Principal 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

Categories:
Año:
1963
Editorial:
Addison Wesley Longman Publishing Co
Idioma:
english
Páginas:
227
ISBN:
0201009013 9780201009019
File:
PDF, 14.25 MB
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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

ADDISON-WESLEY 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

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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 1957-1958,
1958-1959 and 1959-1960.

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 many-valued 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 non-zero 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
·

non-zero.

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 non-zero.
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.

� nanXn-t.

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 non-negative 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 well-known 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 sub-field of C, i. e. the sub-field
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)

= � (z-z)
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 above-mentioned 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, non-discrete, 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 well-defined number

The set of

r

;;:. o for which

is clearly an interval of the half line R+, and this interval is non-empty
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 non-negative 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

=

(S-S.)

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)

=

� nanXn-1

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

-

non-zero 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

n-1--1' (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 nanzn-1•

n�O

,Li n(n - I)anzn-2, 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'n-p
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)n-lun-1 + ...

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)n-1_+ ...
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. e-i1

to

by relation

I

§ 2,

no.

REAL

)

8).

Thus the

and T are inverse.

ei1

FUNCTION

shows that

e-i1

(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 ) n-y2 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(y-f),

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 many-valued

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

vice-versa.

(t) > o

Let D be the open half-plane Re

denotes the real part oft).
value of arg

t

t in this

For any

(recall that Re

(t)

half-plane, 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 half-plane 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'
,
z-z

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 e-fCt>

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) ! (r-r0)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 e-I/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 x-x0

)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

(x-x0)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)

<lx-x01 < •

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)

=

(x-x0)"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)k-k'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)
- ( X-Xo)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

(X-X0 )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

( e-k

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)
+n-1)
+ ... +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 + nn-r) 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 +nn-l
=

�

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
--

=

l-z

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:an-1

+

�an-2 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

e-iz

(

.

eit

resp. sin z =

_

e-iz

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

�

I-Z

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 complex-valued) 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 sub-intervals

[a, b]

described by the parameter

... )

where

a< t1 < t2 < . .. < t._1< t.< b.
y to the i-th 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 sub-intervals as above, with the property that the restriction of
each sub-interval 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 well-defined; 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 non-empty (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(F-G)

=

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 x-axis (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 GREEN-RIEMANN

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 Green-Riemann 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 Green-Riemann 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 dy-y 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 dy-ydx
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 many-valued function (that is to say with
x

many branches) in the plane with the origin excluded.

5·

STUDY OF MANY-VALUED PRIMITIVES

Let

w

be a closed form defined in a connected open set D.

Although

w

has not necessarily a (single-valued) 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

f1-f2 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 sub-interval

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 non-empty {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 complex-valued 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