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co
ANALYSIS
•
.
. .
'-,.,·X.·:;··-.··, .
....
",;" .. • "',' '
.'. . ..-, .
International Series In Pure and Applied Mathematics
S,..,nie,. G. S",.,.... a .... E. B.
eon...'ting Bdj...,.
AlIlfar.: Complex Analysis
BeMer and Oruag: Advanced Mathematical Methods for Scientists and Engineers Bvc/c: Advanced Calculus
B1l.tJClcn and lMaly: Finite Graphs and Networks CMnsy: Introduction to Approximation Theory Ch£rler: Techniques in Partial Differential Equations ('odtli",ton and Uvln••", Theory of Ordinary Differential Eq".t;ons Conle and de B_: Elementary N1!muical Analysis: An Algoritbmic Approach
: Introduction to Partial Dilterential Equations and Boundary Value Problems
DeUman: Mathematical Methods in Physics and Engineering Golomb and Shanka: Elements of Ordinary DilI'erentiai Equations
Hammi",: Numuical Methods for Scientists and Engineers Hildebrand: Introduetion to Numerical Analysis
Hl1IC1elwld#r: The Numerical Treatment of a Single Nonlinear Equation
Kal_. fi'alh. aM Arbib: Topics in Mathematical Systems Theory
LaB... Vector and Tensor Analysis McCtmll: Topology: An Introduction with Applications to Topological Groups
Monk: Introduction to Set Theory Moure: Elementa of Linen Algebra and Matrix Theory M01lI'-a aM I>uria: Elementary Theory and Application of Numerical Analysis
PUJf'I: Matrix Theory and Finite Mathematics
Pip"" and Haruill: Applied Mathematics for Engineers and Physicists RalaIon and R<JbiMtJJitz: A Firet Course in Numerical Analysis
RilIF and ROle: Difiaential Equations with Applications
Rudin: Principles of Mathematical Analysis Shapiro: Introduction to Abetmet Algebra
Simmona: DilI'erentiai Equations with Applications and Historical Notes
SimliUlilt: Introdnction to Topology and Moda n Analysis
Slrubk: Nonlinear Differential Equations
, . ,
:
.., .,~.~" '
. " " .' . . . ., " , '
PLEX AN LYSIS
An Introduction to the Theory of Analytic Functions of One Complex Variable
Third Edition
U..;a:.-!.i!.1'Jl...'..:.:. ",..'.- •......._,,-t ~':;·"~,·i~~
Lars V
Professor Harvard Un iyersity
,
Emeritus
,
.
;
.
•
. .
,McGraw.BlI, I"C'.
YOlk St. Louis San Flanci~ A••cklarui Bogol3.
I.isboD London Madrid ~xico City .Milan
New DeJbi Sill J..ln Sinppoiic
..
• ••
Tokyo Toronto
, . ..
.
, ,-.:;;
, ";':';N
ANALYSIS
•
@) 1m, 1966 by MaOra..-BW, Inc. All rip. ~.
1968 by MaOraw-BW, Inc. A11....,10.......t.
i lillteo! ill the' UDiteo! 8~"" of~. No port 01. &his publication
may be >epIOdueed, stored ill a reIrieval OYI""', or tnnSIPitled, in ....y
for. M by ....y - . .,Inlrollic, m""',anical, pho~ """".Iin.. or
othenrioo, witbout doe paior .,i"on ponDiosiao 01. tile publiober.
161718192021 BRBBRB 969876543210
Tbia bonk ..... ""' in Modo, n SA by MollO&Y," Compooition Compony, Inc. Tbe eclillDll Wec6 Carol Nap. and 8\opben WocIoy;
the ..........ction IUponioor _100 (};Pmp...'b.
(In_tiobal _ _ in pwe ....d applied "'a\lwm,"ti.,.)
IncJudea index.
1. A.IIaIyt\c functioDl. I. Title.
QAlI31.A411 urn
511;'.93
'18-17078
IBM
.... .---..,--,"~ -~,.'-
..•.. ..
"
,'~
....-....
_: .-,..:,.
".,.: "
'.-:"~:,,-;,,"-."•-'1""'..-,
ToEma
,
,'. "
-' ..
,
Contents
.
•
• •
...
xm
1 COMPLEX
1
The AlI.bra of Comp'- Number.
1
.. 1.1 Arithmetio OperatiODB
1
.•.. 1.2 Square Rook
a
.. 1.3 Just.;fieat.ion
4
...... 1.4 Conjuption, Absolute V&lue
6
Inequalitiee
9
Geometric .Repre.."tation of Comples Number.
12
Addition IIDd Multiplication
12
The BiDomi·1 Equation
Iii
Analytic Geometry
17
• 'l'he Spherical Representation
18
2 COMPLEX FUNCTIONS
. . .' ; -, .
.'.. ......'
. : .._..._,;.,......,.,.. : ..'
.-.'... ..
-_..
,'
... .. '-
21
21
22
24
28 30
33
•. 33
· •
36
. ..· ... .
CONTENTS
2.3 Ulrifonn Convergence
35
2.4 Power Series
38
2.5 Abel's Limit Theorem
41
The Esponentiol and Trigonometrie Functions
3.1 The Exponential
42
3.2 The Trigonometric Functions
43
3.3 The Periodicity.
44
3.4 The Ingsrithm
46
CHAPTER 3 ANALYTIC FUNCTIONS AS MAPPINGS
49
I Elementary Point Set Topology
50
1.1 Sets and Elements
50
1.2 Metric Spsces
51
1.3 Connectedness
Ii(
1.4 Compsctness
59
1.5 Continuous Functions
63
1.6 Topological Spaces
66
2 Con/or_lity
67
2.1 Arcs and Closed Cw ves
67
2.2 Analytic Functions in Regions
69
2.3 Confonnal Mapping
73
2.4 Length and Area
75
3 Linear T,..~ormations
76
3.1 The Linear Group
76
3.2 The Cross Ratio
78
3.3 Symmetry
80
3.4 Oriented Circles
83
3.5 Fa.milies of Circles
84
4 Elementary Conformal Mappings
89
4.1 The Uee of Level Curves
89
4.2 A Survey of Elementary Mappings
93
4.3 Elementary Riemann Surfaces
97
CHAPTER 4 COMPI EX INTEGRATION
101
1 Fundamental Tlworemll
101
1.1
101
1.2
104
1.3
105
1.4
109
1.5
112
COIiTIEIiTI ..
Z Cauchy'" IRtegral Formula
114
2.1 The Index of a Point with Roolpect to a Closed Curve
114
2.2 The Integral Formula
118
2.8 Higher Derivatives
120
a LocOI Propertie" of Analytical Funetiona
124
3.1 Removable Singularities. Taylor'. Theorem
124
3.2 Zeros and Poles
126
3.3 The Local Mapping
130
3.4 The Maximum Principle
133
4 The GeneFUI Form of Cauchy's Theorem
137
4.1 Chain. and Cycles
137
4.2 Simple Connectivity
138
4.3 Homology
141
4.4 The General Statement of Cauchy's Theorem
141
4.5 Proof of Cauchy's Theorelii
142
4.6 Locally Exact Diftereotial.
144-
4.7 Multiply Connected Regions
146
5 The Calculua of Rellidlle.
148
5.1 The Residue Theorem.
148
5.2 The Argument Principle
152
5.3 Evaluation of Definite Integrals
154
Harmonic Functiona
162
6.1 Definition and Basic Properties
162
6.2 The Mean-value Property
165
6.3 POll!oon'. Formula
166
6.4 Schwara's Theorem
168
. 6.5 The RelIection Principle
172
•
CilHAPTER 5 SERIES AND PRODUCT DEYELOPMENIS
175
-tc.--~._-'oUler Se,ia £"JHln.iona
175
1.1 Weierotrass'a Theorem
176
".;. 1.2 The Taylor Suies
179
l '-'".,
"
"•',•.,"..-.
.1.
3
".,
-Pa
r
The Laurent
tial FracCio
Sa
na
ies
an
d
FacforiHtioR
184
187
:Partia..l,F-r-a.ct.iooa
187
Infinite PrOduct..
191
O'D'lI1ioaI Prodnota
193
The Gem.... Fun.ction
IG8
SIiIrIiq'. Formula
201
,. . . . . .~.,
. ~,
.
•
. ..
"
"'.'
..
CONTENTS
Entire F..nc:tiona
206
3.1 Jensen'. Formula
'JJIl
3.2 Hadam.ro'. Theorem
208
4 The Riemann Zeta Function
212
4.1 The Product Development
213
4.2 Extension of t(.) to the Whole Plane
214
4.3 The Functional Equation
216
4.4 The Zeros of the Zeta Function
218
$ Normm
219
5.1 Equicontinuity
219
5.2 N()lwality ....d
220
5.3 Arsel·'s Theorem
222
5.4 Families of Analytic Functions
223
5.5 The Classical Definition
225
CHAPTER 6 CONFORMAL MAPPING. DIRICHLET'S
229
1 TIw Riemann Mapping Tlworem
229
1.1 Ststement ....d Proof
229
1.2 Bouvdary Behavior
232
1.3 Use of the Ralection Principle
233
1.4 Analytic Ares
234
Z Conformal Mapping oj PoIyg01l3
2.1 The Behavior at &n Angle
235
2.2 The Schwan-Christoffel Formula
236
2.3 Mapping on & Rectangle
238
2.4 The Triangle Functions of Schwars
241
A Closer Loolc at Harmonie Functiona
241
3.1 Functions with the Mean-value Property
242
3.2 Harnack'. Principle
243
4 TIw Diriehlet Problem
245
4.1 Subharmonic Functions
245
4.2 Solution of Dirichlet'. Problem
248
5 Canonical Mapping. of Multiply Connected Regiana
251
5.1 Harmonie Moosures
5.2 Green'. Function , 5.3 Parellel Slit Regions
252
257
, ,
259
CONTENTS
•
7 ELLIPTIC FUNCTIONS
263
1 Simply Periodic Functio....
263
1.1 Repr""""tation by Exponentials
263
1.2 Th.. Fourier Development
264
1.3 FunctioDB of Finite Order
264
2 Doubly Periodic FunctionB
265
2.1 Tbe Period Module
266
2.2 Unimodular Transformations
266
2.3 Tbe Canonical B8l!is
268
2.4 Geneml PropeJ1ies of Elliptic Functions
270
:1 The Weier8t.....8 Theory
272
3.1 The Weierstrass p-fuootion
272
3.2 The Functions t(z) and q(.)
273
3.3 The Dillerential Equation
276
3.4 The Modular Functi<>n X(T)
277
3.1i The Conf"rmal Mapping by MT)
279
CHAPTER 8 GLOBAL ANALmC FUNCTIONS
283
1 Analytic Continuotion
283
1.1 The WeierstnIM Th=y
283
1.2 Germs and Sbeaves
264
1.3 Sections and Riemann Surfaces
287
U Analytic Continuations ahlDK Arcs
289
1.6 H<>mooopic Cum..
291
1.6 The M"nodromy Theorem
295
1.7 Branch Points
297
.. J Algebroic FunctioR8
300
•
2.1 Tbe Resultant mTwo Polyn<>mials
300
2.2 Definition and Properties of Algebro.ic Functions
301
2.3 Behavior at tbe Critical Points
304
."·· Picard'. Theorem
-
306
•
"-' .~. '
-
3.1 hounlry Valuee
307
0-
Linear Dige,../itio! Eq....cloru
308
',.,. 4.1 Ordinary Points
309
,.2 Rcgn],r BiOl'dpr Points
311
;"" 4.3 Solutions at Infinity
313
4.4 The Hyperge<>metric DilI_ntial Equation
3l1i
·.'.' 4.6 Riemann's P<>int "f View
318
328
Preface
•
Complu A1I(llyaia has successfully maintained its place as the sta.ndsrd
elementary text on functions of one complex varisble. There is, never-
theless, need for a new edition, pa.rtly because of changes in current mathe-
matical terminology. partly because of differences in student preparedness
· and aims.
There aTe no radical innovations in the new edition. The author still
believes strongly in a geometric approach to the b!lSics. and for this reason
the introduetory chapters are virtually unchanged. In a few places,
:throughout the book, it was desirable to clarify certain points thst ex-
f'perience has shown to have been a source of possible misunderstanding or
• •
,-difficulties. Misprints and minor errors that have come to my attention
"have been corrected. Oth~I wise, the main differences between the second
r'~d third editions can be summarized as follows;
•:. 1. Notations and terminology have been modernized, but it did not
necessary to cha.nge the style in any significant way.
Z. In Chapter 2 a brief section on the change of length and area under
mapping has been added. To some degree this infringes on the
self-contained exposition, for it forces thc reader to fall back on
for the definition and manipulation of double integrals. The I•S mm• or.
4 there is a new and simpler proof of the general form of
theorem. It is due to A. F. Beardon, who has kindly ~I witted
to reproduce it. It complements but does not replace the old proof.
has been 1etained and improved.
,. A short
.. on the Riemann zeta function he" been included.
. .
PRI!FACE
This always fascinates students, and the proof of the functional equation
illustrates the UBe of
in a less trivial situation than the mere
computation of definite integrals.
5. Large parts of Chapter 8 have been eompletely rewritten. The
main purpose WIlB to introduce the reader to the terminology of genllB and
sheaves while emphllBizing all the classical concepts_ It goes without
saying that nothing beyond the basic notions of sheaf theory would have
heen compatible with the elementary nature of the book.
S. The author hIlB successfully resisted the temptation to include
Riemann surfaces IlB one-dimensional complex manifolds. The book
would lose much of its usefulness if it went beyond its purpose of being
no more than an introduction to the basic methods and results of complex
funetion theory in the plane.
It is my pleasant duty to thank the many who have helped me by
pointing out misprints, weaknesses, and errors in the second edition.
I am partieularly grateful to my eolleague Lynn Loomis, who kindly let
me share student reaction to a recent
based on my book.
LaTif V. AM/Drs
COMPLEX ANALYSIS
•
1
PLEX NU
•
L THE ALOE.ItA OF COMPLEX NU• •EItI
It is fundamental that real and complex numbers obey the sarne basic laws of arithmetic. We begin Our study of complex fWle-
tion theory by stressing and implementing this analogy.
1.1. ..4ritiunetJe Operations. From elementary algebra the
reader is acquainted with the imaginlJf1l unit i with the property
," = -1. U the imaginary nnit is combined with two real num-
bel8 a, fl by the proce:aaes of addition and multiplication, we
obtain B compw number a + ifl. a and II are the r~al and
';
pari of the complex number. If '" = 0, the number il
to be """,Zy imaginlJf1l; if fl - 0, it is of C01l1Be real. Zero i8
the only number which is at once real and purely imaginary.
Two complex numbers are equal if and only if they have the same
res] part and the 88me imaginary part.
Addition and multiplication do not lead out from the system
of oomplex nnmbers. Ae,nming that the ordinary rules or
. arithmetic apply to complex numbers we find indeed
(1)
(a + ill) + ('"( + sa) - (a + '"() + i(fl + /I)
and
(2)
+ + + + (a iII}(., i.) - (..., - (l6) i(a. fl.,).
In the
IU, identity have made lI'e of the relation " - -1.
It is . . obvioua that division ill
We wi&h to
....,
..,,-
.-,
".--' :,
..;- --
. . ";~<,"'-'"
-___
'~-::'"".,
__~-',:"
",,-', __
-I
__
'-.
':.<',.
,"
.
••:,~':;
__-_,.'
'.._.
-'_,.'.'.•.-.;.:.'-'
1
,'" ... -
- -,'~ -~- '-'~-.~
2
COMPLEX "NA~ VSI.
+ + + ahow that (.. ifJ)/h if) is a complex number.• provided that 'Y + if ¢ O. If the quotient is denoted by x i1/, we must have
a + ifJ = + + (-y i6)(x iy).
By (2) this condition can be written
.. + ifJ - ('Yz - 6y) + i(1x + 'VY),
and we obtain the two equations
a~'Yz-6y
fJ = 3x + 'VY·
This system of simultaneous linear equations has the unique solution
.
x
=
.'Y..",++fJ0"i
fJ'V -ao
11-'1"+3"
+ for we know that '1' 6· is not zero. We have thus the
(3)
+ + + a i(j _ (JI.'Y fJ3 i fI'Y - ai.
+ + + 'Y i6 - 'Y. 01
'1' 01
Once the existence of the quotient has been proved, its value can be found in a simpler way. If numerator and denominator are multiplied
with '1 - io, we find at once
+ + + + (JI. i(j (a ifJ)('Y - i3) (a-y (ja) i(fJy - ai)
+ 'Y+ ii = h ia)('Y - io) =
'Y' + 6"
..
As a special
the reciprocal of & complex number ¢ 0 is given by
1 a-i/J
a+ifJ=a'+fJ"
We note that i" has only four possible valuee: 1, i, -1, -i. They
correspond to values of 11 which divided by 4 leave the remainders 0, 1,
2,3.
EXERCISES I. Find the values of
(1 + 2i)',
I)
-3 + 4i'
2 +i I
3 - 2i '
+ + (1 i)' (1 - ,)-.
