u, for instance, that it is zero when u = o, which is obvious from the series expansion of T(u). PROPOSITION 6. 2. lff ( t ) is a branch of log t in a connected open set D, the Function f ( t) has derivativef' ( t) with respect to the complex variable t, and f' ( t ) In fact, for h = I/t. complex =I= o and sufficiently small, we have f (t + h)  f( t) _f (t + h) f (t) . h and, when e" e' , zz t  efCt+h)_efCt) ' tends to o, this tends to the algebraic inverse of the limit of as z' tends to z = f( t ) ; the limit we seek is then the inverse of the value of the derivative of e' for z = f ( t) , which is equal to efCt> Note. = I It. This result checks with the fact that the derivative of the power I I + series T(u) is indeed equal to __, Definition. U For any pair of complex numbers t =I= This is a many valued function oft for fixed connected open set D is defined as above. ix. o and ix, we put A branch of ta in a Any branch of log t in D defines a branch of ta in D. Revision. Here the reader is asked to revise, if necessary, the power series expansions of the usual functions, arc tan for any complex exponent where log ( I + x) ix and for x x, arc sin complex such that x , etc. [xi< I, Moreover, we consider denotes the principal branch (the function then takes the value I for x ( I + x) " = o) ; the reader should study its power series expansion. 35 POWER SERIES IN ONE VARIABLE 4. I • Analytic Functions of a Real or Complex Variahle DEFINITIONS Definition I. I. We say that a functionf(x), defined in some neighbourhood of x0, has a power series expansion at the point x0 ifthere exists a formal power series S(X) satisfies = � anX" whose radius of convergence is =I= o and which n;;::,o for 1x  x01 sufficiently small. This definition applies equally well to the case when x is a real or a complex variable. The series S(X), if it exists, is unique by no. 8 of§ 2. Iff(x) has a power series expansion at x0, then the functionfis infinitely differentiable in a neigbourhood of x0 because the sum of a power series has this property. If the product Jg oftwo functionsfand g having power series expansions at x0 is identically zero in some neighbourhood of x0, then a least one ofthe functionsf and ..� is identically zero in a neighbourhood of x0; in fact, this is an immediate consequence of the fact that the ring of formal series is an integral domain (§ I, proposition 3. 1). If f has a power series expansion at x , there exists a function g also having a power series expansion at x0 and having derivative g' =fin some neighbourhood of x0; such a function is unique up to addition of a constant in some neigh bourhood of x0; to see why this is so, it is sufficient to examine the series of primitives of terms of a power series expansion of the functionf. We shall consider in what follows an open set D of the real line R, or the complex plane C. If D is open in R, D is a union of open intervals and, if D is also connected, D is an open interval. We write x for a real or complex variable which varies over the open set D. 1. 2. A function f( x) with real or complex values defined in the open set D, is said to be analytic in D if, for any point x0 e D, the function f(x) has a power series expansion at the point x0• In other words, there Definition must exist a number p(x0) > o and a formal power series S(X) = with radius ofconvergence;>. p(x0) and such that f(x) = � an(x  x0)" n;;;:,o � a.Xn n;;;:,o for The following properties are obvious : any analytic function in D is infinitely differentiable in D and all its derivatives are analytic in D. ANALYTIC FUNCTIONS OF ONE VARIABLE The sum and product of two analytic functions in D are analytic in D: that is to say, the analytic functions in D form a ring, and even an algebra. It follows from proposition 6. 1 /f (x) f(xo) 1 of § 2 that, if f (x) is analytic in D, then is analytic in the open set D excluding the set of points o. = Finally, proposition 5. g values in D' and if is analytic in D. 1 of§ 2 gives that, if f x0 such that is analytic in D and takes its is analytic in D', then the composed function g of Let f be an analytic function in a connected set D; if f has a primitive g, ' that is, if there exists a function g in D whose derivative g is equal to j, then this primitive function is unique up to addition of a constant and it . is an analytic function. Examples of analytic functions. Polynomials in x are analytic functions on the whole of the real line (or in the complex plane). A rational function P(x)/Q(x) is Q(x0) o. = analytic. t1ve 2. 1 It will follow from proposition 2. The function arc tan I • analytic in the complement of the set of points 1 • + xz x 1 x0 such that that the function e:& is is analytic for all real x since its deriva . is ana yt1c. CRITERIA OF ANALYTICITY PROPOSITION convergence p 2. I. Let S(X) is =F o. = Let }: anXn be a power series whose radius of n�o S(x) be its sum for lxl < p. = }: anx n n�O Then S(x) is an analytic function in the disc Jxj < This result is by no means trivial. p. It will be an immediate consequence of what follows, to be precise : PROPOSITION lxol < 2. 2 With the conditions of proposition Then the power series p. 2. I, let x0 be such that ( 2. I) has radius of convergence (2. 2) > p  lx01 and for CARTAN Ix  Xol < P  !xol· 3 37 POWER SERIES IN ONE VARIABLE Proof of propositio n For r0 < r < ( 2. 3) h 2. 2. Put r0 = lx01, oc. = la.I. We have p, we have \ 1s<P>(xo) I (r  ro)P < ph, (pp.; .r) ! OCp.tq(ro)q(r  ro)P, p?:oP· q q < h n�O oc. ( O�p�n � p!( n; p) ! (rr0)P(r0)•P ) , n. < h oc.r• < n�O oo. + ( 2. 1 ) Thus the radius of convergence of the series is > r  r0. can be chosen arbitrarily near to p, this radius of convergence is Now let x be such that � ..:.i p, q !x  x01 < p  r0• The double series ( 2. 