+ Z. If Z 0= Z i1l (z and 11 real), find the real and imaginary parts of
z-1 '
•- 1
z + l'
_1 .
Z"
So Sbow that
-1 ± i 3 • ~1 and
2
• =1
for all combinations of signs.
1.%. Squa.... Roots. We she!) now show tbai the square root of II
+ complex number can be found explicitly. If the given number is a i/J.
+ we are looking for a number x ill such that
+ + (x i1/)' = a ip.
Thill is equivalent to the ByBtem of equatioll8
<")
ZI - y~ = a
2zy = /J.
From theee equations we obtain
· .
•
•
+ + + (x' 1/')' - (x' - y')' ","'y' ~ a' P'.
we must have
x' + II' = Va' + pI,
· tbe square root is positive or zero. Together with the first equa.· . (") we lind
z· = l(a + V Ot' + p') 1/' = l( -Ot + Va' + P·)·
that tbeee qnantities are positive or zero regardJeIl8 of the sign
• •
... •.,The equations (5) yield, in general, two opposite values for :t and two
. . .Butth .Be values cannot be combined arbitrarily, for the aeoond
(4) is not II cppeequence of (5). We must therefore be careful
andy BO that their product bas the Bign of p. This leads to the
-±
+. •.jfjJj -a
va . . (fitlo;;' For' .;. 0' theviilues are ± if a ;;: 6, ±i v::;
.. - -' •.·'_. ,.' " .• ". 'C', <i~' _." .'. __
- . " -, ,
..,. ,
"':;:"'::':.: ",.-:<.'.' '"'" ,.....: :":,
.. ,' -,
...
" ' , -.,
.- - - -
if a < O. It is understood that all square roots of positive numbelll are
taken with the positive sign. We have found that the square root of any complex number exists
+ and hall two oppoeite VJLlues. They coincide only if .. ifJ - O. They
are real if fJ - 0, a ~ 0 and purely ima,giuary if fJ ~ 0, .. ;:ii O. In other
words, except for zero, only positive numbers have real square roots and only negative numbers have purely imaginary square roots.
Since both square roots are in ,eneral complex, it is not poBBible to
distinguish between the positive and negative square root of a complex number. We could of course distinguish between the upper and lower
sign in (6), but this distinction is artificial and should be avoided. The corlect way is to treat both square roots in a symmetric manner.
EXERC.SES
I. Compute
Vi,
VI + i,
2. Find the four values of {I-I. S. Compute {Ii and {I-i.
... Solve the quadratic equation
1- i 2
a' + (a + i/3). + "I + i~ = 0,
1.tI. JlUt4/I....tio... So far our approach to complex numbers has been
completely uncritical. We have not questioned the existence of a number
+ system in which the equation'" 1 = 0 hall a solution while all the rules
of arithmetic remain in force.
We ocgin by 1'8('8 !Jing the characteristic properties of the real-number
IlYlltem which • B denote by R. In the first place, R i8 a jield. This
means that addition and multiplication are defined, aatiafying the IJUOCi,-
alive, conlmutGtWe, and
law. The numbers 0 and 1 are neu-
+ tral elements under addition and multiplication, respectively: a 0 = a,
+ a • 1 - .. for all... l\(oreover, the equation of IlUbtraetion fJ z = a
bae alWA)'ll & solution,and the equation of division (lz - .. baa a solution
whenever·fJ ~ O. t
One shows by elementary reasoning that the neutral elements and the
results of 8Ilbtraction and division are unique. Also, every field is an
intefITal domaiA: a{J - 0 if and only if .. ~ 0 or /3 = O.
t We uoume that the reodor ...... working lmowleclp of e1emeatary algebra,
Alth<nqrh tho above ch...ctorizatioD of a field is eomplete, it obvioUlly d_ DO\
coa.vey muah to .. student who is not aJready at Ie 5-t vapeIy lammar trit.b the concept.
..,' ...
COIIPLEX NUIIBERS
5
These properties are common to all fields. In addition, the field R
has an tmkr relatWA ex < (J (or fJ > a). It is most eMily defined in terms
of the set R+ of poMtiDe real numbers: ex < (J if and only if fJ - a e R+.
The set R+ is characterized by the following properties: (1) 0 is not a posi-
tive number; (2) if ex '" 0 either ex or -a is positive; (3) the 8UD\ and the
product of two positive numbers are positive. From theoe conditioDl! one
den ves all the usual rules for manipulation of inequalities. In particular
one finds that every square a' is either positive or zero; therefore 1 .. l'
is a positive number.
By virtue of the order relation the sums 1, 1 + 1, 1 + 1 + 1, ...
are all different. Hence R contains the natural numbers, and aince it is a
field it must contain the subfield formed by all rational numbers.
Finally, R satisfies the following eomplolene88 ctmdition.: every incre8&-
ing and bounded sequence of real numbers has a limit. Let a, < al < a. < . . . < a. < . . . ,and "'''Slime the existence of a real number B
such that ex. <: B for all... Then the completeness condition reqnires the
existence of a number A .. lim. •• a. with the following property: given
any. > 0 there exists a natural number ... such that A - • < a. < A for all .. > ....
Our disell"8ion of the reaI-number systsm is incomplete inasmuch as
we have not proved the existence and uniqueness (up to isomorphisms) of
. a Bylltem R with the postulated properties. t The student who is not
thoronghly (amiliar with one of the constructive procesees by which real
numbers can be introduced should not fail to fill this gap by consulting any
textbook in which a full axiomatic treatment of real numbers is given.
+ + The equation ",I 1 ~ 0 has no solution in R, for a l 1 is always
>positive. ':luppose now that a field F can be found which cont'inB R as a
',aubfieId, and in which the equation:r" + 1 .. 0 can be solved. Denote a
(BOlution by i. Then ,,' + 1 .. (:r + .)(" - .), and the equation
+ !.:r' 1 .. 0 has exactly two roots in F, i and -i. Let C be the subset of + ': consisting of all elements which can be expreosed in the form a i(J + + . real a and (J. This reprnentation is unique, for ex ifJ .. eI i(J'
a - eI .. -i({J - ,8'); hence (ex - eI)' .. - «(J - ,8')', and this is
only if a .. a', fJ .. ,8'.
The 8ubeet C is a subfield of F. In f&Clt, except for trivial verifica-
the Ieader is asked to earry out, this is exactly what was shown
. Sec. 1.1. What is more, the strueture of C is independent of F. For if
.is another field containing R and a root i' of the eqnation ,,' + 1 - 0,
,
, '... '
tAn u.m'p'lIAi.tm. belw!eq two 6.elda is a one-to 0"8 aouwpondenoe which pte
.. '01''''' alid . .. The waid ia !lIed quite. puraIJy to indicate a ........
v
'e
S;;d
.
.. -
.
..
.,
.
.
.
d'leIath·.1h·"
.... OopAidered ilDpocl.nt
..
".
. . """', • . ,
•
_,_-._,:-.'",,•,"._-..••:'\-,"- ':.1.-(,-,,,,.._.•...-",'..,_...."-.._._"._..,,,
,,'"'
.
.
.-....
"_."v<._--_,
'_.'.,.;..,,.~.:.;.-...~-~_..".-,,_.. ..'..,."-".-':.---,._-~....'.
.... ,,','.~-,-_,,_•••• , ........-
.
.
.
-
. ' , ' ,-
,
..... :;..,__ "._ .. ,,_.', .. ,.·~··.-·:;:.,;-','-.,i',;·::·,~:·'·-
,
_~-< - ,.
"
,:-," ..:."(,',.,,. • •"
•
COIIPLEX ANALYS'S
the
+ subset C' is formed by all elements .. i'/J. There is
a one-to-one correspondenee between C and C' which
.. + ifJ
and .. + i'fJ, and this correspondence is evidently a field isomorphism.
It is thus demonstrated that C and C' an) isomorphic.
We now define the field of compln numbers to be the subfield C of an
a.rbitrarily given F. We have just BOOn that the choice of F makes no
difference, hut we have not yet shown that there exists a field F with the
required properties. In order to give our definition a meaning it remains
to exhibit a field F which contains R (or a subfield isomorphic with R)
and in which the equation ",I + 1 ~ 0 bas a root.
There an) many ways in which such a field can be constructed. The
simplest a.nd most direct method is the foUowing: Consider all expre:::ions
+ + of tbe form a ifJ where .., fJ arere·l numbws while theBigns and i are
pure aymbols (+ does not indicate addition, and i is not an element of a
field). These expressions an) elements of a field F in which addition and
mwtiplication &nl defined by (1) and (2) (observe the two dift'erent mean-
i!J&ll {If the sign +). The elements of the pa.rticula.r form a + 10 are seen
+ to constitute a subfield isomorphic to R, and the element 0 i1 satisfies
+ + + the equation x" 1 = 0; we obtain in fact (0 i1» - - (1 10).
The field F bas thus the required properties; moreover, it is identical with
~ conuponding subfield C, for we can write
a + ifJ = (.. + 10) + fJ(O + il).
The existence of the complex-number.field is now proved, and we can go
+ + baCk to the simpler notation a ifJ where the indicates addition in C
and i is a root of the equation ",' + 1 = O. -- . ' ·
EXIERC:ISIES (For students with a h.okground in algebra)
:L Show that the Bylltem of all matrices of the special f6rm
•
a fJ ,
-fJ a
• .
.
G!l!Jibined by matrix addition and matrix multiplication, is isomorpbie to
.the field of complex numbers.
•. 2. Show that the complex-number system can be thought of as the
~d of all .polynomials with real coefficients modulo the irreducible
+ polynomial ",' 1.
1.4. Co.vugation, Ab"olute Yahle. A complex number can be
denoted either by a single letter a, representing an element of the field C, or
+ in the fornl a ifl with real .. ....d fl. Other standa.rd notations &nl + + + z = X iv, r - ~ 1'1, 1.11 = u iv, and when ,.,ed in this conneotion it
•
- -- .
COIIPLEX NUII.EIIS
7
is tacitly undel'lltood that z, y, ~, 'I, ", V are real numbers. The real and
ime.giuary part of a complex number a will also be denoted by Re a, 1m a.
In deriving the rules for complex addition and multiplication we used
only the fact that i! ~ -1. Since -i has the same property, all rules
must remain valid if i is evelY where replaced by - i. Direct verification
+ shows that this is indeed so. The trlVlBformation which replaces .. i/3
by a - ifl is called complex Clmjugation, and a - ill is the clmjugale of
.. + ifl. The conjugate of a is denoted by ii. A number is real if and
only if it is equal to its conjugate. The conjugation is an involulory
transformation: this meaDS that " ~ a.
The formulas
Re
II
=
a+4
2 t
A-a
Ima - 2i
the real and imaginary part in terms of the complex number and its conjugate. By systematic use of the notatioDS a and Ii it is hence
possible to dispense with the use of separate letters for the real and imaginary part. It is more convenient, though, to make free lI'e of both
notations. The fundamental property of conjugation is the one already referred
to, namely, that
a+b~ii+ii
(iij = a. Ii.
The corresponding property for quotients is a coll8equence: if 4Z = b,
then Iii = Ii, and hence (bla) - Ii/a. More generally, let R(a,b,c, .••)
stand for any rational operation applied to the complex numbers a, b, c,
.... Then
R(a,b,c, . • .) = R(4,Ii,c, . • .).
As an application, consider the equation
+ + . . . + coli" CIZ"-I
c.-~ + c. = o.
U t' is a root of this equation, thtn f is a root of the equation
cor + c¢' I + . . . + c._~ + c. = O.
r .
tn particIIJ8I', if the coefIicients are reo/, rand are roots of the same equa-
li$D, and we have the familiar theorem that the noDnlal roots of an eque -
~OD with real ooeflicients occur in paire ofoonjugate roots.
+ . The product 44 - a l fJ' is alays positive or zero. Its Donnega-
.. the modllZua or
of the
DUID-
•.a; it
- ne't&t1Iljnoloi,i"
bJ'
•
CO.~LE. ANALYSIS
the fact that the modulus of a real m.mber coincides with its numerical
value taken with the positive !lign.
We repeat the definition
. ad ~ lal',
where lal ~ 0, and observe that 141 ~ lal. For the absolute value of a
product we obtain
and hence •
labl' ~ ab· iili ~ abclii ~ a4bb ... lal'lb\", labl = la\ • Ibl
since both are ~ o. In words:
TM absolute value of a product is equal to tM product of tM abeol~
of tM f~tor8.
It is clear that this property extends to arbitrary finite products:
la,a, .•• a,,1 = la.1 . 11121 ••• 1a,,1·
The quotient alb, b F 0, satisfies b(a/b) = a, and hence we have also
Ibl . la/bl GO lal. Of
The formula for the absolute value of a Bl1m is not as simple.
la + bl" = (a + b)(<< + b) = ad + (ab + 1>4) + bb
or
(7)
la + bl' = lal" + Ib\" + 2 Re abo
The conesponding formula for the difference is
(7')
la - bl" = lal" + Ib\" - 2Re ab,
and by addition we obtain the identity
(8)
10 + bl' + la - hl" = 2{lal' + Ibl').
We find
EXERCISIS
L Verify by calculation that the values of
z
.
.
z'+ 1
+ for Z = :E. ilJ and II = :E - ilJ are conjugate.
2. Find the absolute values of
+ + .) -2i{3 .)(2 + 4J)(1
and
(3+41)(-1 +2.)
(-1-1)(3-1) .
COIIPLEX NUMBERS
•
L Provetb&t
a-b 1-a/) =1
+ + if either lal - lor Ihl = 1. What exception must be made if lal - Ibl = 1? 4. Find the eonditions under which the equation az hi c = 0
in one complex unknown has exaetiy one SDlution, and compute that SDlution.
L Prove lAgrange's identity in the complex form
1.$. Ineq..aliti.... We shall now prove SDme important inequalities
which will be of consta.nt use. It is perhaps well to point out that there
is nO order relation in the complex-number system, and henee all inequalities must be between real numbers.
From the definition of the absolute value we deduce the ineq"alities
(9)
-Ial ;l!! Re a ;l!! lal -Ial :ii 1m a :;; lal·
The eqnality Re a = lal holds if and only if a is real and ;;:; O.
If (9) ill applied to (7), we obtain
, and hence
,
" (10)
la + bl" ~ (Ial + lb/)'
la + bl ;l!! lal + Ibl·
i This ill called the triang~ iMqUGlitll for reMOns which will emerge later.
, By induction it can be extended to arbitrary BUmB:
;'
Ii.. '(11)
la, + a. + . . . + aal ~ la,1 + laol + .'. . + laal.
"
"I:.. _
The abeolute 1IGl~ 0/ a &'Um Ut at tI163t equal to IAe 114m o/1Ae ab,olute
;~ ,
o/Ihe It:rmo.
, The reader ill well aware of tbe importance of
(11) in the
.,: ~ case, and we .haJJ find it no Ie. imporlant in the theory of complex
t:;.-. bnmbu
t', Let\JJl detbjmine a.ll
of equality in (11). In (10) the equality
r holds if and only if ali ~ 0 (it is convenient to let c > 0 indicate that
f ill is real and~). If b pi 0 , oondition can be written in the
: form Ibll(a/b) ~ 0, and it ill bellee!!CtiliWlleDt to alb ;; O. In ' '
"1 ' .
", ",
," ,
. ' .. . .- .'-
10
COMPLEX ANALYSIS
we proceed 88 follows: Suppose that equality holds in (11); tben
la.1 + la,l + ... + 1...1= I(a. + a,) + a, + ... + ...i :;; la. + a,l + la,l + . . . + 1...1:;; la.1 + la,1 + . . . + la.l.
Hence la. + a.1 = la.1 + 11101, and if a, ~ 0 we conclude that a./a, iii: O.
But the nnmbering of the terms is arbitrary; thus the ratio of any two nonzero terlll8 must be positive. Suppose conversely that this condition
is fulfilled. ABsuming that a, ~ 0 we obtain
. . . + ...1= la.l· 1 + ~ + . . . + ~
41
o.
1 + a, + . . . + ~
a,
a,
= la,l
1 +
llaa..11
+
.
.
.
+
I...
a.
= la.1 + la.1 + ... + la.l.
To sum up: 1M sign of equalitylwlds in (11) if and mUy if 1M ratio oj any
huo nonzero terms is pOBitive.
By (10) we have also
lal = I(a - b) + bl ~ la - bl + Ibl
or
la! - Ibl ~ la - bl·
For the Same reason Ibl - lal :;; la - bl, and these inequalities can be
combined to
(12)
la - bl .. lIal - Ibll·
Of course the same estimate can be applied to la + bl.
A special case of (10) is the inequality
.613)
la + i.61 ~ lal + IIlI
which expressee that the absolute value of a complex number is at most
equal to the sum of the absolute values of the real and imaginary part. Many other inequalities whose proof is less immediate are a1AO of fre '
.ent llBe. Foremost is Caudal!3 inequality which states that
,da,b, + ... + a.b.I' :;; (ja,l' + ... + 1...I')(lb1I' + ... + Ib.l")
n,. in shorter notation,
•
t t J. a eonven;eot s..mmation index and, nsed sa & subscript, cannot. be O()nfqzed
:"ith the bnaFnary unit.. It. e--,.,ms poin~lesa f,o bcm ita 13".
, ,,'
, ,"
,,' "
'".,,:,.~,:
COIIPLEX NUIIBERS
11
To prove it, let;>. denote an arbitrlll'Y complex number. We obtain
by (7)
(15)
. . "
"
k k k,-1 '-I 140 - ;>.ii,I' = 1401· + 1;>.1' Ib,I' - 2 Re}; }; 11,'1>,.
i-I
~_1
This expreBBion is ~ 0 for all;>.. We can choose
for if the denominator should vanish there is nothing to prove. This choice is not arbitrary, but it is dictated by the desire to make the
(15) as small as possible. Substituting in (15) we find, after
Bimplifieations,
which is equivalent to (14).
From (15) we conclude further that the sign of equality holds in (14)
if and only if the a. are proportional to the ii,.
Cauchy's inequality can aIao be proved by means of Lagrange's
'. identity (Sec. 1.4, Ex. 4).
EXERCISES
,
I. Prove that
,
, ,
a-b
1- db
<1
\' If lal < 1 and 1"1 < 1.
I' " 2. Prove Cauchy's inequality by induction. t , " U 1a.1 < 1. >-. ii:; Ofori = 1, •..• nand;>., +;>.. + •.• +}... = I,
Callow that
,: J
,"',.
!}...a, + }..... + . . . + }...o.! < l.
.. ,,
4.
Show
-
t
'!.
)
a
'-
t
'.
t
b
ml
' -'.
are complex 111!mberu satisfying
-
'.
'." -.,
.. ,." . , .
.•.. ,,- : .~,
'.'
,
,
.,.i,:,·,;,,:I· .....,-t+,!-+oI-2Iol· " '. '"
.,•"~-.-.. .
>..' - .. --,: ... , , -
'--'
_'-:;','.: ..;, -:' -..-,.,,<-: .-:";',:: ' '::,<:_::.,:::~":J::.:('~.~:-:;7~,~~::L-:.
::.'~.:~ 'i~'- ~,,~,;,~ ,..:
,
-
,
-...":".....-,
, ,- ..'.:: . -,~,-",'- ;-,.-'
12
COIIPLEX ANALYS ..
if and only if lal ;:;; lei. If this condition is
and
values of Izl?
what am the smallest
2. THE CEOMETRIC REPRESENTATION OF COMPLEX NUMBER.
With respect to a given rectangular coordinate system in a plane, the
+ complex number a = a ill can be represented by the point with coordi-
nates (a,Il). This representation is constantly
and we shan often
speak of the point a as a synonym of the number a. The first coordinate
axis (z-axis) takes the name of real aai8, and the second coordinate axis
(y-axis) is called the imaginary 1IriB. The plane itself is referred to as the
compln plane.
The geometric representation derives its usefnlness from the vivid
mental pictures 1W3ociated with a geom~tric language. We take the point
of view, however, that all conclusions in analysis should be derived from
the properties of real numbers, and not from the axioms of geometry.
For this reason we shall nee geometry only for descriptive
and
not for valid proof, unIe8B the language is so thinly veiled that the analytic
interpretation is self-evident. This attitude relieves us from the exigencies
of rigor in cODllectioD with geometric considerations.
2.1. C.ometric Addidora orad
The addition of com-
ple:J: nlJmbers MIl be visualized as vector addition.. To this end we let a
. number be
not only by a point, but also by a vector
from the origin to the point. The Dumber, the point, and the
vector will all be denoted by the same letter IJ. As usual we identify all
vectors which can be obt.ained from each other by parallel displacements.
Place a second vector b so that its initial point coincides with the end
point of a. Then a + b is represented by the vector from the initial point
01 a to the end point of b. To cOnstruct the difference b - a we draw
+ bo~ vectors a and b from the I!&me initial point; then b - a points from
the end point of a to the end point of b.
that a b and a - b
are the diagonals in a parallelogram with the sides a and b (Fig. 1-1)•
. An additional advantage of the vector repreeentation is that the length
of the vector a is equal to lal. Hence the distance between the points a and b is la - bl. With this interpretation the triangle inequality
la + bl ;:;; tal + Ibl and the identity la + bll + la - bl' ~ 2(lal' + Ibll )
become familiar geometric theorems.
The point a and its conjugate «lie By I ometrically with
to the
real
The symmetric point of a wi~ respect to the imaginary exis is
P". 1-' Veetc>r addition.
-4. The four points a, -ii, -a, 4 are the vertices of a rectangle which is BY rometnc with re8pect to both &Xes.
In order to deri"" a geometric infAlrjlletation of the product of two complex numbers we introduce polar coordinafAls. Ifthe polar coordinafAls of the point (a,/J) are (r,'P), we know that
a=rCOll'P
fl ~ rain 'P.
+ Hence
we
can
write
a =
a + ifl =
•
r{c08 'P
i RiD ",). In thia trigo-
nometric fO''IIl of a complex nllmber r is alway8 £; 0 and equal to the
modulua lal. The polar angle", is called the argument or amp/iltMie of the
complex number, and we denote it by arg a.
+ Consider two complex numbera II, ~ r,(coB '1" i sin tpJ and + a, - r,{c08 '1'1 i sin 'P')' Their product can be written in the from + + II,/It = T,rll{coa "', cos 'PI - ain 'P,8in 'PI) .(ain '1',_ 'PI C08 '1" sin '1'.)].
By meana of the addition tbeoreI1l8 of the cosine and the sine this
lion can be simplified to
,,. (Ui) We recogoize that the product haa the modulus "T, and the argument
+ ,, '1'1 '1'" The latter result is new, and we expreBB it through the equation
(17)
f·
'.' It is clear that. this formula can be extended to arbitn.ry products, and
L," -- We can therefore state:
:1; . -
TM lD'IIumem of II product ia eqUIIllo lite aum of lite argument8 of lite
'. fat:tma.
:
Tbia is fundamental. The rille that we have just formwated give/! a
; deep and IIne*pcdted justification of the geometric rep_tation of com-
i }>lex numhera. We must be fully aware, however, that the manner in
;:.·wLioh we have
..tlIe!OImula (l7).vioIatee our prineiplell.
,~":.,\¥... '
't:,i.
~;":.
.
. .
.'
"
.-:.... . . -- '. -
.. -:':"~'
, .'",
.. .
... - ," .
. ." ' --.: ,' '..'-~". "..'
' ',
.
.,."...,'".".
.
..'
,
".
'
.
'-
,- '-' ,
In the
.' "
... .' "
first place the equation (17) is between tlnglu rather than between num-
bers, and secondly its proof rested on the III!e of trigonometry. Thus it
remains to define the argument in analytic terms and to prove (17) by
purely analytic means. For the moment we postpone this proof and
shall be content to dismlf", the
of (17) from a less critical
standpoint.
We remark first that the argument of 0 is not defined, and hence (17)
has a meaning only if til and a. are #- O. Secondly, the polar angle is
deterillined only up to multiples of 360°. For this reason, if we want to
interpret (17) numerically, We must ag.ee that multiples of 360° shall not
count.
By Dltl&llB of (17) a simple geometric construction of the product ala.
r.a.o be
It folloWD indeed that the
with the vertices
0, 1, til is similar to the triangle whoae vertices are 0, a., ala.. The points
0, 1, tI" &lid a. being given. this similarity detbzmines the point ala. (Fig.
1-2).ln the case of division (17) is replaced by
(18)
argaa-., = arg a. - arga,.
The geometric construction is the eame, except that the similar triangles
are now 0, 1, tI, and 0, tI./tll, al.
Reliltlrk: A perfectly acceptable way to define angles and arguments
would be to apply the familiar methods of calculns whioh permit US to
e:xpreA8 the length of a circular arc SII a definite integral. This leads toa
correct definition of the trigonometric functions, and to a computational
proof of the addition theorems.-
The reason we do not follow this path is that complex analysis, as
PlO. w, vector multiplication.
COMPLEX NUMBERS
l'
opposed to real analysis, offers a much more direet approach. The clue
lies in a direet oonneetion between the exponential function and the trigonometrie functions, to be derived in Chap. 2, Sec. 5. Until we
reach this point the reader is asked to subdue his quest for complete rigor.
EXIRelns
1. Find the symmetric points of a with respect to the lines which
bisect the angles between the coordinate axes.
2. Prove that the points 4., /It, 4. are vertices of an equilateral triangle
4: a: a: + + + + if and only if
= a./It a,a. aall•.
1. Suppoee that II and b are two vertices of a square. Find the two
other vertices in all poesible cases.
4. Find the center and the radius of the circle which eircuID8Cribes
a., a.. the triangle with vertices /Is, ExprMl the result in symmetric fom•.
11.11. The Binomial Equation. From the preceding results we derive
+ that the powers of a = r(cos'P i sin 'P) are given by
(19)
+ a" = r>(COII "" i sin Ikp). •
This formula ;8 triyially valid for .. = 0, and 8ince
, it holds also when 11 is a negative integer.
..,.,,'',
For r = 1 we obtain de M oivr6'8 !I1rnl1l1a
(20)
, .
which provides an extremely simple way to
"teI.,r.ms of cot! 'P and sin 'P.
.
cos Ikp and 8in 1Itp in
'. To find the nth root of a complex number a we have to solve the
+ :., Biipposing that a ¢ 0 we write a = ,(cos 'P i sin "') and
z = p(008 B + iBin B}.
(21) takes the fonn
p"«()011 ,,/} + fain.,) - r{cos tp + i sill 'P).
, ,
"
,
"""""-',
-",--, . - , ".''-,-'" ,
'"-".-"'-,,".c.'..'.--" ,"-
.' ,
.... - - ,'.;. ,"
:.',.
18
COM PLE!X ANALYSIS
This equation is certainly
obtain the root
A' if p. = r and = 'P. Hence· we
00.8'nP- + I•S.lnnt-p I
where .y;: denotes the positive nth root of the positive number r.
But this is not the only solution. In fact, (22) is also fnJfiJ1ed if n8
diJfelll from " by & multiple of the full anp. If ang\eB are eJ:preBBed in
the full angle is 2Ir, and we find that (22) is Il&tisfied if and only if
8 _ ! + k. 2Ir,
n
n
k is any
. However, only the values k = 0, 1, • • . • n - 1
give
values of r. Hence the complete solution of the equation
(21) is given by
+ + i sin ! k ~ I k .. 0, 1, ... , A-I.
n
n
Tllenl (Ire n nth roots of any compiez number ¢ O. Tiley 1r.avs Ute modulus, and their arguments arB equally epaced.
Geometrically, the nth roote are the vertices of a regnlar polygon with n sides.
The (I = 1 is particularly important. The roots of the equation z" - 1 are called nth roots of unity, and if we set
(23)
Cd = C.OB2 r + ,.8.1D -2Ir
n
A
all the roote can be
by 1, (oJ. (oJ', ••• , ",_1. It is alAO quite
evident that if Va denotes any nth root of (I, then all the nth roots can be
exp..-d in the fOlln ",. . Va, k = 0, 1, . . . , n - 1.
EXERCI. .,
L Eqlreee C08 31', COB 4." and sin 5., in terms of COB ., and sin .,.
+ + + . . . + Z. Simplify 1 C08 I' 008 2.,
+ . . . + ain 2"
sin RIp.
COB RIp
sin ., +
I.
the fifth and tenth roots of unity in algebraic form•
.. If til is given by (23), prove that
+ ... + + . . . + 1
",tA
",<_1)1 = 0
for any integer It. which is not a multiple of n.
COMPLEX NUMBERS
17
S. What ill the valUll of
1 - ... + .... - ... + (-1)-",(0-1)'1
1.3. A.nalytic GfH1_try. In classical anaJytic geometry the equation
of a loous ill
as a relation between:l: and y. It can juat as well
be
in tenruJ of z and i, sometimes to distinct advantage. The
thing to remember is that a complex equation ill ordinarily equivalent to
two rea1 equations; in order to obt.ain a genuine locua these equations
should be
the same.
For instance, the equation of a circle is III - 01 ~ r. In algebraic
form it can be rewritten as (z - a){l! - 4) ~ r". The fact that this equa-
tion is invariant under complex conjugation is an indication that it
repreeents a single real equation.
A Bkaight line in the complex plane can be given by a parametric
equation z '" a + bt, where a and b are complex numbers and II ,& 0; the
+ parameter t nIDS through all real values. Two equations II '" II bt and + II ~ a' b't represent the same line if and ouly if a' - a and h' are real
multiples olb. The Ii"'" are parallel wbenever II' ja a ree ' multiple of b, and they are equally directed if b' ill a pOBitive multiple of b. The direc-
tion of a directed line can be identified with arg b. The sngle between
+ + • = a bt and It ... a' b't is a.rg b'/b; observe that it depends on the
order in which the line. are named. The lines are orthogonal to each
other if b'/II is purely imaginary.
Problema of finding int..reeetioDS between lines and circles, parallel
or orthogonal Jines, tangents, and the like usually become exceedingly'
simple when expreaaed in complex form.
An ineqlla1ity I_ - al < r
the inside of a circle. Similarly,
+ a d irooted line II - a bt determines a right half plane consisting of all
pointuwithIm {. - a)/b < oand a left half plane with 1m (11- a)/b > O.
· An easy argument shows that this distinction is independent of the
·
.' p.,.,.metric representation.
.... aXERC.SES
·
•
L When
+ + <II: bi C = 0 represent a line?
z. Write the equation of an ellipse, hyperbola, parabola in complex
fOlm.
·
.'
.. Prove that the
of a parallelogram bisect each other and
~t the dial'luals of a rhombus are orthogonal.
· " .... Prove .u·lytieilUy that themidpointa of parallel choMs to a circle
··Iie on a
.petpmidicmJar to the chorda.
" ' - t ' · ", L BhDWttr.pt aD .~ that'pus t.hroilgh II and 1/4 intereect the
·
.. -.'.It" -u"l"_1·""
.""~ .'," . """ ", . ';. .- ,.
. ..
-. , '
#.: .
. . :.;'.
.- " ". ,
.....,.-..,.
- -,
• •
... . ,
:,-,,"
.... .... ,'-',
:,- '"
' . . .'.
. -.-- .....~..,.'·1,,;,~t
, " ._'"., ",",'.-.,•..,'.,
-""'~-·'<V!:
"
1.
COMPLEX ANALYSIS
2.4. The Sphericol Representation. For many purposes it is useful to
extend the system C of complex numbers by jntroduction of a symbol 00
to repre!!ent infinity. Ita connection with the finite numbers is estab-
+ + lished by setting a 00 ~ 00 a == 00 for all finite a, and
/)'00 == oo'b== 00
+ for all b ~ 0, inchuJing b == 00. It is irnpOSllible, however, to define
00 00 and O· 00 without violating the laws of arithmetic. By special
convention we shall
write alO = 00 for a ~ 0 and bloo = 0
for b ~ 00.
In the plane there is no room for a point
to 00, but we
r.an of comw introduce an "ideal" point which we call the point at infinity.
The points in the plane together with the point at infinity form the
extended COtI'plex plane. We agl ee that every straight line shall
through the point at infinity. By contrast, no half plane shall contain
the ideal point.
It is desirable to introduce a geometric model in which all points of
the extended plane have a concrete repre!lentatiw. To this end we con-
sider the unit sphere S whose equation in three-dimensional space is
x~ + x~ + x: = 1. With every point on S, except (0,0,1), we can associ-
ate a complex number
(24)
+ z; x, 1 -
ix,
x, J
and this conespondence is one to one. Indeed, from (24) we obtain
and hence (25)
Izl" = x~ + x: = 1 + x.,
(1 - x.)' 1 - x.
x, =
Izlt Izl'
+ -
1
t'
Further computation yields
x, = 1z++lIzll'
(26)
z-j
x, == i(l + Izl')'
The
ean be completed by letting the point at infinity
couespond to (0,0,1), and we can thus teglll'd the sphere as a repre-
!!entation of the extended plane or of the exteDded number system. We
note that the hemisphere x. < 0 c<Jtlesponds to the disk Iz\ < 1 and the
COMPLeX NUMBERS
1.
° hemisphere %. > to its outside Izl > 1. In function theory the sphere
S is referred to as the Riemann sphere.
If the complex plane is identified with the (%.,x.)-plane with the
%,- and %.-axis
to the real and imaginary axis, respectively,
the transCormation (24) takes on 'I simple geometric ml'8.ning. Writing
+ ,. = Z iI/ we caD verify that
(27)
z:I/:-l = Z,:ZI:Z. - 1,
and this means that the points (z,I/,O) (z"z.,x.), and (0,0,1) are in a
straight line. Hence the
is a central projection from the
center (0,0,1) 88 shown in Fig. 1-3. It is called a lllereograpkic pr(}jeetion.
The context will make it clear whether the stereographic projection is
regarded 88 a mapping from S to the extended complex plane, or via _Ba.