3). Its sum can therefore be calculated by regrouping the terms in an arbitrary manner. this sum in two different ways. h a. n! h (O�p�nP· (n 1 r (p + q) ! ap+q(xo)q(x  xo ) P P'• q'• is absolutely convergent by n?;0 Since > p  r0• 1 ) p. We shall calculate A first grouping of terms gives (x  x0)P(x0)•P ) = h a.x• n?;0 = S(x); another grouping gives Formula ( 2. 2) follows from a comparison of these two and this completes the proof. Note I. than p The radius of convergence of series lxol· S(X) Then S(x) I ix  I  = = ( 2. 1 ) may be strictly larger Consider, for example, the series I . I  Z X I I  for iXo ( I Ix!< = h (iX)•. n�O I. Choose a real number for . X x0  Z I  ixo )1 � = _.:.i . i• n?; o ( I  ixo)•+l x0, so we have (X  Xo) . " ANALYTIC FUNCTIONS OF ONE VARIABLE This series converges for [x  x0[<VI + (x0)2 and V I + (x0)2 is strictly greater than 1  f xoi· .Note 2. Let A(r) = � [an[rn for n�O r< P· From inequality (2. 3), we have (2. 4) for [x[ � r0 < r< p • .Note 3. If x is a complex variable, we shall see in chapter II that any func tion which is differentiable is analytic and is consequently infinitely diffe rentiable. The situation is completely different in the case of a real variable : there exist functions which have a first derivative but no second derivative (one need only consider the primitive of a continuous function which is not differentiable). Moreover, there exist functions which are infinitely differentiable but which are not analytic; here is a simple example: the functionf(x), which is equal to zero for x o and to eI/z• for x =I= o, is infinitely differentiable for all x; it vanishes with all its deri vatives at x o so, if it were analytic, it would be identically zero m some neighbourhood of x o, which is not the case. = = = THEOREM. In order that an irifinitely differentiable function of a real variable x in an open interval D should be analytic in D, it is necessary and si+ffecient that any point x0 e D has a neighbourhood V with the following property : there exist numbers M and t, finite and > o, such that for arry x e V and any integer p ;;>. o. Indication of proof. The condition is shown to be necessary by using inequality (2. 4). It is shown to be sufficient by writing a finite Taylor expansion of the functionf(x) and using (2. 5) to find an upper bound for the Lagrange remainder. 3· PRINCIPLE OF ANALYTIC CONTINUATION THEOREM. Let f be an analytic function in a connected open set D and let x0 e D. The following conditions are equivalent : o for all integers n ;;>. o; a) f<nl(x0) b) f is identically zero in a neighbourhood of x0; c) f is identically zero in D. = 39 POWER SERIES Proef. b) It is obvious that and implies for all n c) implies Suppose ONE VARIABLE IN We shall show that a). is satisfied. 0 > o with the convention that j< > c). a) series expansion in powers of coefficients � J n. n (x0) = f. b) = o But f (x) has a power in a neigbourhood of x0 and the are zero; thus f (x) is identically zero in a neighbour hood o f x 0 which proves Suppose conditions (x  x0) implies a) We have then J<n>(x0) b) b). To show that f is zero at all points of is satisfied. D, it is sufficient to show that the set D' of points xeD in a neighbourhood efwhichf is identically zero is both open and closed of b), thus, since D is connected, D' will be equal definition of D' that it is open. It remains to be proved that, if is in the closure of D', then x0eD'. at points arbitrarily close to x0 (D' is not empty because to D). It follows from the However,J<n>(x) x0 eD o for eachn >o = (in fact, at the points of D'); thus J<n>(x0) because of the continuity of J<n>; this holding for all above that f (x) is identically zero in a neighourhood of x0• = o implies as n >o Thus x0eD', which completes the proof. COROLLARY I. The ring ef analytic functions in a connected open set D is an integral domain. For, if the product Jg of two analytic functions in D is identically zero and if x0e hood of D, then one of the functions f, g is identically zero in a neighbour x0 because the ring of formal power series is an integral domain. But, iffis identically zero in some neighbourhood of x0, thenfis zero in the whole of D by the above theorem. COROLLARY 2. (Principle of analytic continuation) 1J f and g in a connected open set then they are identical in D. The D two anarytic functions coincide in a neighbourhood ef a point ef D, problem ef anarytic continuation is the following : given an analytic function h in a connected open set D' and given a connected open set D containing D', we ask if there exists an analytic function f in D which extends h. Corollary 2 shows that such a function f is unique if it exists. 4• ZEROS OF AN ANALYTIC FUNCTION Let f (x) be an analytic function in a neighbourhood of x0 and let f (x) = � n;?:0 ( an xx0 )n be its power series expansion for sufficiently small Ix that f(x0) = o and that f (x) J  x0 . Suppose is not identically zero in a neighbourhood of x0• ANALYTIC FUNCTIONS OF ONE Let k be the smallest integer such that converges for sufficiently small function such that g(x0) =/= near enough to we have x0, f (x) = ak =/= Ix  x0J o. 1+5 The series and its sum g(x) o in some neighbourhood of (xx0)kg(x), g(x0) =/= The integer k > o thus defined is called the for the function f. VARIABLE is an analytic x. Thus, for x o. order of multiplicity of the zero x0 (4· 1 ) , where g(x) It is characterized by relation is analytic in a neighbourhood of x0• The order of multiplicity k is also characterized by the condition If k = 1, we call Relation f(x) =/= o x0 a zero. simple and continuity of (4· 1 ) o for In other words the point zero of the function f(x). If k ;>. 2, we call g(x) <lxx01 < • x0 imply (e > x0 a multiple zero. o sufficiently small). has a neighbourhood in which it is the unique PROPOSITION 4. 1. Iff is an analytic function in a connected open set D and if f is not identically zero, then the set of zeros off i's a discrete set (in other words, all the points of this set are isolated). For, corollary 2 of no. 3 gives that f is not identically zero in a neigbour hood of any point of D, so one can apply the above reasoning to each zero off. In particular, any zeros of the function compact g. subset of D contains only a 5· MEROMORPHIC FUNCTIONS Let f and g suppose that finite number of be two analytic functions in a connected open set D, and g is not identically zero. f(x)/g(x) is defined x0 ofD such that g(x0) # o, The function and analytic in a neighbourhood of every point that is to say, in the whole of D except perhaps in certain isolated points. Let us see how f(x)/g(x) behaves in a neighbourhood of iff(x) is not identically zero, we have which is a zero of g(x); f(x) = (xx0)"fi(x), a point x0 POWER SERIES IN ONE VARIABLE where k and k' are integers with k ;>. o and k' > o, f1 and g1 are analytic in some neighbourhood of x =I= x0 x0 with x0, but near to f_(x) g(x) (x = f1(x0) =I= _ o and g1(x0) =I= o; hence, for 1x x0)k  k' f ( )_ g1(x) The function h 1(x) =f1(x)/g1(x) is analytic in a neighbourhood of we have that 1o h1(x0) =I= o. x0 and Two cases arise: k ;>. k'; then the function (x xo)kk'h1(x) is analytic in some neighbourhood of x0 and coincides with f(x)/g(x) for x =I= x0• Hence the extension off/g to the point bourhood of x0 and admits 20 x0 as x0 is analytic in a neigh a zero if k > k'. k < k' : then I f(x)  ( XXo)k'k h1(x), g ( X) _ We say in this case that is called the to + oo. " infinity" x0 is a order of m ultiplicity pole of the pole. As x tends to x0, I�(�} I tends We can agree to extend the function fg / by giving it the value at x0• We shall return later to the introduction of this unique number infinity, denoted lff(x) of the functionf/g; the integer k' k oo. analytic and has x0 as a zero of order k > o, then of order k of I x0 is clearly a pole If (x). Definition. A meromorphic function in an open set D is defined function f(x) which is defined and analytic an the open set D' to be a obtained from D by taking out a set of isolated points each of which is a pole off(x). In a neighbourhood of each point of D (without exception),'j can be expressed as a quotient h(x)/g(x) of two analytic functions, the denominator being not identically zero. The sum and product of two meromorphic functions are defined in the obvious way : the meromorphic functions in D form a ring and even an algebra. In fact they form a field because, iff(x) is not identically zero in D, it is not identically zero in any neigh bourhood of any point of D by the theorem of no. 3; sof(x) is then analytic, or has at most a pole at each point of D and is consequently meromorphic in D. PROPOSITION 5. I. The derivative f' of a meromorphic function fin D is mero morphic in D; the functions fand f' have the same poles; if x0 is a pole of order k ofj, then it is a pole of order k + I off '. EXERCISES For, f' is defined and analytic at each point of D which is not a pole off x0 It remains to be proved that, if Moreover, for x near x0, J (x) g(x) being analytic with and as g1(x0) =I= o, x0 is is a pole of j, x0 is also a pole of f'. I (XX0 )kg(x), = g(x0) =I= o, k > o. Hence, for a pole off' of order k + x =I= x0, I. Exercises 1. Let K be a commutative field, X an indeterminate and E the algebra of formal power series with coefficients in K. = K[[X]] For S, T in E, define d(S ' T) �  o ( ek T, if S if S =I= T, = and w(S  T) = k. a) Show that d defines a distance function in the set E. b) Show that the mappings (S, T) + S + T and (S, T) +ST of E x E into E are continuous with respect to the metric topology defined by d. c) Show that the algebra K[X] of polynomials is everywhere dense in E when considered as a subset of E. d) Show that the metric space E is complete. (If (S.) is a Cauchy sequence in E, note that for any integer on n for sufficiently large m > o, the first m terms of S. do not depend n. ) e) Is the mapping S + S' (the derivative of S) continuous? 2. Let p, q be integers ;;;> 1 1. Let S1 (X) be the formal series + x + x2 + . . . + x· + ... , and put a) Show, by induction on (I) 1+P+l!JP n, that + +1) +n1) + ... +P(P 1) ... (p n! 2! = (p+1) ... (p+n) , n! 43 POWER SERIES IN ONE VARIABLE and deduce (by induction on Sp(X) (2) where b) ( �) (3) � = ·�o (P + nnr) x·, h!(k � h) ! denotes the binomial coefficient Sp (X). Sq(X) Use the expansion p), = SP+q (X) to show that + l r ) (P+q+n n +r ) (P + ll+r ) (q +nnl = � o,,;1,,;n (which is a generalisation of (r), the case when q = r). 3· Find the precise form of the polynomials pn in the proof of proposition 7. I' § r, for n .::::;;;: 5 and calculate the terms of degree .::::;;;: 5 of the formal (compositional) inverse series of S(X) 4. Find the radii of convergence of the following series.: (lq l < r), � q•'z• ·�o � nPz• (p integer> o) , n�O � a.z•, with Oz..+1 b2n a2n+l, a2n n�O ) a b) c) = where 5. I I I X Xa + Xs + ... + ( r)P  X2P+l +.... 3 5 2p+r = a b and are real and o = n;;;;:,. for o, <a, b <r. Given two formal power series S(X) � a.x· and � (a.)PX", V(X) = ·�o T(X) = � b.X· (b. '/= o) , ·�o let U(X) (where p = n�O is an integer). p(U) and, if p( T ) 'f= = = � a.b.X•, n�O W(X) = Prove the following relations: (p(S))P, p(V ) ;;;;:,. p(S). p(T), o, p(W) < p(S)/p(T). � (a./b.) X n�O EXERCISES Let a,b and c be elements of radius of convergence of the series 6. S(X) = I C, c not an integer < o. What is the · + 1). (b + 1) x2 ... +ab x +a(a + c 2!c(c + l) . +a(a+ l) . .