In the spherical representation there is no Himple interpretation of
addition and multiplication. Its advantage lies in the fact that the point
at infinity is no longer distinguished.
It is geometrically evident.that the stereographic projection traQ8-
forms every straight line in the z-plane into a circle on S which
through the pole (0,0,1), and the
is also true. More generally,
any circle on the sphere corresponds to a~le or straight line in thez-plane.
To prove this we observe that a circle on the aphere liee! in .. plane
+ + + a,%, a,xl a,xl - ao, where we can 8'sume that at a~ + ai - 1
° and ;:i! ao < 1. In terms of z and i this equation takes the form
+ + + a,(,. i) - ati(,. - t) a.(W - 1) = ao(W 1)
or
+ (a. - a.)(z· + y') - 2..,z - 2a.y + ao a. - O.
a. a. a, a. For ~ this is the equation of .. circle, and for = it represents
a straight line. CODveISely, the equation of any circle or straight line
N
•
•
• r, ,
•
!9-:,'-
. . .~W,
~>
iI<' __
• • • •
. .
. " ,
. .
.
.
."""
.'
'
.
. . '.
- .- ,
cen be written in this f()ml. The ooi1espondence is coll8equently one
to one.
'
It is easy to calculate the distance d("t) between the stereographic
of IS and i. H the points on the sphere are denoted by
(ZI,3:"Z.), (z;,:/:;,z;), we have first
(ZI - za' + (z, - zJ' + (z. - Z~)I ~ 2 - 2(z1:l:; + x.z; + Z1:l:').
From (35) and (36) we obtain eIter a short computation
z,Z; + z,(z,; ++1Z)1(::l1:; + I') - (z - 2)(:1 - 2') + (1aI' - 1)(lz'I' - 1)
..
(1 + W)(1 + Iii")
. ~
(1 + lal1')+(1 I+z!'l)a('lI')+-la'21Ia) -
" •
•
Aa a result we find thet
•
(28)
d(z,.') .. ....,
For II - GO the corresponding formula is
d(z, GO)
~
VI
2
+
"Izli..
EXERCISES
J. Show that z and :I
to diametrically opposite points on
the RiP-mann sphere if and only: if Ii.' = -1.
I. A cube has its v!lrlices on the sphere S and its edges parallel to the
coordinats axes. Find the stercographic projections of the vertices.
J. . problem for a regUlar tetrahedron in general position.
... Let Z, Z' denote the stereographic projectiollB of z,:I, and let N be
the north pole. Show that the triangles NZZ' and Nze are similar, and
use this to derive (28).
s. Find the radius of the spherical image of the circle in the plane
whoes center is a and radius R.
, '.
. ,
2
PLEX FUNCTI s
1. INTRODUCTION TO THE CONCEPT OF
ANALYTIC FUNCTION
.
The theory of functions of a complex variable aims at extending
calculus to the complex domain, Both differentiation and inte-
gration acquire new depth and signifiMIIll8; at the sam" time the
range of applicability becomes radically
Indeed, ouly
the analytic Dr holomorphic functions MIl be freely differentiated
and integrated. They are the Dnly true "functions" in the sense
of the French "Thoorie des fonctions" or the German
"Funktionentheorie.'J
Neverthelees, we sball use the term "functiDn" in its modem
meaning. Therefore, when stepping up to oomplex nwnbers we
haw to eonsider four different kinds of functions: real functions
of a real variable, reAl functions .nf a complex variable, complex
functions of a real variable, and complex functions of a complex
variable. As a practical matter we
that the letters. and ID
,ball always denote complex variables; thus, to indicate a complex
function of a complex variable we use the notation ID = !(z). t
The notation 11 - J(z) will be Jlsed in a neutral manner with the
UIlderstMding that z and 1/ can be either rea! or complex. When
_ want to indicate that a variable is definitely restricted to real
values, _ shall usually denote it by t. By these
we
* .. t Modem .tudOllla &Ie "ell .ware that f It"Ddo for the runntjpn aDd J{a)
··for. -'01. fn..,OD. &.e,_, "",')eta are taAitioDally minded and
,
wntJnue1O,~·.,·,·t"tba"DoteOiiJf..)." :;- '
..•••
•
t.
~r;
.. ... -"'(,-"---",',-.. .., .: ..' .
.. , ~
.
.
-,,:,',:-.:
;:,;
_.';-:.;":
..
'~';::~;·~'..~~f;;i-'-i-'";.:.;,:: 0;
:'
,
: .:";::'.:: -:. :.:,
21
..
:-
.
.:.'
.
','., ",-,,"-
ZI
COIIPUX AIIALYSI.
+ do not wish to cancel the earlier convention whereby .. notation z = x iy
automatically implies that x and 11 are real.
.
It is CBsentia! that the law by which a function is defined be formulated
in clear and unambiguoup terms. In other WOrdB, aD functioDB mut be
weU defined and consequently, until further notice, Mngl£-oalued. t
It is 1Wl necessary that a function be defined for aD values of the
independent variable. For the moment we shall deliberately under-
emph&Bive the role of point set theory. Therefore we make merely an
informal agreement that every function be defined on an open 1Iflt, by
which we mean that if I(a) is defined, then f{x) is defined for all " suffi-
ciently cloBe to a. The formal treatment of point set topology is deferred
until the next chapter.
1.1. Limits and Continuity. The following basic definition will be
adopted:
TM Jundion f(,,) itt 84id to have the limit A IJII " tmda to a,
(1)
.....lim I(x) = A,
iJ' . only if the loUOOIing is I!:m:
Far tJVeTY • > 0 there exittls a number II > 0 with the property tIwl
1/(:) - AI < ,for all values of x such tIwllx - al < II and:r >" a.
. . . !'-.,. '
. "'l'!rla definition makes decisive liRe of the absolute value. 8ince the
notion'of absolute value has a meaning for complex as well 88 for real
we can UBe the same definition regardless of whether the variable
the function J(,,) are real or complex.
..........'. . an alternative simpler notation we sometimes write: f(x) --> A for
. ' ''l'bere are some familiar variants of the definition which correspoud
. . eere where a or A is infinite. In the real. case we can distinguish
+'" the limits
and - "', but in the complex case there is only
~infinite limit. We truet the reader to formulate COllect definitions
tci'Cover all the possibilities.
'.. '" 'The well.known results concerning the limit of a'sl1m, .. product, and
a ,qilotient continue to hold in the complex
Indeed, the proofs
depend only on the properties of the absolute value expreesed by
labl - lal . Ibl and la + bl :!Ii lal + Ibl.
t We eb·D IIOmetim e& IISD the plecm'.mc term .in;. ,.sd/flAditJlt& to un_line
that the f1"'efiou has cmly one yalue for: elM vr'u of the ~
..
.. .
: -..... : .,: ..
COMPLEX FUNCTIONS
Condition (1) is evidently eqwvalent to
(2)
.l.i.m... i(x) = A.
From (1) and (2) we obtain
.li-m4 Re I(x) = Re A
(3)
......lim 1m I(x) = 1m A.
(1) is a con""quence of (3).
The function I(x) is said to be continuous at a if and only if
.....lim I(x) = I(a). A conlin"""" ,unction, without further qualification,
is one which is continuous at all points where it is defined.
+ The sum/(x) g(x) and the product/(x)g(x) oftwo continuousfune-
tions are continuous; the quo\ient I(x)/g(x) is de1ined and continuous at
a if and only if I/(a) 'J"f o. If I(x) is continuous, so arc· Be I(x), 1m I{x),
and 11(x)l.
,
The derivative of a function is defined as a particular limit and can be
'; considered regardless of whether the variables are real or complex. The
forn1a.l definition.is
" ,
, ' (4)
I'(a) = lim I(x) -/(a).
..... z-Q
The usual rules for forming the derivative of a sum, a product, or a
". quotient are all valid. The derivative of a composite function is deter-
mined by the chain rule.
There is nevertheless a fundamental difference between the
of a
',real and a complex independent variable. To illustrate our point, let
/fII) be a r«Jl function of a complex variable whose derivative exists at
,"
Then I'(a) is on one side real, for it is the limit of the quotients
+ I(a 1) -/(a)
Ii.
,
Ii. tends to zero through real values. On the other side it is also the
of Uw quotients
I(a + ih) - lea)
=
..
iii.
118 such purely imaginary. Therefore f(G) must be zero. Thus a
of & oomiMex variableeitber hIIII the derivative zero, or eLte does DOt 1IliiIt.' , "~"'" .. '
...... .. ,.'. , ,
,
,- •. , .,,:..: ':' -': .,:- -' '_ ' -' :.;_" ._"r. _.. . ,,',_ ": .,. .. . . ..-..;.::_~.:, ;'._~,'
-
,
'"
. -
- ,~
. ' - , - ' ,
.",'
~,':, ;,;.'',~"
,-~
.. -
c.
,.
',..
'.'-';:''0
.,;".;.;:":"'..H."'
".•'
'._
'.:'"-
-
"
.,
...·-,k-,":....,
.
;"",'~'
. " .
.......
24
COMPLEX ANALYSIS
of a complex function of & real variable CM be reduced to the
real
+ II we write z(t) = :I:(t) iy(t) we find indeed
+ ret) = :1:'(1) iv'(t), .
and the existenee of s'(I) is equivalent to the simuItaruloU8 existenee of
:e'(0 and TI(t). The complex notation hIlS nevertheless certain formal
advantages which it would be unwise to give up.
In contrast, the existence of the derivative of a complex function of a
complex variable has far-reaching consequences for the structural proper-
ties of the function. The investigation of these consequences is the central theme in complex-fllnction theory.
1.2. Ancdytfe Functio...... The class of aMlgticfunctioot is fom.ed by
the complex fllnctions of a complex variable which
a derivative
wherever the function is defined. The term holDlltorphic fumlirm is nsed
with identical meaning. For the purpose of this preliminary investiga-
tion the reader may think primarily of functioU8 which are defined in the
whole plane.
The 811m and the product of two analytic functions are again analytic.
The same is true of the quotient f(z)/g(%) of two analytic functions, pro-
vided that I/(z) does not vanil!h. In the general calle it is
to
exclude the points at which g(z) = O. Strictly speaking, this very typi-
cal case will t.hus not be included in our considerations, but it will be clear
that the results remain valid except for obvious modifications.
The definition of the derivative can be rewritten in the form
fez) + _ lim fez 11) - fez) •
~o
h
All a first consequence fez) is
continuous. Indeed, from
+ + f(1l 11) - fez) - h· (f(z h) - f(z»/II we obtain
+ lim
~o
(J. (z
h) - f(z)) = 0 . fez) ... o.
+ If we write fez) = v(s) w(z) it follows, moreover, that v(z) and II(Z)
are both continuous.
The limit of the difference quotient must be the same regardless of the way in which h approaches lIero. If we choose re.a\ values for h, then the imaginary part 'I is kept constant, and the derivative becomes .. partiaJ derivative with respect to:e. We have thus
COMPLEX FUNCTIONS
Similarly, if we substitute purely imaginary values ik for h, we obtain
fez}
_
lim I(z + i~)
~o
ik
-/(z)
""
-
i
~
ay
~
-
i
~
ay
+
~. ay
It follows that I(z) must satisfy the partial difterential equation
(5)
~
iIz
-
-ia~y
whioh resolves into the real equations
•
au av
(6)
-iIz= -aJ y
are the Caw:hy-Riemmr.n differential equations which must be
satisfied by the real and imaginary part of any analytic function. t
We remark that the existence of the four partial derivatives in (6) is
implied by the existence of fez). Using (6) we can write down four form81Jy difterent expressions for fez); the simplest is
fez} - ~az +i~a.z
For the quantity If(z)I' we have, for instance,
If(z)I' = aaxu •+ aauu' = aaxu' + aaxv' = aauxaavy
The last expression shoWl! that If(z)I' is the Jacobian of u and v with
respect to z and II.
We shall prove later that the derivative of an analytic function is
itself analytic. By this fact u and v will have continuous partial deriva-
tives of all orders, and in particnl.... the mixed derivatives will be equal.
Using this information we obtain from (6)
a'v a'v
AU=8z.+ay.-O
a'u a",
Av=az·+ayt=O.
A function u which satisfies LGp/<M:e', equation Au = 0 is said to be
laarmtmie. The real and
part of an analytic function are thus
hi monic. If two .harmonic functions u and " satisfy the Cauchy-
lliemann equations .(6),. then v is to be the coniWl'lle IIormonie lu-
tA--"·
..M. ibe
met.ria
•
~ :. ..'
,.-.....
_
..
-
-," .
.. , . - ." - -'
. . .-, • ••
--:
__-' _.":'-.:__..,.:.: :...:,,;.:.."~.:.'.,-:, ,..,..:', -...'.':!i- ...''.-~"_\..',,":'.:"-" ;..1).:..,.',~!'_;r-'.-,-:. .',~ "'-'..-".. .':,:''.;.:'~l. :,-,''~-.''":,'c'.,:.t-..":.:'-"
.
:,)':.. . ""
'-'"
.,""' ':"
.'-.;
"i-;,:, "~,:;",,
.
.",
':"""-" '" " ' " .
i.~.>"::'. '-' :,'~'~'.'-'
'-
.- .....-..,,"' ~ .-'
"•.'," '.. ,.~~,
tibn of u. Actually, v is detem';ned only Up to an additive coDStant, 80
that the Il8e of the definite article, although traditional, is not quite aceu-
rate. In the same sense, u is the conjugate h8Jmonic function of - •.
o
This is not the place to iliscuss the weakest conditions of regularity
which CILll he imposed on harmonic functions. We wish to prove, how-
+ ever, that the function 1£ ill determined hy a pair of conjugate har-
monic functions is always analytic, and for this purpose we make the
explicit assumption that 1£ and v have continuousfil"ilfnlrder partial
derivatives. It is proved in calculus, under exactly these regularity con-
ditions, that we can write
az a + + u(x. 1.,1/ k) -
u(x,1/) =
au I.
+
au
Y k
+
6,
ii + + . v(x h,y k) - .(z,1/) = II +~ k + to,
where the remainders "" "
tend
to
zero
more
-
rapidly
than
I.
+
ik
in
the
+ + + lienee that 6,/(1. ik) -+ 0 ILlld ../(1. iTe) -+ 0 for I. ik ..... O. With
+ the notation fez) = u(x,1/) iv(z,y) we obtain by virtue of the rela-
tions (6)
+ + J(z I. ilc) - fez} =
~
iJz
+ i
~
ax
(h + ik) + I. + iI,
and hence
,
+ + + + fun f(. I. ilc) - f(z) = ~ i ~.
.H~ ,0
h. ik
ilz ax
.e conclude that f(l) is analytic. -,.-- If u(x,1/) and v(x,1/) have rontmOOWl jiTBt-order partial derivatives wh.ich
j,aliwf1/ the Cauchy-Riemann differential equation" then J(z) = u(z) + w(z)
_.. tmallltic with rontinU0U8 derivative /'(.), and COIWerMIII.
The conjugate of a harlllonic function can be found by integtation, and in simple caseo the computation can be made explicit. For inst;a.nr,e,
U = ",' - y' is harmonic and au/ax = 2x, au/ay = - 21/. The conju-
'pte function must therefore satisfy
i-aN", ~ 21/, aay. = 2",.
-
-
+ From the first equation v = 2:I:y <p{y), where tp(Yl is a function of 1/
alone. Substitution in the second equation yields tp'(y) - O. Hence
.,(Y) is a
and the most general
function of ",' - 1/' is
+ + 2:I:y e where" is a consta.n~. Obeerve tha~:r:" - 1/' 2izy - z". The
analytic function with the real pa.-t "," - 11' is- hence z' + te.
- . --,-".',
COIIPLEX FUNCTIONS
There is an interesting formal procedure whieh throws considerable
light on the nature of analytic functions. We present this procedure
with an explicit warning to the reader that it is purely fonnal and does
not
any power of proof.
.
Consider a complex function/(.:z:,II) of two real variables. Introducing
+ the complex variable z = .:z: + i,l and its conjugate z = .:z: - iy, we 'can
write x ~ t(z i), y = -!i(z - I). With this change of variable we
can consider 1(.:z:,II) as a function of z and li which we will treat as inde-
pendent variables (forgetting that they are in fact conjugate to each
other). If the rules of calculus were applicable, we would obtain
-iillzl=21- -ailx-l ' 1. ~ i-Jf ,
-iiJlfl'
-21-
+aiJxf
1•
-iJf
~
·
These expressions have no convenient definition as limits, but we can
nevertheless introduce them as symbolic derivatives with respect to z
and!. By comparison with (5) we find that analytic functions are ch......
acterized by the condition ill/ilt = 0. We are thus tempted to say that
an analytic fMetion is independent of I, and a function of 0 alone.
This formal reasoning supports the point of view that analytic funll-
tions are true functions of a complex variable as opposed to functions
which are more adequately de.cribed 88 complex functions of two real
variables.