(a + n l ) .b(b + l) .. (b n!c(c + 1) ... (c + n  l) . n  l) x· +. .. + Show that its sum S(z), for fzJ < p(S), satisfies the differential equation z(1 z)S" + (c(a + b + l)z)S'  abS 7. Put Let S(X) s. = a0 + = o. Li a.X• be a formal power series such that p(S) = ·�0 · · + · l a., t. =  (s0 n+ 1 + s1 + · · for + s.) · n > === 1. o, and put V(X) = 2i t.X•. U(X) = 2i s.X•, n�O Show that : (i) p(U) = p(V) 1  = lz 8. Let S(X) = ,.�o l, (ii) for all fzl < l, ( Li a.z•) = Li s.z•. n�O n�O Li a.X• be a formal power series whose coefficients are n�O defined by the. following recurrence relations : a0 = o, a1 = l, a. = oi:an1 + �an2 for n > 2, where a, � are given real numbers. a) Show that, for n > 1, we have fa . [ <;; (2c)•1 where c =max (fa!, l�I, 1/2) and deduce that the radius of convergence p(S) =F o. b) Show that (1 az �z2)S(z) = z, for lzl < p(S), and deduce that, for fzl < p(S), S(z) = c) z l az  �z2 · Let z1, z2 be the two roots of �X2 +aX  I = o. By decomposing 45 POWER SERIES IN ONE VARIABLE the right hand side of (1) into partial fractions, find an expression for the an in terms of z1 and z2 and deduce that p(S) =min <lz1I• lz2D· (Note that, if S(X) =S1(X) . S (X), then p(S) >min (p(S1), p(S 2) ) . ) 2 g. Show that, if x, y are real and n is an integer >o, then � O(;p(;n � O(;p(;n sin (px + y) =sin cos (px + y) =cos (Use cos(px + y) + 10. 1 1. (2 (2 i sin (px + y) ) + y) !!__x + y sin n !!__x n +1 2 sin x/sin �. 2 +1 x/sin�, 2 2 =ei<P"'+Y) =eir (ei"') P. ) Prove the following inequalities for z e C : Show that, for any integer n >1 and any complex number z, (1 + )n =1 + z + 2,;;p,;;n (1) ... (1 ) z and deduce that n I n zP ' I p. n ( 1 + _£) · n�oo e' = lim r2. p I £.J � n n Show that the function of a complex variable z defined by cos z = e;, + 2 eiz ( . eit resp. sin z = _ eiz 2i ) is the analytic extension to the whole plane C of the function cos x (resp. sin x) defined in § 3, no. 3, Prove that, for any z, z' e C, cos (z + z') =cos z cos z'  sin z sin z', sin (z + z') =sin z cos z' + cos z sin z'; cos2 z + sin2 z = I. r 3. Prove the relations � x < sin x TC < x for x real and o < x < 7t/2. EXERCISES =x + ry with 14. Let (i) Show that z x, y real. (x + 01) 12 =sin2 x + (x + 0>) 1 2 =cos2x + !sin !cos sinh2y, sinh2y; az (ii) determine the zeros of the functions sin az, cos number =F o) ; (iii) Show that, if l � az l .:;;; (N. B. 15. 7t sn cos� Slll'ltZ cosh'lty and I  l cosh a . sm az ,;;;::: sin'ltz""' . Slnh 'lt <a< 7t and is a positive integer, n . I z = ++ ry, 2 for . n (where a is a real (n ___!__) ( 1)' + 2 for n+2 z By definition, cosh =cos (iz), sinhz = isin Let I be an interval of the real line R. (iz).) Show that, ifj(x) is an analytic function (of a real variable but with complex values) in I, it can be extended to an analytic function in a connected open set D of the complex plane containing I. 16. (or:n), (�n) (i) Let be two sequences of numbers with the following properties : M> a) there is a constant o such that lor:1+or:2+···+a.l<; M b) the �n (Introduce Sn =or:1 + > 1, · cients such that that the series 1] p(S) • n · is · · n>1, >�. > · · · . and write be a formal power series with complex coeffi =1, and that � a.xn n""O �1>�2> · + or:. Let S(X) = � anX n�o (ii) [o, n are real > o and Show that, for all for all �an n""O is convergent. Use (i) to show uniformly convergent in the closed interval of R, and deduce that lim o<;�1 � a.x• """o = � a•. """o 47 POWER (iii ) Let S(X)= }": n�1 SERIES X • n2 IN ONE VARIABLE now and let D be the intersection of the open / disc lzl < 1 and of the open disc lz I I< constant a such that I. Show that there exists a S(z) + S(1 z)=a log zlog (1 z) for ze D, where log denotes the principal branch of the complex logarithm in the half�plane Re(�)> o (which contains D). (Note tha�,. if z e D, then log (I z)= T(z) with T(X) because of proposition 6.  (log dog ( 1 z)) k X.S'(X), of § 3, and that proposition 6. I d = = log ( 1 z) z � IZ for 2 of § 3 gives zeD. Finally, use (ii) to show that a= a (Cf. chapter v, § 2, no. � 1 /n2,  (log 2) 2 = }": 1 /n22•1. n�t 2, the application of proposition 2. 1.) ) cHAPnR II Holomorpbic Functions� Cauchy's Integral 1. Curvilinear Integrals GENERAL THEORY I. We shall revise some of the elementary ideas in the theory of curvilinear R2• Let x andy denote the coordinates in R2� A differentiable path is a mapping integrals in the plane ( I. I ) t  y( t ) [a, b] into the plane R2, such that the coordinates x(t) and point y(t) are continuously differentiable fU:nctions. We shall always suppose that a< b. The initial point of y is y ( a ) and its end point is y(b). If Dis an open set of the plane, we say that y is a differen of the segment y(t) of the tiable path of the open set Dif the function y takes its values in D. A dijferential form in an open set Dis an expression w = P dx + Qdy whose coefficients P and Q are (real or complexvalued) continuous functions in D. If y is a differentiable path of D and define the integral .£w w a differential form in D, we by the formula J: lb r'(CJ>), (I)= .. where y*(w) denotes the differential f(t) = formf(t) dt defined by P(x(t), y( t ))x' ( t ) + Q(x(t), y(t))y'(t); 49 HOLOMORPHIC FUNCTIONS, CAUCHY'S INTEGRAL in other words, y*(w) is the differential from x=x(t), y=y(t). Thus, deduced from by the w change of variables Consider a1 < u now a continuously which is such that u + t ( u) t(a1)=a, t(b1)= b. ( 1. 1 ) is for The composed mapping of u+ y(t(u)). y1• It defines a differentiable path by t=t(u) function and the mapping ( I. 2) by differentiable < h1 (with a1 < b1), whose derivative :e'(u) 'is always > o and change of pa rameter. the mapping ( 1. 