By similar fonnal arguments we can derive a very simple method
which allows us to compute, withont use of integration, the analytic
function I(z) whoae real part i8 a given ·harmonic function u(.:z:,I/). We
remark first that the conjugate function I(z) has the derivative zero with.
respect to 0 and may, therefore, be considered as a function of !; we
denote this function hy J(I). With this notation We can write down the
identity
+ + u(.:z:,y) - t!/(.:z: il/) l(x - il/)].
It i8 reasonable to expect that this is a formal identity, and then it holds
even when z and 1/ are complex. If we substitute z = z/2, 1/ - z/2i,
iie obtain
+ 1I(z/2,0/2i} - l!f(z) J(O)]. •
Since /(%) is only determiMd up to a purely imaginary constant, we may
• weU aliSume that 1(0) ill real, which implies J(O) = u(O,O). The funD-
tlon /(0) can thus be computed by means of the fannula
1(') = 2u(a/2, z/'a) - u(O,O)• .
A pqrely imegiDMY oo.8ten t.':an ~ &d~~at-·"iU.:
. In this form the method is definitely limited to functions u(%",I/) wMeIl
~ ,
, . .' - ' - " . -
U
COMPLEX ~N~LYSIS
are rational in :I: and tI, for the function must have a meaning for com-
plex values of the argument. Suffice it to say that the method can be extended to the general caee and that a complete justifi, cation can be
gi• ven.
EXERCISI;S
L If g(w) and fez) Me analytic functions, show that g(J(z» is also
analytic.
:&. Verify Canchy-Riemann's equations for the functions Zl and t l •
+ I. Find the most general harmonic polynomial of the form azl + + b:l:"y czyt dy'. Determine the conjugate harmonic function and the
corresponding analytic fnnction by integration and by the formal method.
... Show that an analytic function cannot have a constant absolute
value without reducing to a constant.
,
.. Prove rigorously that the functionsj(z) and'"'(B"') Me simultaneously
analytic.
L Prove that the functions u(z) and uti) are simultaneouslyharmonic.
7. Show that a harmonic function satisfies the fOfmal differential
equation
aa,'a"i = o.
1.3. PolynomialB. Every oonstant is 8D analytic function with the
derivative o. The simplest nonconstant analytic function is Ie whose
derivative is 1. Since the sum and product of two analytic functions are '
again analytic, it follows that every polynomial
(7)
pe,) = a. + a,z + . . . + a,r
is an analytic function. Its derivative is
P'(z) = 4, + 2a.z + ...
The notation (7) shall imply that a,. F 0, and the polynomial is then
said to be of
1l. The constaDt 0, considered as a polynomial, is in
many respects exceptional and will be excluded from our oonsiderations.t
For 1l > 0 the equation pe,) = 0 has at least one root. This is the
~ed /nDdamentaI theorem of algebra which we shall prove later.
If pea,) ... 0, it is shown in elementary algebra that pe,) = (z - a,)p,(z)
where P,(z) is a polynomial of deglee 1l - 1. Repetition of this
finally leads to a complete factorization
(8)
P(z) = a,.(z - ",){z - al) • . . (z - ...)
,
t For formal reaSDns, if the constant Oiarel&rded M a polynomi&l, itadell'ee is.
equal to - ...
COMPLEX FUNCTIONS
21
where the (I., aI, • . • , .... are not
distinct. From the fac-
toriz"tion we conclude that P(z) does not Vlmish for any value of z
different from a" a., ... ,..... Moreover, the factorisation is uniquely
determined except for the order of the factors.
If exactly h of the (lj coincide, their common value is called a zero of
P(z) of the order h. We fiud tbt the 8Um of the orders of the zeros of a
pcIynom;a] is equal to its degJee. More simply, if each zero is counted
as ma.oy times as its order indicates, a polynomial of degJee n has exactly
n zeros.
The order of a zero a can also be determined by consideration of the
8ucceeeive derivatives of P(z) for z = a. Suppose that a is a zero of
order h. Then we can write P(z) = (z - a)·P.(z) with P.(a) ¢ O. SUIl-
ceseive derintion yields Pea) = pI(a) = . . . - pel-li{a) = 0 while
pel)(a) ¢ O. In other words, the order of a zero equals the order of the
first nonvanishing derivative. A
of order 1 is called a simple zero
and is characterised by the conditions Pea) = 0, pI(a) ¢ O.
As an application we shall prove the fonowing theorem, known as
L'l.laJI' ~
Theorem 1. If allll~08 oj a poll/flOmiol P(z) lie in a hnlJ plane, !hell all zeroB oj the derivative pI (z) lie in the same hnlJ plane.
From (8) we obtain
(9)
+ . . . + P'{z) _ 1
-,---1--=._.
P(II) - z - a,
z- ....
Suppose tbt the half plane H is defined as the part of the plane where
1m (z - a)/b < 0 (see Chap. 1, Sec. 2.3). If ... is in H and z is not, we
have then
But the imegin&Iy parts of reciprocal numbers have opposite sign.
Therefore, under the same
1m bez - ...)-1 < O. If this is
true for all k we conclude from
r - 1m bPP('(sII}
< L" b
1m Z-Cl,t
0,
I-I
.
and coDBequently P' (_) ¢ O.
In a sharper fonnwation the theorem tells WI that the smaUest convex
polygon tbt contains the aeroe of P(.) also contains the zeros of PI(Z).
COIIPLEX ANALYSIS
1.4. RatiolMJl Functions. We turn to the ""_'Al of a rational function
P(z)
(10)
R(z) = Q(z)'
given lIB the quotient of two polynomials. We
and this is ell89n-
tial, that P(o) and Q(z) have no common factors and hence no common
zeros. R(z) will be given the value QD at the zeros of Q(z). It must
therefore be considered as a function with values in the extended plane,
and as such it is continuous. The zeros of Q(z) are called poles of R(z),
and the order of a pole is,by definition equal to the order of the corre-
sponding lero of Q(z).
The derivative
(11)
R ' (• )
_ -
P'(z)Q(o) - Q'(z)P(z)
Q(.).
•
,
exists only when Q(z) ~ O. However, as a'rational Iunction defined by
the right-hand member of (11), R'(z) has the same poles as R(z), the order
of each pole being increased' by One. In case Q(z) has multiple zeros, it should be noticed that the expression (ll) does not appear in reduced
form. Greater unity is aOOieved if we let the variable z 88 well 88 the values
R(z) rang<! over the extended plane. We may define R( QD) as the limit of R(o) as z ....... , but this definition would not determine the order of a zero or pole at ... It is therefore preferable to consider the function R(l/z). which we can rewrite as a rational function R,(z), and set
R( QD) = R,(O).
n
R,(O)
=
0
or
.. ,
the
order
of
the
zero
or
pole
at
..
is
defined
.-
as
the
order of the zero or pole of R,(z) at the origin.
With the notation
R
(.)
=
ab..++
a,z
b,z
+ +
......
++ab."....-.
we obt.ain
•
where the power ..' • belongs either to the numerator or to the denomi-
nator. Accordingly, if m > 11 R(z) has a zero of order III - 11 at QD. if
III < 11 the point at .. is a pole of order,1I - tn, and if III = 11
R(oo) = a./b. ~ 0,00.
, ,
-, . ''."".'..,...
COMPLEX fUNCTIONS
31
We can now count the total number of zeros and poles in the extended
plane. The count shows that the number of zeros, including thoee at 00,
is equal to the gIeater of the numbers m and n. The number of poles is
the same. This common number of zeros and poles is called the ~ of the rational funetion.
If a is any constant, the function H(,,) -' Q has the 8&me poles as R(o),
and consequently the same order. The zeros of R(o} - Q are roots of
the equation R(o} - a, and if the roots are counted as many times 88 the
order of the zero indicates, we can state the following result: A rational function. R(o) of order p 11M p zer/HI and p pole8, and every
equation. R(z} = a has e:MCtIy p roota.
A rational function of order 1 is a linear fraction
8(z) - ; t~
with a8 - fh ;F O. Such fractions, or linear
will be
studied at length in Chap. 3, Bee. 3. For the moment we note merely
that the equation." = 8(z) has exactly one root, and we find indeed
z = 8-1(w) =
aw
-'\'V
I-+fJa
•
The transformations 8 and 8-1 are inverse to each other.
The linear transformation z + a is called a parallel tr~, and
l/z is an ~simI. The fOlmer has a fixed point at CD, the latter inter-
changes 0 and 00.
Every rational function has a representation by parti6l ff'tJdiorul. In
order to derive this representation we
first that R(z) has a pole
at DO. We carry out the division of P(z) by Q(z) until the
of the
remainder is at mOllt equal to that of the denominator. The result can
be written in the form
(12)
+ R{z) = G(z} H(z)
where G(z) is a polynomial without constant teno, and H{.) is finite at DO.
The
of G{:.) is the order of the pole at GO, and the polynomial G(z)
is called the ri~ part of R{z) at ...
Let the distinct finite poles of R(z) be denoted by fJ., fJ.. • • • , fJ••
+} The function H fJi
is a rational function of r with a pole at r = "'.
By une of the decomposition (12) we can write
•
"',,,-,:. '
• • .,• ';
•
- ,"-,..,,--," - -.·,''tj , '.,", ,"--' ,"- , ". ' - . ' " ,. - ,- ,'- ,.". ~-. . -,- •• ',
32
COMPLEX ANALYSIS
or with a change of variable
R(o) = Gj
1
z - fl;
+H;
Here G;
1
• - fl;
is a polynomial in s ~ fJ; without constant term, called
the singular part of R(z) at flj.
The function Hi
1
z - /l;
is finite for
z = flj.
C(lnsider now the expression
(13)
I• . R(z) -
G(z) -
Gj
;-1
---'1--".
Z - fJ;
This is a rational function which cannot have other poles than fl., fl"
. . . , fl. and ... At z = fJ; we find that the two terms which become
infinite have a difference Hj z ~ fJ; with .."finite limit, and the same
is true at ... Therefore (13) has neither any finite poles nor a pole at ...
A rational function without pole. must reduce to a constant. ""d if this constant is absorbed in G(z) we obtain
(14)
L R(.) = G(z) +
• Gj
1 z - fJ ••
,. - 1
'
This representation is well known from the calculus where it is use<! as .. technical device in integration theory. However. it is only with the
introduction of complex numbers that it becomes completely SIlccessf!!1.
EXERCISES
1. Use the method of the text to develop
0'
z' - 1
and
1
+ z(z 1)'(0+ 2)'
in partial fractions.
.
2.. If Q is a polynomial with distinct roots "'1. . • . ....... and if P is a
polynomial of d"gree < fl. show that
3. Use the formula in the preeedingexe..,ise to prove that there existe
a unique polynomial P of degI ee < fI with given values c. at the pointe
ott (I.agrange·s interpolation polynomial).
.,"-.:'
..•...,: -'~.' '..-",....~..,....- ...
COIIPl.EX FUNCTIONS
... What is the general form of a rational function which has absolute
value Ion the cirrJe Izi ~ 11 In particular, how are the zeros and poles
related to each other?
s. If a rational fllnction is real on Izl = I, how are the zeros and poles
oituated?
6. If R(z) is a rational function of order n, how large and how small can the order of R'(z) be?
Z. ELEMENTARY THEORY OF POWER SI!RIES
Polynomials and rational functions are very special analytic functions.
The easiest way to achieve peater variety is to form limits. For instance, the slim of a convergent aeries is such & limit. If the terms are functions of a variable, so is the sum, and if the tern.s are analytic functions, chances are good that the 8Um will also be analytic.
Of all aeries with analytic u,nns the power aeries with complex
coefficients are the simplest. In this section we study only the most
elementary properties of power aeries. A strong motivation for taking
up this study when we are not yet equipped to prove the most general (thoae that depend on integration) is that we need power aeries
to construct the exponential function (Sec. 3).
2.1. Sequences. The sequence {a..1 r has the limit A if to every. > 0
there exists an n. such that Ia.. - AI < • for n ~ no. A sequence with a
finite limit is said to be ~, and any sequence which does not con-
verge is divergent. If lim. , . ... = .. , the aequence may be said to
diverg4 II) infinity.
Only in rare C88eS can the convergence be proved by exhibiting the limit,
so it is extremely important to make use of a method that JK>lmits proof
of tbe existence of a limit even when it cannot be determined explicitly.
The test that serves tbi. purpose hears the name of Cauchy. A seqilence
. will be called
or a Cav.chy ~, if it satisfies the follow-
ing condition: given any I > 0 there exists an n. such that I... - a..1 < •
whenever n ~ n. and m ~ nO. The test reads:
A "'tIIcnu i. Cl)f/.vergent i/ and I)f/.ly i/ it il a Cauday.eq_.
The neceeaity is immediate. If a.. --+ A we can find n. such that < ./2 for n iii!: n.. For tn,n ~ n. it fonows by the triangle
that Ia.. - a..1 :i Ia.. - AI + Ia.. - Al < •.
The 8ufficiooey is closely connected with the definition of real num-
.\I:eIs, 8!ldOIll! way in which real numbers can be introduced is indced to
c!)ndition.However, wewiah to lise
bou~e!lrqonOto"'~uenc:eof real num.
f -~
'
,
." .
r-... - . -.-."<;.' .... '.,- . .- . -:.-. ... ..". -'\.!'.' - -. ,:.,..."...-.•. .. ......;., >" ••_,_;.,,',, . _,~"-.-.,. '''''"_-/:'-_.''"''.;.-,~: :-,..c_,_,,'"' • .-
.,'..
" .<, ":-.--.;.,....__,'.:
.'
-'
':",'_.-""''.,:'0"-"::",~.'-:"-"1\<"'""-,,~:',-;,
,
.'
-';,.,'-,,' _,..- '-_a,.,'-,.',, -,~,._,•" '~ ~',__., ..;. -_,,,-..-....
-,,-h,~.'".",','-
.,-
..
-m..-'-..-'-~"''-.',-,'"'-~,'-.,
, ..•.•" .>. ":):"n•"•",.'" ,
.
.'.,
'A--"".-",-.
'·...k.~,".-'',,,.~-...'..."..-~'..';;.•.,;."y.~-,_.;.-'.;.--.-,-_--.',..,''...-,',, -_...'
-,',
•. ._,.~.
34
COIIPLEX ANALYSIS
The real and imaginary parts of a Cauchy sequence are again Cauchy aequence., and if they converge, SO does the original aequence. .. For this
reason we need to prove the sufficiency only for rel!l sequences. We use the opportunity to recall the notions of limes IlUperior and lime8 inferior.
Given a real sequence \a.lr we shall set a. = ma.x \a., ... , ...1, that
is, a. is the greatest of the numbers a" .... ,.... The sequence la.lr is + nondecn-smng; hence it has a limit A, which is finite or equal to 00.
The nnmber A. is known as the leasl upper bouM or IlUpre",um (l.u.b. or IlUp) of the numbers ...; indeed, it is the least Dumber which is ~ all a:". Construct in the same way the least upper bound A. of the sequence \... ,: obtained hom the original sequenee by deleting a" • • • , .....
It is clear that \A.I is a nonino easing sequence, and we denote its limit
+ by A. It may be finite, GO, Or - GO. In any case we Mite
...... A=limsup ....
It is easy to characterUe the limes superior by its properties. If A is
+ ., finite and I > 0 there exists an n. such that A.. < A
and it follows
+ that "'.:;; A... < A & for .. $; .... In the opposite direction, if
"'. :;; A - • for n ~ "0, then A.. :;; A - ., which is impossible. In other words, there are a.rbitrarily large n for which > CI_ A -.. If
+ A = CD there are a.rbitrarily large ..., and A = - CD if and only if ".
tends to - CD. In all cases there caonot be more than one number A with
these properties.
The liMes inferi()r cao be defined in the Mme ma·nner with inequalities
reversed. It is quite clear that the limes inferior and limes superior will
be equal if and only if the sequenceconverges to a finite limit or diverges
+ to GO or to - GO. The notations are frequently simplified to lita and
The reader should prove the following relati9ns:
+ + + li'"m- a. lim fl. ;li lim (... (1.) :> lim ... lim (1. + + + lim a. lim fl. ;li ffiii (... fl.) ~ lim ... lim fl•.
Now we return to the sufficiency of Cauchy's condition. From
+ • I... - a..1 < • we obtain 1...1< 1....1 for n $; n., and it follows that
A = lim ". and a = lim ... are both finite. If a ;of A choose
(A .;.. a)
0=
3
and deu,m.ine a corresponmng no. By definition of a and A there exists
+. an a. < a
and an a. > A - 0 with m,n i1:; no. It follows that
A - a - (A - ....) + (.... - a.) + (a. - < a) 3.,eontrarytothech.oi.~
of.. Henee a - A, and the sequence converges.
.
." .-
....
", '"
-'...
".,
CO.PLEX FUNCTIONS
35
2.2. Series. A very simple appli~tion of Cauchy's condition permits
\18 to deduce the convergence of one sequence from that of another. If it
is true that lb. - b.1 :; 1"- - ""I for all pairs of subscripts, the sequence
Ib.1 may be termed a c<mtroction of tbe sequence 1",,1 (this is not S standard term). Under this condition, if {a.l is a Cauchy sequence, 80 is {b.l. Hence convergence of {a.l implies convergence of {b.}.