2) is y1 is deduced from y f1(u) du deduced from w We say that The differential form equal to f (t(u)) t'(u) du, by virtue of the formula giving the derivative of a composed function. The formula for change of variable in an ordinary integral thus gives the equation lw=jw. T Tt ;: In other words, the curvilinear integral w does not change its value if the differentiable path y is replaced by an<'ther which is deduced from by change of parameter. y We can, then, denote paths deduced from one another by change of parameter by the same symbol. Take now t == t(u) defined for b, t(b1) =a (the description a continuously differentiable function a1 < u < h1, but such that t'(u) < of the segment is reversed). o, t(a1) = We then see that Jw=r, 1 w. r say therefore that we have made a change of parameter in changes the orienta tion of y; the effect of this is to multiply Subdivide the interval number of subintervals [a, b] described by the parameter ... ) where a< t1 < t2 < . .. < t._1< t.< b. y to the ith of these intervals; the mapping lw r 50 Let r i t into which by a  I. finite [t., b], be the restriction of it is clear that = i (Jw). i=I y; J: w y We II.I.I CURVILINEAR INTEGRALS; PRIMITIVE OF A CLOSED FORM This result leads to a generalization of the idea of a differentiable path · A piecewise differentiable path is de.fined to be a continuous mapping 2 y:[a,b]�R , such that there exists a subdivision of the interval [a,b] into a finite number of subintervals as above, with the property that the restriction of each subinterval is continuously differentiable. to y We define The sum on the right hand side is independent of the decomposition. The initial point of y1 is called the initial point of y and the final point of Y•+i is called the final point of We say that a path is y. closed if its initial and final points coincide. A closed path y can also be defined by taking, instead of a real parameter t varying from Example. a to b,a parameter (J which describes the unit circle. Consider, in the plane 2 R , the perimeter (or ' boundary ') of a rectangle A whose sides are parallel to the coordinate axes. rectangle is the set of points (x,y) The satisfying Its boundary consists of the four line segments =a2, =b2, x =a1, y =bl, x y b1 <Y < b2, a1 < x < a2, b1 <Y < b2, a1 < x < a2• For this boundary to define a piecewi!je differentiable closed path necessary to stipulate the'sense of description chosen. y,it is We agree always take the following sense of description : y increases from x decreases from _v decreases from x increases from Thus the integral J. w b1 to a2 to b2 to a1 to b2, a1, b1, a2, along the side x =a2, b2, side x a1, side y =b1• along the side y along the along the = = is welldefined; it does not depend on the choice of the initial point of y because it is always equal to the sum of integrals along the four sides, each described in the sense indicated. 51 HOLOMORPHIC FUNCTIONS, CAUCHY'S INTEGRAL PRIMITIVE OF 2. DIFFERENTIA L FORM A LEMMA. Let D be a connected opere set of the plane. Any two points a e D and b e D are the initial and final points, respectively, of some piecewise differentiable path in D. (Briefly this says that a and b can be joined by a piecewise differentiable path). Proof. Each point c e D is the centre of a disc contained in D can be joined to each point of this disc by a piecewise differentiable path contained Suppose that in D, for instance, a radius. can be joined to to a ae D is a given point; if then any point sufficiently near to a, c c can also be joined because of the previous remark; thus the set E of points of D which can be joined to a is On the other hand, E is closed in D; because, open. if c e D is in the closure of E, of previous remarks, so c can c can be joined to some point of E because be joined to the subset E of D is nonempty (as so it must be the whole of D. By hypothesis, D is connected; a. a e E) and is both open and closed, This completes the proof. Let D again be a connected open set in the plane and let y be a ,piecewise differentiable path contained in D with initial point Let F be differential form w = dF; (2. I ) l dF D, and final point = F(b)  F(a). if the differential dF is identically the function F is constant in D. Given a differential form in a connected open set D, we investigate <d whether or not there is a continuously differentiable function dF = in D such that equivalent to w. If w = P dx + i'lF (2. 2) by Such a function F, if it exists, is called = Q dy, the relation dF d(FG) = a primitive of the form is In this w. F since A necessary and sufficient condition that a differential form has a primitive in D is that y contained in D. shows that = w o. PROPOSITION 2. I. I. F(x, y) Q. case, any other primitive G is obtained by adding a constant to Proof. b. then we have the obvious relation It follows from this and the lemma that, zero in a continuously differentiable function in D and consider the a J w = o for arry piecewise differentiable closed path � The condition is necessary because, if r (t) .J� = 0 w w = dF, relation (2. I ) whenever the initial and final point� of y coincide . II. I .2 CURVILINEAR INTEGRALS;· PRIMITIVE OF A CLOSED FORM 2. The condition is sufficient. For, choose a point (x0, y0) e D; any point (x, y) e D can be joined to (x0, y0) by a piecewise continuously differentiable path y contained in D (by the lemma); the integral the choice of y because the integral of by hypothesis. J; (I) does not depend on round any closed path is zero (I) Let F(x,y) be the common value of the integrals J: (I) along paths y in D with initial point (x0, y0) and final point (x, y). We shall show that the function F so defined in D satisfies relations Give x a small increment h; the difference (2. 