An infinite series is a formal infinite sum
(15)
1I1+a.+··· +11.+ .. ...
Associated with this series is the sequence of its partial sums
B. - a, + a. + . . . + a...
The series is aajd to converge if and only if the cotlesponding sequence is
conveJ&ent, and if this is the case the limit of the sequence is the sum of
the series.
Applied to a aeries Cauchy's convergence test yields the following
condition: The aeries (15) converges if and only if to every , > 0 there
exists an n. such that la.. + II" 11 + . . . + a..+,1 < • for all n ~ n. and
p ~ O. For p = 0 we lind in particular that 111.1 < I. Hence the gen-
eral term of a convergent series tends to aero. This condition is
,
but of course not 1I11fficient.
If a finite number of the terms of the aeries (15) are omitted, the new
series converges or diverges together with (15). In the C9lJe of conver-
+ gence, let R. he the Slim of the series which begillB with the term a"."
Then the SUm of the whole series is S = a.. R•.
The series (15) can be compared with the series
(16)
11111 + la.1 + . . . + 111.1 + . . .
fomltlli by the absolute values of the terms. The sequence of partial
sums of (15) is a contraction of the sequence con esponding to (16), for
III. + a..+1 + . . . + a..+o\ :; 111.\ + 1a..+,1 + . . . + 11.1" ,,1· There-
fore, convergence of (16) implies that the original series (15) is convergent.
A series with the property that the series forllled by the abeolute values
of the terms converges is said to be oollOlut611l Clll'Wtrgent.
",eJIee. 1.J. Uniform eo....
Consider a sequence of functions f.(",),
aDiii!fined on the _e set E. If the sequenile of values If.("'» con-
~ for eveJj z that belcmgs to E, then the limit f(z) is egeiu a function
,>
'
dn·B';';" By'ileliDition,
if
c>
0
and
'"
. .
'toE· ·exiots aDllti sucb
that 1/.(z) -/(z)1 < • for fa ~ 1It, but llti is allowed to depend on z.
,. . .... ...... .. ... .., ,
":
.
,.)....
" ",
. ',.'.,."..,...........,.,.',; .' '
' ' ,,.. .'..". '
:'
,
COMPLEX ANALYSIS
For instance, it is true that
.. lim 1+-1 x=x •
+ for all x, but in order to have 1(1 l/n)x - xl = lxl/n < s for n l?; n.
it is necessary that n. > lxi/eo Such an no exists for every fixed x, but
the requirement cannot be met s; multaneously for all x. .
We Bay in this situation that the sequence converglls pointwise, but
not uniformly. In Positive fonnulation: TIuJ sequence (f.(x) I convergea
.....iJolfnly to J(x) on IIuJ aet E iJ to every , > 0 ~re exiaIB an n. BUCk thai.
If.(x) - f(x)1 < • for aU n l?; no and aU x in E.
The most important consequence of uniform convergllnce is the
following:
TIuJ limit Iunaion f1j a uniformly c_rgenl sequence f1j conIinuOUI
funclion& ia itself conIinuoua.
.
Suppose that the functions f.(x) are continuous and tend uniformly
to f(x) on the set E. For any • > 0 we are able to find an n such that If.(x) - l(x}1 < e/3 for all zin E. Letxo be a point in E. Becausef.(x} is continuous at Xo we can find Ii > 0 such that II.(x) - f.(x.) I < _/3 for all
x in E with Ix - x.1 < a. Under the same condition on x it follows that
+ IJ(x) - l(xo)1 ~ I/(x) - [.(:c)1 I/.(x) -J.(x.)1 + If.(x.) - l(zo)1 < "
and we have proved that f(z) is continuous at :c••
In the theory of analytic functions we shall find uniform convergence
much more important than pointwise converg1lnce. However, in most
eases it will be found that the convergence is uniform only on a part of
•
the set on which the functions are originally defined.
"
Cauchy's necessary and sufficient condition has a counterpart for
uniform convelgllnce. We assert:
The sequence (f.(x) I c07lllergea unilorllllyon E if and only if to every a > 0 ~e exial8 an no BUCk thall/.(z) - f.(x}1 < afar all m,n l?; n. and aU
zinE.
The neceMity is again trivial. For the sufficiency we remark that
the limit function fez) exists by the ordinary form of Cauchy's test. In
the ineqnality If.(z) - I.(x) I < & we can keep n fixed and let m tend to
GO. It follows that 1/(:e) - f.(x) I ~ e for n l?; n. and -all z in E. Hence
the convergence is uniform.
For practical use the following test is the most applicable: If a
sequence of functions 11.(:&) \ is a contraction of a convergllnt sequencc of
constants (0,,\, then the sequencc (f.(:e)1 iSllniformly convergent. The
hypothesis means that 1/..(x) - I.(x)\' ~ law. - 0,,1 on E, and the, con-
.
.
.'
-. . . d '
COMPLEX FUNCTIONS
n
clusion follows immediately by Cauchy's condition.
In the case of series this criterion, in a somewhat weaker fonn, becomes
particularly simple. We say that a series with variable terms
j,(1') +ft(1') + . . . +1.(1') + . . .
has the series with positive terms
a,+a.+··· +a.+'"
for a majortmt if it is true that 1/.(x)1 ~ M a. for some eonstant M and
for all sufficiently large ..; converaely, the first series is a minoran! of the
second. In these circumstances we have
I/.(x) +1.+,(1') + ... +1.+,(x)1 ~ M(a. + a.+l + ... + "-+.).
Therefore, if the majorant
the minorant oonverges uniformly.
This condition is frequently refened to as the WeierstI'dB8 M lest. It has
the slight weakness that it applies only to series which are also absolutely
convergent. The general principle of contraction is more eomplicated,
but has a wider range of applicability.
EXERCISES
I. Prove that a convergent sequence is bounded.
...... 2. If lim z. ~ A, prove that .l.i.m......!. (z, + z. + . . . + z.) = A·
3, Show that the sum of an absolutely convergent seriee does not
change if the terms are rearranged.
4. Disc"'" completely the convergence and uniform convergence of
the sequence Inz-Ir.
50 Discuss the uniform convergence of the series
for real values of x.
c. If U = v, + u. +
...
,
V
=
v, + v. + ...
are
convergent
oeries, prove that UV = u.v, + (u,v, + UoV.) + (v,v, + u,v, + v,v.) + ...
provided that at least One of the oeries is abeolutely convergent.. (It is
easy if both mes are abeolutely convergent. Try to arrange the proof so
economically that the absolute convergence of the second series is not
needed.)
.-.. . -,
- ";'
.-"
'.
"'
:'::';":,:-
-;'-.'-
;- ....~.-..
'
. ,-,'.,:';,,_.;.:-.:,"-;.:~,:,"'.::,-'""'.;:,:':--',. :':':-:..!,~;..:,;'.,.::,-~,..j,~:':;:-"."":"""",-"~,.~,-',">","""."-''':''''.~~\-'"+-'''- '''~_''''''''':~'''-'''''-:'';'''.'-"-'
COMPLEX ANALYSIS
:4.4. P01fIer Series. A power seriu is of the fonll
(17)
a.+a.z+a,z' + ... + a.z" + ...
where the coefficients a- alld the variable z are complex. A little more
generally we may consider series
z., which are power series with respect to the center but the difference is so
slight that we need 1I0t do so in a formal manner.
As an almost trivial example we consider the geomelric IlerU8
1+"+.'+,,, +1"+ • • •
whose partial sums can be written in the form
1 + z + . . . + z"-' = 1l--1z".
Since I" -> 0 for \z\ < 1 and \1"\ ~ 1 for \z\ ~ 1 we conclude that the
geometric series converges to 1/(1 - z) for Izi < I, diverges for Izl ;;,. 1.
It turns out that the behavior of the geometric series is typical.
Indeed, we shall find that every power series converges inside a circle and
diverges outside the same circle, except that it may happen that tltEl
aeries converge. only for z = 0, Or that it converges for all values of z. More precisely, we shall prove the following theorem due to _.A. bel:
-
Theorem 2. For ellery power aeria (17) tAere eziMa a number R, 0 ~
R ;l! 00, called 1M rodiua oj
, IlIith 1M JollOVJing properlie8:
(i) .TM 8eMa ronv...gu absolutely Jor ellery Z fDith \z\ < R. IJ 0 ;:;;
p < R 1M convergen<:e is uniJ•• HI Jor \z\ ;:;; p.
(ii) IJ Izl > R 1M leall8 oj 1M senu are unbounded, and 1M IleN8 i&
conaequently divtrgtllt.
,
(iii) In 1.1 < R 1M aum of 1M aeria is an analylic fufldioo. TM
derivative can be obtained by lermwiBe differtlltiation, and 1M derived aeria
haa 1M same radius oj CUTIIIergenee. " , The circle ItI .. R is called the circle oj _ ...ge'/lC6; nothing is claimed
about the convergence on the circle. We shall show that the assertionein
the theorem are true if R is chosen according to the formula
(18)
-- l/R = lim sup v\a-I.
-
.-•
--C_."'.
--.-..,
COMPLEX FUNCTIONS
31
This is known as Hadamard'sfof'WIlJla for the radius of convergence.
If 1.1 < R we can find p 80 that 1.1 < " < R. Then IIp> l/R, and by the definition of limes superior there exists an 110 such that 1a.1 1/" < 1/p,
1a.1 < IIp"forn ~ n•. Itfollowsthat!a"z"1 < (1'lIp)"forlargen,sotbat
the power series (17) bas a convergent geometric series as a majorant,
and is consequently convergent. To prove the uniform convergence for
Izi ;:;; p < R we cboose a 1" with p < l < R and find 10,,"1 ~ (P/l)" for
n.. n S:
Since the majorant is convergent and has coutant term~ we
conclude by Weierstrass's M test that the power series is uniformly
convergent.
If I_I > R we choose I' 80 that R < " < 1.1. Since III' < llR there are arbitrarily large n such that 1a.11I· > 1/p, 1a.1 > l/pa• Thus la"zal > (1.1/1')' for infinitely many n, and the terms are unbounded.
r •
The derived series 110,,,---1 has the same radius of convergence,
1
+ because Vn --+ 1. Proof: Set Vn = I 3,. Then ba > 0, and by use + + of the binomial theorem n = (1 6.)" > 1 ! n(n - 1)8~. This gives
i! < 2/n, and hence i, --+ O.
For I_I < R we shall wtite
•
r + •
/(.) = aoZ- = B.(.) R.(.)
•
where
.-. () + + . 8•• =a. al'
..
+ a._•I·1ZR,.(') =
•
4 ~
aoZ••
and also
r •
!t(z) = na"zo-I = lim B~(Z),
1
......
We have to show that 1'(.) = ft(-). Cl'nsider the identity
(19)
» B.(.) - Sa("'~ _ 8~('.) + (.~(".) - ft(••
Z - 20
+ RaC.) - R.(••) ,
.2 -.I.
where we a&IIUIlle thah ". " and 1.1. 1••1 < I' < R. The last u,lin eat> be
.. . ... . ..
•
'o}t;-.
Il10(.......'
+
.......'...
+
+ et-'), . '.
,. .,
.
•
:;""'": :.,', ';',";", >.c, .,, .. :.'-' "
:,
" ,:",'
---~"::'-;"~:'
,~:."~" l'i.l;;'i~ ',
,', ','"
','. ,'" -, ';;,.','":-':
,,:::,..~ '.':
, ';, _
-:...x~.:'
"."
"
,
.
:
•
.
. .-,-,..' •.
'.i:.;:.
<:~-',';.F.....:";."~;.'.;-'.,.
,).~:•..
:',.::"';.~-;
,
,
'
'
, ,-'-,'.-:.:-
l(!:-);.....).
",,:,:;, :. ... ,,\-,.,.•.'!.vl"('
~,."-,':r~:'':,,'".I"~""""1..~j:
..
CO ...LEll ANALYSIS
and we conclude that
The expression On the right is the remainder term in a convergent series. Hen"" we can find no such that
~•.(8) - R.(z.) <!
z - z.
3
for 11 ~ 11..
There is also an 11, such that Is~(z.) - 1.(z.)1 < ./3 for 11 ~ 11,.
Choose a fixed 11 ;0:; 110, 11.. By the definition of derivative we can lind
o> 0 such that 0 < II - %.1 < 0 implies
I 8.(Z) Z -
8.(Z.)
Zo
-
&•' (z".j
< •
3- a
.
When all these inequalities are combined it follows by (19) that
I(z) ..,. I(zo) _ /t(Zo) <.
z - %0
when 0 < Iz - z.1 < o. We have proved that {(zo) exists and equals
it~OJ.
Since the reaeoning can be repeated we have in reality proved much
more: A power series with positive radius of convergence has derivatives
of all orders, and they are given explicitly by
I(z) = ao + a,z + a,z' + . ; .
f(z) = a, + 2a.z + 3a,z' + ...
. . . . + + + ... f"(%) = 2,.. 6a,z 12aot'
. . . .. . .. . . . .. . .. . . . . .. . . .. .. . . . . . .
.
+ + + . . . f Ci) (z) = k '.a.
(k+1)1 I!
a,+12
(k+2)! 21
a~tZ' -T
In particular, if we look at the last line we that a. = jC"(O)/Ic!, and the power series becomes
J(z) = J(O) + + {(O)z {,(OJ Zl + ... + J(O) (0) ZO + ...
2 !
11!
"
Thie is the familiar Taylor-Maclaurin development, but we have proved it
only under the assumption that/(z) has a power
development. We
do know that the
is uniquely determined, if it exists, but
the main part is still
namely that every analytic function has a
Taylor development.
•
COMPLEX FUNCTIONIJ
41
EXEReI SES
L Expand (1 - z)-, m a positive integer, in powel'8 of z.
2. Expand ~::13 in powel'8 of • - 1. What is the radius of
convergence?
.. Find the radius of convergence of the following power series:
L L:r, L L L "Pz',
n!zo, q"%"(lql < 1), %"'
.. If %a.z" has radius of convergence R, what is the radius of con-
vergence of %/J,.!t.? of ~~Z"?
S. If f(z) = %a.%", what is %n'II,,"'?
.. If %110%" and %b..' have radii of convergence R 1 and R., show that
the radius of convergence of %a.bo%" is at least R.R,.
7. If Iim.__ 111.1/111.+11 = R, prove that %a.z" has radius of COn-
vergence R.
- .. For what values of • is
•
convergent?
•
t. Same question for
J.5. Abel'. Limit Theorem. There is a secoild theorem of Abel's which refere to the case where a power series conwrges at a point of the circle of convergence. We lose no generality by 888Uming that R = 1 and tbat the convergence takes place at z = 1.
- !- Theorem 3. If}; a. convergea, IAen f(ts) = 0.J" tenda tD J(1) aa z
o
0
IIppr0acAe8 1 in """" II UNJfI tMt 11 - zl/(1 - 1_1> remllim bounded.
HemIIrk. Geometrically, the condition means that" stays in an angle
< lSO° with vertex 1, symmetrically to the part (- 00 ,1) of the real axis.
It is euatomary to sa,y that the approach
place in a SUlk IJfI9k.
- Proof. We may assume,~t.I a. ":" 0, for thisean bea.ttNned by adding
"""',.
-
-. -
.
• •
.. . ...... .. '" .. .. ".. '-'" , .. -
.
. ,
.'.
"':,' . .. , ,_.' ... ,:,' ,
'...-.....,.", ;..,--~',.-,''. ._.:_,•
•'.,.>. ,_-"-
"-,'--,:'.
,:,"-_';".
,_-
.'
-'
.''. " "
...'.:..-,".
' ,
,
'-".,',.,"_~-,,
.,,'.',~",' "
:.,..~,.,
.
,
_"~··__·,:."-"'·A':':.""_"'_;
___
' ~-_"'_~"··~, '"">·'-_ ',.:.-_·,"~.,"-:.,·.'"•""~.~,,",'."
',
'
,''_"l- Q~","...',,''..:.~".,".c.".:".',.,·.
.....-
,
,-'
.._.t-..~~
42
COIIPLEX ANALYSIS
+ + ... + a constant to a.. We write Sa = a. a,
tIa and make use of
the identity (summation by parts)
8a
=
(Z) =
8.(1
4.
-
+ aiZ + + z) 8,(,
... + a.r = - Zl) + ...
+ 8. (81 - 80)' + 8._,(,-' -
+
r)
... + + s.r
(8.
-
s__ . ) z ·
+ + ... + + = (1 - ,) (80 8iZ
8._1"-') S,$'•.
But 8.," -> 0, so we obtain the representation .
L •
I(z) = (1 - z) Sal'. o .
We are assuming that \1 - z\ ;:;; K(l - It\), say, and that 8. -> O.
Choose m so large that \s,l < 8 for 11 !5;. m. The remainder of the
series 2:8,$', from 11 = m on, is then dominated by the geometric series
•L..• \z\" = _\z\"/(l - \zl) < _/(1 - \z\}. It follows that
\/(z) \ ;:;; \1 - z\1 ..L-1 8~i I+ KE. o
The first tenn on the right cail be made arbitrarily small by choosing z sufficiently close to 1, and we conclude thatf(z) -> 0 when z ..... 1 subject to the stated restriction.
3. THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
The person who approaches calculus exclusively from the point of view of
real numbers will not expect any relationship between the exponential •
function e" and the trigonometric functions cos z and sin z. Indeed, these
functions seem to be derived from completely different sources and with
djil'erent purposes in mind. He will notice, no doubt, a similarity between
the Taylor developments of these functions, and if willing to use imaginary
+ arguments he will be able to derive Euler's lonnula e;Z = cos Z i sin z
as a formal identity. But it took the genius of a Gauss to analyze its full
depth.
'
With the preparation given in the preceding section it will be easy to
Jefine e', cos z nnd sin z for complex z, and to derive the relations between
these functions. At the same time we can define the' logarithm as the
inverse function of the eXpOnential, a.nd .the logarithm leads in tum to the
correct definition of the argument of a complex number, and hence to the
nongeometrie definition of angle.
S.l. TIuJ Exponential_ We may begin by defining the ~ial
function as the solution of the differential equation
COIIPLEX FUNCTIONS
(20)
1'(&) = 1(,,)
with the initial value 1(0) = 1. We solve it by setting
I(z) = a. + a,z + . . . + a"Z" + . . . I'(z) = a, + 2asz + ... + + 1Ia"z...., ...
If (20) is to be satisfied, we must have a.-I = 1ICI., and the initial condition gives IJo = 1. It foTIows by induction that a. = 1/1I!.
The solution is denoted by e' or exp II, depending on purely typo-
graphical cODsiderations. We must show of CoUl"l!e that the series
(21)
e'
=
1
+
IZi
+
Zl
2!
+
...
+
srt
ni
+
...
converges. It does 80 in the whole plane, for Vni..... 00 (proof by the
reader).
It is a consequence of the differential equation that e' satisfies the
addition tIoearem
(22)
ee I i = e- . ~.
+ Indeed, we find that D(e" e-') = e' . e-' e' • (-e-') - O. Hence
eo . e' 'is eo constant. The value of the constant iafound by setting .. = o.
+ We conclude that e" e-' = e<, and (22) follows for Z = fI, C = a b.
Remark. We have used the fact thatj(z) is constant ilf'(2) is identically
zero. This is eertainly 80 il!is defined in the whole plane. Foril! = u+ W
we obtam. -aa"u, =
-clu =
iJy
-iIIJ =
iJz
i-iIJIJy -
0,
and the real version of the theorem
shows that! is constant on every horiJJOntal and every vertical line.
As a particul&r case of the addition theorem eo· r ' = 1. This showa
that e' ill r&eVeJ zero. For real :z: the aeries development (21) ahows that
e" > 1 for:z: > 0, and since eo and .,. are reciprocals, 0 < e" < 1 for:z: < O.
The fact that the aeries bas real coefficients shows that exp il is the complex
conjugate of eJql z. Hence 1_"1" = e'" r" - 1, and le<i"1 = e'.
1.2. The Tri601Wmemc Funcrio7U. The trigonometric functions are
defined by
+ of< r i , .
IV - .,..
(23)
COs .. =
2 ,SID2= 2': .
Substitution in (21) showB that they have the aeries developments
,
',," .. ", " _~ ~ •• p". "',~ ·,w:f.I.,."W ...'.""""~ -.-".".-.~ _"'.
. . " .,. .1 . '.:,"
... . .'.. ..
'~,
.
.• '. ,' , '.'.,--'-'.. , ••,r
COMPLEX ANALTII18
.
,I Zi
81DZ=Z-3j+51- .••
For real z they reduce to the familiar Taylor developments of cos x and sin x, with the significant diJference that we have now redefined these
functions without use of geometry. From (23) we obtain furthe.r Euler's formula
e"=cosz+isinz
•
as well as the identity
+ 80S' z sin' Z = 1.
It follows likewise that
DC08.= -sinz, Dsinz=cosz.
The addition formulas
cos (a + b) = cos a cos b - sin a sin b
+ + sin (a b) = cos a sin b sin a Co! b
are direct consequences of (23) and the addition theorem for the exponential function.
The other trigonometric functions tan z, cot %, sec %, cosec z are of
secondary importance. They are defined in temlll of cos z snd sin • in the
customary manner. We find for inetance
.~H _ c ..
+ tan z = -. '. eU
e-<-.
Observe that sJl the trigonometric functions are rational functions of ....
•
EXERCISES
+ 1. Find the values of sin i, cOS i, tan (1 J').
+ 2. The hyperbolic cosine and sine are defined by cosh % = (e' e-')/2,
sinh % = (e' - ,,')/2. Express them through cos it, sin iz. Derive the
addition formulas, and formulas for cosh 20, sinh 2z.
+ + S. Use the addition formulas to separat~ cOS (x ill), sin (x ill) in
real and imaginary parts.
... Show that
,
+ Icos -I' = sinh'lI .+ cos' x = cosh'lI - sin' '" = ~ (cosh 21/ cos 2;1;)
and
lsin _" = sinh'1I + sin' x = cosh'lI - COS' x ~ ~ (cosh 211 - C08 2;1;).
8.8. TMP.,.iodidcy. Weeaythat/(z)bae.theJl'llMJdcif/(8+'O:) -/~)
COMPLEX FUNCTIONS
for all z. Thus a period of e' satisfies eO>' = 6', or e' = 1. It follows that
c = i .. with real ..; we prefer to say that .. is a period of e'-. We ohaIi
show that there are periods, and that they are all integral multiples of a
positive period "'"
Of the many ways to prove the existence of a period we choose the
following: From D sin II = cos II ~ 1 and sin 0 = 0 we obtain sin II < 1/
for y > 0, either by integration or by use of the mean-value theorem. In
the same way D cos 1/ = - sin II > -II and cos 0 = 1 gives cos 1/ > 1 11'/2, which in tum leads to sin 1/ > y - 1/'/6 and fiDally to cos 1/ < 1 -
va + y'/2 1/'/24. This inequality show. that cos < 0, and therefore va there is a II. between 0 and with cos II. = O. Because
+ COS'1/. sin'lI. = 1
we have sin II. = ±1, that is, ..... = ± i, and hence ."" - 1. We have
sho wn that 411. is a period.
Actually, it is the smallest positive period. To see this, take 0 < II < II.. Then sin II > y(1 - y'/6) > 11/2 > 0, which shows that cos 1/ is
+ strictly decreasing. BecaUse sin 11 is positive and COS'1/ sin'1/ ~ 1 it
follows that sin II is strictly increasing, and hence sin II < sin 1/. = 1.
The double inequality 0 < sin II < 1 guarantees that ei • is neither ± 1 nor
±i. Therefore e"· .., 1, and 4y. is indeed the smallest positive period.
We denote it by ....
Consider now an arbitrary period "'. There exists an integer n such
+ that n",. ~ '" < (n 1)"". If .. were not equal to ""'0, then '" - n",.
would be a positive period < .... Since this is not possible, every period
must be an integral multiple of "'"
The 81IUIl/es/ plMitive period of fi' is denoted bll 2..-. In the course of the proof we have shown that
en/S = i,
These equations demonstrate the intimate relationship between the nUrn-
bers e and 11'.
When 1/increases from 0 to 2..-, the point 111 = e'· describee the unit circle
Iwl = 1 in the positive sellll6, namely from 1 over i to -1 and back over -i to 1. For every wwith Iwl = 1 there is one and only one 1/ from the
half-open interval 0 ~ II < 2..- such that w = e'v. All this follows readily
from the elltablisbed fact that cos II is strictly decreasing in the "first
.i. quadrant," that is, between 0 and 11'/2. Froman algebraic point of view the mapping w= establishes a
M'IOIIIOI'JIM8"~ .between the additiw group of real numbers and the
muitipUClMive"aroupof,«lOinp!eJ; Rumbas with ab80lutevalue 1. The
~ IIf·tM;·
. ..~~. fo"i\V~ by all integral
mul~ 211&.
,', ,. ,-,-
.... . . ", ..: '... ~..
.
'
"
.
:
:
"
:..:
'
.
,' ..);~~. "
~,:'
,"~"':~I.'..,)..:.:~.~::.\,'..
'. i .. . ': "":
" '.....' , '
.' . .. ,
. ',.,,:.. , '." , :, ,'._
...
COIIPI,.I!X ANALYSIS
3.4. The Logarithm. Together with the exponential function we must slso study its inverse function, the logarithm. By definition, z - log w is a root of the equation 6' = w. First of all, since 8' is always ,-!O, the number 0 MIl no logarithm. For w ... 0 the equation.- - w is equivalent to
(24)
.,. = Iwl, .
.;0 = 1It/lwl.
The first equation has a unique solution % = log Iwl, the r.allogarithm of the positive number liDl. The right-barul member of the second equation
(24) is a complex number of absolute value 1. Therefore, &8 we have just
seen, it has one and only one solution in the interval 0 ;;; 'II < 2-r. Inaddi-
tion, it is satisfied by all 'II that differ from this solution by an integral
multiple of 2-r. We see that every comple:e number other than 0 MIl i.nJiniW.'II
man'll
whiM di.ffer from each other btl multip!e8 of 2-ri.
The imasjnary part of log w is also cal.led the argument of w, arg w, and
it is interpreted geometrically 88 tbe angl6, measured in rsdiallB, between
the positive real axis and the balf line from 0 through the point ID. Accord-
ing to this definition the argument has infinitely many values whicb differ
by multiples of 2-r, arul
log ID = log Iwl + i arg lit.
With a change of notation, if Izl = rand arg z = 8, then II - re". This
notation is 80 convenient that it is used constantly, even wben the expo-
nential function is not otherwise involved.
By convention the logarithm of a positive number shall always mean
the real logarithm, unIese the contrary is stated. The symbol a', where
a and b are arbitrary complex numbers except for the condition a ... 0, is
always interpreted as an equivalent of exp {b log a). If a is restricted to
positive numbers, log a shall be real, and a> has a single value. Otherwi..,
e·..... log a is the complex logaritbm, and a> has in general infinitely many values
which differ by factors
There will be a single value if and only if b
is an integern, and then a" can be interpreted as a power of aora-'. Ifbis
a rational number with the redueed form pig, then a> has exactly g values
and can be represented as W.
The addition theorem of the exponential function ,clear.ly implies
•
+ log (%,%.) = log B, log z, + arg (z,z.) = arg %, arg z"
,
but only in the sense that both sides represent the same infinite !let of
complex numbers. If we want to compa.te a value on the left with a value
on the right, then we can merely assert that they differ by a multiple of
2-ri (or 2-r). (Compare with the remarks in Chap. I, Sec. 2.1.) ", .,\
",:, -'.'," '"." '-'~''''-':-. ,
COMPLEX FUNCTIONS
.7
Finally we discuss the
the equation
cosine which is obtained by solving
+ COB ~ .. ~ (e" rI') = w.
6" This is a quadratic equation in with the roots
~.. =- to ± VWt - 1,
and
:e =arc cos w - -ilog (w ± ";w' - 1).
We can also write these values in the form
+ arc cos w = ± i log (w v'w' - 1),
+ for w v'w' - 1 and w - Vw' - 1 are reciprocal nnmbers. The
infinitely many values of arc cos w retIed the evenneBII and periodicity of
eos :e. The
sine is most eflSi1y defined by
arc
•
SID
UI
=
f2r "
-
arc cos w.
It is worth emphasizing that in the theory of complex analytic func-
tions all elementary transeendental functions can thus be
through t;* and its
log z. In other words. there is
one elementary transcendental function.
..XIRCIIII
L For real 1/, show that every remainder in the series for COS 1/ and
sin 1/ has the 88me sign as the leading term (this generalizes the inequali-
va. ties used in the periodicity proof, Sec. 3.3). 2. Prove, for instance, that 3 < fr < 2
i, : i IL Find the value of e' for z = -
ri, rio
.. For what values oiz is t;* equal to 2, -1, i, - i/2, -1 - i, 1 + 2i?
.. Find the real and imaginary parts of exp (e')•
.. Det6t1line all values of 2', i', (-1)".
"I. Determine the real and
parts of %".
L ExpreBII arc tan w in terms of the logarithm.
,to Show bow to define tbe "angles,f in a triangle, bearing in mind that
they $hould lie between 0 and fr. With thiB definition, prove that the 811m
of the angles i, fr.
Jt. Show tba\ ,he lOOts of the binomial
.. - aarethever-
recuJar tiUI of a
"'< . ,"
po!;y&Oll (equal eidee and
",. "
,,',"."...-.,
" . ""'",:,":..,~"'".,'' .
',':.
. . '.'
"
",'"-h",;';,~,;'':,'"::;~'-i'-"'~";:.:'
,i:,'''.",.,',>'-',:·.,.
.
.l'.:.~.'Y.;.~:".,
" .::':.,;.'. "..:.'" :~..
'. -
,-
.',i-·:".-.:-·,":·./,~i,:·,,-.:.:-:~':-, '~,,,;":,.'
3
ALYTIC FUNCTIONS
AS APPINGS
A function tD - fez) may be viewed 88 a mapping which repre-
sents a point z by its image tD. The purpot!e of this chapter is to
study, in a preliminary way, the special properties of
defined by analytic functions.
In order to carry out this program it is desirable to dewlop
the underlying concepts with sufficient generality, for otherwise
we would soon be forced to introduce a great number of ad hoe
definitions whose mutual relationship would be far from clear.
Since present-day students are exposed to abstraction and gen-
erality at quite an early stage, no apologies are needed. It is
perhaps more appropriate to sound a warning that gJeatest possi-
ble generality should not become a purpose.
In the firat section we develop the fundamentals of point set
topology and metric spaces. There is no need to go very far, for
our main concern is with the properties that are essential for the
atudyof analytic functions. If the student feela that he is already
thoroughly familiar with this material, he ahould read it only for
terminology.
The author believes that proficiency in the study of analytic
functions requires a mixture of geometric feeling and computa-
tioDAlakill. The second and third aections, only loosely connected
with the firat, are expressly designed to develop geometric feeling
by way of detailed study of elementary mappings. At the 88me
time we try to stress ri&or in geometric thinking, to the point
where the geometric imap b. ecomes th.e suide but not the found....
tion of reaa9J'jDg~
•
.. - - "
'
.'---_",.' -
•..,. . . .
",'_ .-,-_••,•'•."••.,,'
. -•" . .
-.."", _,-_-,_"w,
--""
"'~_"
.-...... .. •-.. . . . :.,.-
' "
-,., .,'
.'
.
.
•
.
•
•
•
.
.
•
.
. _.;:._' .~_
•
•
•_• c_
.
.
•
•
_.·
_~
.......•.•.•.•.
~'.:,.'..-;,.~',
••
CO..PLEX ANALYSIS
:L ELEMENTARY POINT SET TOPOLOGY
The brancb of mathematics which goes under the name of t01W/ogy is con· cerned with all questions directly or indirectly related to continuity. The term is traditionally used in a very wide sense and without strict limits. Topological considerations are extremely important for the foundation of tbe study of analytic functions, and the first systematic study of topology
was motivated by this need.
The logical foundations of set theory belong to another discipline. Our approach will be quite naive, in keeping with the fact that all our applications will be to very familiar objects. In this limited framework no logical paradoxes can occur.
1.1. Sets ..nd Elements. In our langnage a lUll will be a collection of
identifiable objects, its ekments. The reader is familiar with the notation
'" e X which expresses that", is an element of X (as a rule we denote sets
by capital letters and elements by smallletteJ:s). Two sets are equal if
and only if they have the some elements. X is a subset of Y if every ele-
ment of X is also an element of Y, and this relationship is indicated by
X C Yor Y J X (we do not exclude the possibility that X = Y). The
empty set is denoted by !If.
A set can be referred to as a space, and an element as a poiR!.
Subsets of a given space are usually called point sets. This lends a
geometric lIavor to the language, but should not be taken too literally.
For instance, we shall have occasion t., consider spaces whose elements a....
functions; in that case a "point" is a function.
The imerlNJCtioo of two sets X and Y, denoted by X () Y, is formed by
all points which are elements of both X and Y. The unioo X U Y con·
sists of all point.s which are elements of either X or Y, including those which
are elements of both. One can of course form'
and union
of arbitrary collections of sets, whether finite or infinite in number.
The compl£menl of a set X consists of all points which are not in X;
it will be denoted by ~X. We note that,the complement depends on the
totality of points under consideration. For instance, a ""t of leal numbers
has one complement with respect to the real line and another with respect
to the complex plane. More generally, if X C Y we can consider the
relative complement Y ~ X which oon$ists of all points that are in Y but
not in X (we find it clearer to lise this notation only when X C Y)•
.
It is helpful to keep in mind the diatnDUtive lowl
XU (Y ("\ Z) = (X V Y) {"\ (X U Z) X () (Y U Z) = (X ("\ Y) U (X () Z) .