2). F(x + h, y)  F(x, y) is equal to the integral J (I) along any path contained in D starting at (x, y) and ending at (x + h, y). In particular, let us integrate along the line segment parallel to the xaxis (which is possible if F(x + h,y)  F(x,y) = lx+h jhJ is small enough) : P (e,y) de, and consequently, if h =F o, As h tends to o, the right hand side tends to P(x, y) because of the conti nuity of the function P. Hence we indeed have bF = bx bF We could prove . . by propos1t10n 2. 1. = p (x,y). Q(x, y) similarly. This completes the proof of Consider in particular the rectangles eontained in D whose sides are parallel to the axes (we mean that the rectangle must be entirely contained in D, both its interior and its frontier). rectangle, we must have J (I) = o If y is the boundary of such a for the differential form (I) to have a primitive in D. This necessary condition is not always sufficient as we shall see later. Nevertheless, it is sufficient when D is 'simply connected' (cf. no. 7). For the moment we '.shall confine ourselves to proving following : PROPOSITION 2. 2. Let D be an open disc. 1J J (I) = o whenever y is the boundary of a rectangle contained in D with sides parallel to the axes, then a primitive in D. (I) has 53 HOLOMORPHIC Proof. Let point ofD. at (x,y), (x0, y0) FUNCTIONS, ' CAUCHY S INTEGRAL be the centre of the disc D and let There are two paths y1 and y2 starting at each of which is composed of two sides ofthe rectangle (w;ith sides parallel to the axes) whose opposite corners are figure 1]. (x, y) be a general (x0,y0) and ending (x 0 , y 0) Thus this rectangle is contained in D and [J and (x, y) J, � J. = w. [see Let (><o.Y) Yz ( ><, y} Y2 (><o,Yo) Fig. F(x, y) y, Y1 (><,Yo) 1. be the common value of these two integrals; then we can show, as above, that bF  bx = p , 3· THE GREENRIEMANN bF b y  Q' .. ' h proves the propos1t1on. wh1c FORMULA This formula, in some sense, generalizes relation ( 2. 1 ) : instead of relating the value of and ordinary integral to values of a function, it relates the value of a double integral to that of a curvilinear one. Let A be a rectangle with sides parallel to the axes, let y be its boundary and let P(x, y) and Q(x, y) be continuous functions defined in a neighbourhood D ofA, the functions . . bP b . contmuous partla . l derivat1ves  and . Q havmg . by bx The GreenRiemann formula can then be written (3. I ) 1 P dx + Qdy = T Proof. ( (�Q.; bbP)l) dx dy. ( J J A. _bx We shall prove for instance that 1 Qdy T = b (( Qdx dy. jjA. bx We know that the double integral of the continuous function calculated as follows 54 : bQ bx can be A CURCILINEAR INTEGRALS; PRIMITIVE OF However, to y gives 1°· bQ dx bx ' = Q(a 2, y)  Q( a i. y); which is. precisely equal to This completes the proof. CLOSED FORM II.I.3 integrating this with respect 1 Qcry. 1 The GreenRiemann formula is valid for more general domains than rectangles, but we shall leave this question aside for the moment. Qdy be a differential form in a connected bP and suppose that the partial derivatives and � exist and are by bx PROPOSITION 3. open set D, continuous in D. I. w Let = P dx + Then the relation (3. 2) w is a necessary condition for open disc. Proof. From formula to have a primitive in (3.1), condition ' it is also sufficient if D is an D; · (3.2) implies whenever y is the boundary of a rectangle contained in disc, this implies that if J: w = w D; that i w = o if D is an open has a primitive (proposition 2. 2). Conversely, o whenever y is the boundary of a rectangle A contained in D with sides parallel to tl}e axes, we have n( (b _bQ) JJA by bX dxdy P Moreover, this implies relation (3. 2). for any such rectangle A. i"f . . o = the contmuous fiunct10n bP bQ  by bx For, . . . D, th ere . is not I"dent1ca11y zero in will be some point of D in a neighbourhood of which it is > o, say, · and consequently the integral 11(bPby A ) bQ dxcry bX  · · will also be > o for a rectangle A contained in this neighbourhood, contrary to hypothesis (3· 3). Proposition 3. 3 is thus proved. 55 HOLOMORPHIC FUNCTIONS, 4· INTEGRAL CLOSED DIFFERENTIAL FORMS Definition. w = P dx + Qdy, with continuous coefficients closed if any point (x0, y0) e D has an open neigh We say that a form P and Qin an open set D, is bourhood in which w has a primitive. bourhood is a disc with centre and CAUCHY'S 3 immediately imply : PROPOSITION 4. We can assume that such (x0, y0). a neigh Therefore, the results of nos. 2 A necessary and sufficient condition for a differential form I. with continuous coefficients in ( w = o whenever y is the .J l boundary of a small rectangle contained (with its interior) in D with sides parallel to the axes. lf we also assume that P and Q have continuous partial derivatives of the first order, then (3. 2) is a necessary and su.fficient condition for w to be closed. w D to be closed is that We know from proposition 2. 2 that any closed form in an We shall now give an example of a closed form pnm1t1ve. w open disc has a in a connected open set D which has no primitive in D. PROPOSITION complex plane 4. 2. C. Let D be the open set consisting of all points z =I= o of the w = <k/z is closed in D but has no primitive. The form For, in a neighbourhood of each point z0 =I= o, there is a branch of log and this branch is, in the neighbourhood of z0, a primitive of dz/z. is closed. To show that w Hence has no primitive in D, it is sufficient to find a closed path yin D such that l dzz =I= o. In fact, let y be the unit circle l centred at the origin and described in the positive sense. we put z = e11 with t z w running from o to 2'1t; • [ w, 1 "d t, dz z dz= ie11 dt, To calculate we have  =' and consequently l dzz = 12". dt (4. 1) t 0 l = . 2Z'lt =f= O. This completes the proof. In the preceding example, the form imaginary part of w. dx + i dy dz = x+ry z the differential form w is complex. Let us now take the Since = x dx +y dy + x dyy dx i xz+y2 x2+y2 , CURVILINEAR INTEGRALSj PRIMITIVE OF A CLOSED FORM It has no primitive because is closed in the plane with the origin excluded. we have by (4. 1) l x dyydx 2 2 x +y T II.1.5 = 2'1t if "f is the unit circle described in the positive sense. In fact, m is the differential of arc tan L, which is a manyvalued function (that is to say with x many branches) in the plane with the origin excluded. 5· STUDY OF MANYVALUED PRIMITIVES Let w be a closed form defined in a connected open set D. Although w has not necessarily a (singlevalued) primitive in D, we shall define what primitive of w along a path "f of D. Such a path is continuous mapping of the segment_ I= [a, b] into D; we is meant by a defined by a do not assume differentiability in this context. Definition. let w Let "f: [a, b] +D be a path contained in an open set D,, and be a closed differential form in b. cribing [a, b]) is called a primitive of A continuous function f ( t) (t des along w "f if it satisfies the following condition: (P) for any 't' e [a, b] there exists primitive F of w in a neighbourhood of the point r(T) e D such that · F( r(t)) (5. 1) for t near enough to THEOREM a 1. = f(t) 't'. Such a primitive f always exists and is unique up to addition of constant. Proof. First of all, if f1 and f2 are two such primitives, the difference f1(t)f2(t) is, by (5. 1), of the form F1(r(t))F2(r(t)) in a neighbourhood of each 't' e [a, b]; since the difference Fi  F 2 of two primitives of w is constant, it follows that the function f1(t) f2(t) is constant in a neigh bourhood ofeach point of the segment I. We express this by saying that f1f2 is locally constant. However, a continuous locally cons tant function on a connected topological space (the segment I= [a, b] in this case) is constant. Indeed, for any number u, the set of points of the space where the function takes the value u is both open and closed. the function It remains to be proved that there exists a continuous functionf(t) satis fying conditions (P). Each point 't' e I has a neighbourhood (in I) 57 mapped by y ' CAUCHY S INTEGRAL FUNCTIONS, HOLOMORPHIC into an open disc where w has a primitive F. Since I is compact, we can find a finite sequence of points a = to < ti < ... < tn < tn+l such that, for each integer into an open disc i where o U; in which w = b, < i < n, y maps the segment [ti, ti+1] has a primitive Fi. The intersection Vin Ui+ 1 contains y(t;+1) so it is not empty; it is connected, so Fi+1  F is constant in U; n U;+1 · We can then, by adding a suitable constant to each F;, arrange, step by step, that F;+ 1 coincides with F1 in U; n Ui + 1. Then, we let f(t) be the function defined by f(t) = F;(y(t)) for t E [t;, t 1+1]. It is obvious that f(t) is continuous and satisfies condition is clear when is different from the '! t; the latter (P); and the reader should verify it when " is equal to one of them. Note. y Suppose that is piecewise differentiable, in other words, that there is a subdivision of I such that the restriction of [t; , t;+i] it is is continuously differentiable. y to each subinterval Then the integral by definition Iff is a primitive along x, we have by formula l w is defined; (2 . I ) 1(1) =f(t;+1) f(t;) , T; whence, by addition, J (5. 2) This leads t o a definition o f thesis of differentiability of w J y =f(b) f(a). w : fo r a continuous path we take relation y, without the hypo (5· 2) as the definition, which is valid because the right hand side does not depend on the choice of primitive f along PROPOSITION 5. I 27ti 1 y. lf y is a closed pa th which does not pass through the origin, ldz is an integer. · T � · CURVILINEAR INTEGRALS; PRIMITIVE OF Proof. w = �Z is a closed form. In z A CLOSED FORM the proof of theorem 1, 11.I.6 we supposed each F; to be a branch of log ;:,. Thus f(b)  f(a) is the difference between two branches of log z at the point y (a) y (b) , and, consequently, is of the form 27tin, where n is an integer. = COROLLARY. 1  1x dy  y dx .\2 y2 � 27t T jx The quantity T +  � ; dx X + . . an integer (the same integer as above) . . zs is often called the variation of the argument of the point z x + ry when this point describes the path y (whether y is closed or not). = 6. HOMOTOPY For simplification, we shall only consider paths parametrized by the segment I = [o, 1) . Definition. We say that two paths lo: i+ n and Y1: I+ D having the same initial points and the same end points (that is to say y0(o) = y1(o), y0( I ) = y1 ( I )) are homotopic (in D) with fixed end points, if there exists a continuous mapping (t, u) + 8(t, u) of 1 x'I· into D, such that (6. I ) �o(t� o) ='t0(t), y0(o) = y1(o), � o(o, u ) = o(t, 1 ) = y1(t), o(r, u) y0(1) = = Y1(1). For fixed u, the mapping t + o(t,u) is a path y0 of D with the same initial point as the common initial point of y0 and y1 and the same end point as their common end point. Intuitively, this path deforms continuously as u varies from o to 1, its end points remaining fixed. There is an analogous definition for two closed paths y0 and y1: we say that they are homotopic (in D) as closed paths if there is a continous mapping (t, u) + o(t, u) of Ix I into D, such that (6. 2) � �(t, o) = y0(t), ( o(o, u) = 0(1, u) o(t, 1 ) = y1(t), for all u, (thus the path y. is closed for each u). In particular, we say that a closed path y0 is homotopic to a point in D if the above holds with y1(t) a constant function. 59 HOLOMORPHIC FUNCTIONS, CAUCHY'S INTEGRAL THEOREM 2. then If y0 and y1 are two homotopic paths of D with fixed end points, r wJ10 for any closedform w 1 w T• in D. If y0 and y1 are closedpaths which are homotopic THEOREM 2'. as closed paths then rw J1. for atry closedform  r J1. w w. These two theorems are consequences of a lemma which we shall now state. First of all, here is a definition : Definition. Let (t, u)  S(t, u) be a continuous mapping of a rectangle (6. 3) a'< u< b' into the open set D, and let following the mapping S is a w be a closed form in D. continuous function f(t, u) primitive of A w in the rectangle satisfying the following condition : (P') For any point (r, u ) of the rectangle, these exists a primitive F of w in a neigh bourhoodof o(r, u ) such that F(o(t, u)) = f(t, u) at any point (t, u) s�fficiently near to (r, u ) . LEMMA. Such a primitive always exists and is unique up to addition of a constant, This lemma is, in some sense, an extension of theorem 1. it in an similar way. We shall prove By using the compactness of the rectangle, we can quadrisect it by subdividing the interval of variation of and that of u by points t by points t; uh in such a way that, for all i,j, the small rectangle, which is the product of the segments [ti> t1 + 1], [uh ui+1J, is mapped by o into an open disc U1,i> in which w has a primitive F1,i. Keep j fixed; since the intersection U1,in U1+1.i is nonempty {and connected), we can add a constant to each F1,i (j fixed and i variable) in such a way that F1,j and F1+1,j coincide in U1,in U1+1,j; we then obtain, for ue[uh uH1], a functionjj(t, jj(t, u) Hencejj(t, u) is = F1,j(o(t, u) u)) when continuous in the rectangle a< t< b, 6o such that, for all i, we have II. I. 7 CURVILINEAR INTEGRALS; PRIMITIVE OF A CLOSED FORM and it is a primitive of to this rectangle. w following the mapping oh the restriction of 1i Each function jj is defined up to the addition of a constant; we can therefore, by induction onj, choose these additive constants in such a way that the functions u = and jj(t, u) jj+1(t, u) are equal when Finally, letf(t, u) be the function defined in the rectangle (6. ui+i· 3) by the condition that, for all j, we have f(t, u) = when fj(t, u) This is a continuous function which satisfies conditions a primitive of w following the mapping o. (P') and is indeed The lemma is thus proved. Let 1i be a continuous mapping satisfying conditions (6. I) Proofof theorem 2. and let f be a primitive of following o. w constant on the vertical sides t = o and t It is obvious that f is = a I of the rectangle I X I. Thus we have f(o, o) =f(o, 1 f( 1 , ), o ) =f( I , 1 ) and, since j' w = f( 1, o) J  f(o, o), To .. w = f( 1, 1 ) f(o, I ), Tt theorem 2 is proved. 1 The proof of theorem 2 is completely analogous; one uses a mapping o satisfying (6. 7. 2). PRIMITIVES IN Definition. A SIMPLY CONNECTED OPEN We say that D is simply connected SET if it is connected and if in addition any closed path in D is homotopic to a point in D. THEORE M. 3. Any closed differential form has a primitive in D. 1 For, from theorem 2 , we have J in D, which implies by proposition w = 2. 1 w in a simply connected open set o for any closed path y contained that w has a primitive in D. In particular, in any simply connected open set not containing closed form D dz/z has a primitive; in other words, simply connected open set which does not contain o. log o, the z has a branch in any Examples of simply connected open sets. We say that a subset E of the plane is staffed with respect to one of its points a if, for any point z e E, the line segment joining a to z lies in E. 61 H O LOMORPHIC ' CAUCHY S FUNCTIONS, INTEGRAL is starred with respect to one of its points a is simply connected : u between o the homothety of centre a and factor u transforms D into itself; as u Any open set D which for, Dis obviously connected; moreover, for each real number and I, decreases from I to o, this homothety defines a homotopy of any closed curve to a point. convex In particular, a open set D is For, a convex open simply connected. set is starred with respect to any of its points. not In contrast, the plane with the origin excluded is for example, the circle since the integral zero (cf. relation Jz dz ( 4 .I ) ) . [z[ = simply connected : is not homotopic to a point in C I <!!::_ z of the closed form  lo( along this circle is not The reader is invited to prove the equivalence of the following four proper ties (for a connected open set D) as an exercise : a) D is simply connected; izi <:;; b) any continuous mapping of the circle to a continuous mapping of the disc c) !zl I = I into D can be extended into D; any continuous mapping of the boundary of a square into D can be extended to a continuous mapping of the square itself into D. d) if two paths of D have the same end points, then they are homotopic with fixed end points. 8. THE INDEX OF A CLOS ED PATH Definition. Let y be a closed path in the plane C and let a be a point of C which does not belong to the image of denoted by I(y, a), y. (3· I ) : J zd 2 i Proposition 5. I The index of y with respect to a, is defined to be the value of the integral gives that the index z · a I(y, a) is an integer. By referring back to the definitions, we see that, in order to calculate the index, we must find a continuous complexvalued function and such that then we have ef<1> I(y, PROPERTIES = f(t) defined for o <:;; t <:;; I y(t)  a; a) =j(I) f(o). 27tZ OF THE INDEX ) If the point a is fixed, the index I(y, a) remains constant when the closed path y is continuously deformed without passing through the point a. This follows directly form theorem 2' of no. 6. I II. I .8 CURVILINEAR INTEGRALS; PRIMITIVE OF A CLOSED FORM 2) If the closed path 1 is fixed, the index I (1, a) is a locally constant function of a when a varies in the complement of the image of 1. The proof is the same as for I ) . It follows tha