._...' -
ANALYTIC FUNCTIONt At MAPPING'
"
and the De Mtwgan laws
~(X V Y) = ~X fi-Y
-(X fi y) = -X V -Yo
These are purely logirAl identities, and they have obvious generalizations to arbitrary collections of sets.
1.:1. Me",", Spaces. For all considerations of limits and continuity it is essential to give a precise meaning to the terms "sufficiently near" and "arbitrarily near." In the spaces Rand C of real and complex numbers, respectively, such nearness can be expressed by a quantitative condition
Iz - yl <.. For iostance, to say that. a set X contains all :t 1lU1/lIMntiti near to y meaDS that there exists an $ > 0 suoh· that z E X whenever Iz :"'111 < II. Similarly, X c01llains poifItIJ a,bil,ariltlma, to tI if to every • > 0 there exists an z E X such that Iz - 111 < •.
What we need to describe neam""" in quantitative terms is obviously
a d~ d(z,y) between any two points. We say that a set S is a marie
3pQU if there is defined, for every pair Z E S, 11 E 8, II nC)Dnegative real num-
ber d{Z,II) in such a way that the followiog conditions are fulfilled: 1. d(z,y) = 0 if and only if x = y. 2. dCy,z) = d{z,y).
+ 3. d{z,.) ~ d(x,y) dCy,')'
The last condition is the triangle inequality.
For instance, R and C are metric spaces with d(x,y) = 13: - til.
The n-dimensional euclidean space Re is the set of real ...tuples
x = (x" • . . ,xe)
! •
with a distance defined by d(3:,y)' = (Xi - y.)'. We rcr&! tbat we
1
have defined a distanee in the extended complex plMe by
(8ee Chap. I, Sec. 2.~); Bioce this represents the euclidean dista.nee between
the stereoglapbic images on the Riemann sphere, the triangle inequ.ljty is
obviously f"lIi1Ied. An example of a function space is given by C[a,bl,
the set of all continuous functions defined on the ioterval a ;:ii z ~ h. It
beOOmes a metric space if we define dist··"ee by dC/,g) = max If(z) - g(x) I·
In '. of distance, we introduce the following termjnology: For
any. > 0 and~YII e8,.thB!I8~ B(V,4}ota,U x e~with II(z,l/) < 'is r&!ed
"',
. ," ., ,.J.
" , "~: __ ' : . • _. _",,",
.'
_..
, ", _.
52
CO"~LI!X ANALYSIS
the baa with center y and radius 8. It is also refeIled to as the oS-neighbor-
hood of y. The general definition of neighborhood is as follows:
A Bet N C 8 ia called II neighborhood of YES 'f it contain8
•
In other words, a neighborhood of y is a set which contains all points
sufficiently nAIIT to y. We use the notion of neighborhood to define
open, set:
Definition 2. A sel is open if it ia II neighborhood of each of iU element8.
The definition is interpreted to mean that the empty eet is open (the
condition is
heeanee the set has no elements).. The following is an
innnediate consequence of the triangle inequality:
Every baa ia an IJPell Bet.
Indeed, if Z E B(y,Ii), then 6' = a - d(y,z) > O. The triangle in-
equality shows that B(z,f,) C B(y,a), for d(x,z) < 8' gives d(x,y) < 8' +
d(y,z) '" o. Hence B(y,a) is a neighborhood of z, and since Z was any
point in B(y,f) we conclude that B(y,a) is an open set. For gJ eater em-
phasis a ball is sometimes referred to as ILIl open ball. to distinguish it
from the cklsed baa fonned by all XES with d(x,y) :i! 8.
In the complex plane B(z.,f) is an OpM diak with center z. and radius 8;
it consists of all complex numbers z which satisfy the strict inequality
Iz - z.1 < o. We have ju$ proved that it is an open set, and the reader
is urged to interpret the proof in geometric tefillS. The complement of an open set is said to be ckI81ld. In any metric
space the empty set and the whole space are at the same time open and closed, and there may be other sets with the same property.
The following properties of open and closed setH are fundamental:
The intersecliun of II finite number of open 8e18 i$ open.
The uniun of any collectiun of open seiIt i8 open. The union of II finite number of cl08ed sels is clolted. The inter8ectiun of any collection of closed 8818 is closed.
The proofe are so obvious that they can be left to the reader. It
should be noted that the last two statements follow from the first two by
use of the De Morgan laws.
There are many tenus in common usage which are directly related to the idea of open sets. A complete list would be more confusing than helpful, and we shall limit ourselves to the following: i!'lel'ior, cklBurll,
ANA~YTIC FUNCTIONS AS MAPPINGS
b&ufldary, ez!eriM.
(i) The interior of a set X is the largest open set contained in X. It
exists, for it may be characterized as the union of all open sets ex. It
can also be described as the set of all points of which X is a neighborhood.
We denote it by Int X.
(ii) The closure of X is the smallest closed set which contains X, or
the intersection of all closed sets "JX. A point belongs to the closure of
X if and only if all its neighborhoods intel/loot X. Tbe clOlllll'e is usually
denoted by X-, infrequently by Cl X.
(iii) The boundary of X is the closure minus the interior. A point
belongs to the boundary if and only if all its neighborhoods intersect both
X and ~X. Notation: Bd X or ax.
(iv) The exterior of X is the interior of ~X. It is also the oomph,. ment of ~e closure. As such it can be denoted by ~X-..
e e Observe that Int X X X- and that X is open if Int X = X, e e e closed if X- = X. Also, X YimplieslntX lnt Y,X- Y-. For
added convenience we shall alSl) introduce the notions of iaolated point
and auumulatioo poi"'. We say that :I: ~ X is an isolated point of X if :t
has a neighborhood whose intersection with X reduces to the point x. An accumulation point is a point of X- which is not an isolated point.
It is clear that z is an accumulation point of X if and ouly if every neighborhood of x contains infinitely many points from X.
EXERCISES
L If S is a metric space with distance function <1(:1:,1/), show that 8
+ with the distance function a(z,1/) = d(:I:,1/)/[1 d(x,1/)] is also a metric
space. The latter space is bounded in the sense that all distances lie
under a fixed bound.
2. Suppose that there are given two distance functioll8 d(z,1/) and
tlt(x,1/) on the same space S. They are said to be equivalent if they deter·
mine the same open sets: Show that d and il, are equivalent if to every
e > 0 there exists a & > 0 such that d(z,y) < & implies d,(z,1/) < " and
vice versa. Verify that trus condition is fulfilled in the preceding exercise.
.. Show by strict application of the definition that the closure of
Iz - z.\ < aislz - z~ ~ a.
.. If X is the set of complex nnmbers whose real and
parts
are rational, what is Int X, X.,., aX?
s. It is sometimes typographically simpler to write X' for ~X. With
this notation, how is X'-' related to X? Show that X-'-'-'-' = X-'-'.
.. A set is said to be discrete if all its points are isolated. Show that
a
""tin discrete. .
".,-
.-.
R
-.-
-,o'
r
C .
is
c'o-u" nta.ble.
7. Shol" that the. accmriulation points
of
. ,
any set
form
a
closed
set.
. .
<...;,.: ,'.-."--.'
,-" - -,",-"~"
•
. :'.'~
- ',-:-
COMPLEX ANALY81S
1.1. Conneeledneo. If E is any nonempty subset of a metric space S
;Fe may consider E as a metric space in its own right with the same dis-
tance function d(:Z:,lI) all on all of S. Neighborhoods and open sets on E
are defined as on any metric space, but an open seton E need not be open
when regarded as a subset of S. To avoid confusion neighborhoods and
open sets onE are often
to as relative neighborhoods and relatively
open sets. As an example, if we regard the closed interval 0 ;:it :z: ;$ 1 as
a subspace of R, then the semiclosed interval O·:::i! :z: < 1 is relatively open,
but not open in R. Henceforth, when we say that a subset E has some
specific topological property, we shall always mean that it has this prop-
erty as a 8Ilbspace, and its 8ilbspace topology is called the relative topology.
Intuitively speaking, a space is connected if it consists of a Bingle
piece. This is meaningness unless we define the statement in terms of
The easiest way is to give a negative characterisation: 8 i8 not
if tllMe uiat3 II pa,tition 8 = A V B into Opel~ aOOset3 A lind B.
It is understood that A and B are di8joint and nfnIemPIy. The connected-
neN of a space is often used in the following manner: Suppose that we are
able to construct two complementary open subsets A and B of 8; if S is
conn'Mlted, we may conclude that either A or B is empty.
A subset E C S is said to be connected if it is connected in the rela-
tive topology. At the risk of being pedantic we repeat:
Definition 3. A aOOHt of II 'IMtric. $p4C6 i8 conneded if it cannot be r6Fe-
86fIted (JI the union of tfDO disjoint ,ellltively open 8eta none of fDhillll. i8 6"'pl'/l.
If E is open,· a subset of E is relatively open if and only if it ie-open..
Similarly, if E is closed, relatively closed means the same as closed. We
can therefore state: An open ad i8 connected if it cannot be decomposed into
lIDO opm ad8, and II closed 8e1 i. connected if it C/lnnot be decompow ink) tlDO
c/ow Beta. Again, none of the sets is allowed to be empty.
Trivi&! examples of connected sets are the empty set and any set
that consists of a single point.
In the case of the real line it is possible to name all connected sets.·
The most important result is that the whole line is connected, lind this is
indeed one of the fundamental properties of the real-number system.
An i1lterval is defined by an ineqllality of one of the four types:
+ .. a < ~ < b, a;:it ~ < b, a < ~ ~ b, II. ~ :z: ;:it b. t For· a - - GO or
b -
this includes the semi-infinite interv&!s and the whole line.
,
t We denote open iRte.vals by (0,6) and cloBed intervals by [...61. Another common
practice i•.to denote open intervals by Ia,b[ and 8emic1.....l intervals by ]a,bl or 1..,6[,
It is alwa undentood that" < 6.
.
ANALYTIC FUNCTIONS AS MAPPINGS 55
Theorem 1. The nonemply connuted ~ oj the real line or, the intervaU.
We reproduce one of the classic'" proofs, based on the fact that any
monotone sequence has a finite or infinite limit.
Sup~ that the rett.lline R is represented 88 the union R ~ A V B
of two disjoint closed sets. I f neither is empty we rAD find a' E A and
b, E B; we may !l8SUme that a' < bt. We bisect the interval (o"b,) and
note that one of the two halves has its left end point in A and its right end
point in B. We denots this interv'" by (a2,b.) and continue the process
indefinitely. In this way we obtain a sequence of nested inte'."'s
(a.,b.) with a. E A, b. E B. The sequences (a.1 and (b.1 have a common
limit c. Since A and B are closed c would have to be a common point of
A and B. This contradiction shows that either A or B is empty, and
hence R'is connected.
With minor modifications the same proof applies to any interval.
Before proving the converse we make an important remark. Let E
be an arbitrary subset of R and call a a lmDer bound of E if co ~ It for all
:u E. Consider the set A of all lower bounds. It is evident that the
complement of A is open. As to A itself it is easily seen that A is open
whenever it does not contain any largest number. Because the line i.
connected, A· and its complement cannot both be open unless one of them
is empty. There are thus three possibilities: either A is empty, A con-
tains a largest number, or A is the whole line. The largest number a of
A, if it exists, is called the greatest IOt.Der bound of E; it is commonly
denoted as g.l.b. x or inf x for x E E. If A is empty, we ""glee to set
+ a ~ - 00, and if A is the whole line we set a ~ 00 • With this con-
vention every set of real numbers has a uniquely determined greatest lower bound; it is clear that a ~·+oo if and only if the set E is empty.
The 1to8! upper bound, denoted 118 tu.b. x or sup x for x E E, is defined in a
corresponding manner.t
Returning to the proof, we assume that E is a connected set with the
greatest lower bound a and the least upper bound b. All points of E lie
between II and II, limits included. Suppose that a point ~ from the open
interval (11,11) did not belong to E. Then the open sets defined by:t: < ~
and:t: > f cover E, and because E is connected, one of them must fail to
meet E. Suppose, for instance, that no point of E lies to the left of t·
Then ~ would he a lower bound, in contradiction with the fact that II is the
greatest lower bound. The opposite assumption would lead to a similar contndietiol), and we cooclude that ~ must belong to E. It follows that E
is an open, cJ.08ed, or ...
interval with. the end points II and b; the
are aDd fi . , . . . __ II .;,' ..;:~.,
~.
",.,'-
to be1nCluded. --,.-
II
COMPLEX ANALYSIS
In the course of the proof we have introduced the notions of gteatest
lower bound and least upper bound. If the set is closed and if the bounds
are finite, they must helong to the set, in which cere they are called the
minimum and the maximum. In order to he sure that the bounds are
finite we must know that the set is not empty and that there is some
finite lower bound and some finite upper bound. In other words, the eet must lie in a finite interval; such a set is said to be botmded. We have
proved:
Theorem 2. Any cloaed and bounded _ p t y 3d of real numbers htJg
a minimum and a mazimum.
The structure of connected sets in the plane is not nearly 80 simple as in the MOO of the line, but the foUowing charact.eJUation of open connected sete contains essentially all the information we shall need.
Theor..m 3. A - p l y open 36/ in tile plane ill conmcted if and only
if any tVlO of ita points can be joined by a polygon which lies in tile 8Ot.
The notion of a joining polygon is so simple that we need not give a
formal definition.
We prove first that the condition is neceBBarY. Let A he an open con-
nected oot, and choose a point a fA. We denote by A, the subset of A
whose pointe can be joined to a by polygons in A, and by A. the subset
whOile pointe <,annot he 80 joined. Let us prove that A, and A. are both
open. First, if a, fA, there e. jsis a neighborhood \z - a,\ < a contained
in A. AU points in this neighborhood can be joined to a, by a line seg-
ment, and from there to a by a polygon. Hence the whole neighborhood
is contained in A " and A, is open. Secondly, if a. fAt, let \z - a.[ < •
be a neighborhood contained in A. If a point in this neighborhood could
be joined to a by a polygon, then a. could be joined to this point by a line
segment, and from there to a. This is contrary to the definition of A.,
and we conclude that A. is open. Since A was connected either A, or
A. must be empty. But A, contains the point a; hence A. is empty, and
all pointe can be joined to a. Finally, any two points in A can he joined
by way of a, and we have proved that the condition is neceaeary.
For
we remark that it is even poaeible to join any t.to points
by a polygon whose aides are parallel to the coordinate axes. The proof
is the same. • .In order to prove the sufficiencY we MOUrne that A has a representa-
tion A = A. V Alas the union of two disjoint open sete. ChOOllea, fA"
as E A. and suppose that these pointe can be joined by a jIOlygon in A.
. , .' ,
ANALYTIC FUNCTIONS AS .APPINGS
II
One of the sides of the polygon must then join a point in A, to a point in
A" and for this re'son it is ,mffieient to consider the case where a, and a.
are joined by a line segment. This segment has a pa.rametric representa-
tion z = a, + I(a, - a,) where t nms through the interval 0 :;; t :;; 1.
The subsets of the interval 0 < t < 1 which conespond to points in At
and A., respectively, are evidently open, disjoint, and nonvoid. This
contradicts the conneetednel!8 of the interval, and we have proved that the condition of the theorem is sufficient.
The theorem generalizes easily to Ro and Co.
Definition 4. A fI01Iem¢lI connected open set i8 called a regiqn.
By Theorem 3 the whole plane, an open disk I.. - al < p, and a half
plane are regions. The same is true of any 6-neighborhood in R". A region is the-more dimensional analogue of an open interval. The closure of a region is called a cloaed region. It should be observed that di1ferent regions may have the same closure.
It happens frequently that we have to analyze the structure of sets which are defined very implicitly, for instance in the course of a proof. In such cases the first step is to decompoee the set into its maximal connected comptmenl3. As the name indicates, a component of a set is a connected subset which is not contained in any larger connected subset.
If E is the given set, consider a point 0 E E and let C(a) denote the
union of aU connected subsets of E that contain a. Then C(o) is sure to
contain a, for the set consisting of the single point a is connected. If we
.can show that C(a) is connected, then it is a maximal connected set, in
other words a component. It would follow, moreover, that any two
components are either disjoint or identical, which is preeisely what we
want to prove. Indeed, if e E C(o) (\ C(b), then C(o) C C(e) by the
definition of C(e) aod the conneotedness of C(o). Hence a" C(e) , and by
the same reasoning C(e) C C(a), so that in fact C(/I) = C(e). Similarly C(b) = C(e), and consequently C(a) = C(b). We call C(/I) the component of a.
Suppose that C(a) were not conneeted. Then we could find relatively
open sets A, B '" JJ Buch that C(a) - A V B, A (\ B = S. We may
8SS'llnp. that II" A while B contains a point b. Since b EC(/I) there is a
connected set E. C B wbinb contains II aod b. The representation
B, .. (E. (\ A) V (8. f"\ B) W(Mlid be a decomposition into relat.iveIy
open subsets, and
B.n A, beE.f"\ B Dei~
would be
emptY'. 'tbis is a
and we oonclude that
.... ,-- ,.
. .
:. ',. ""--'"
....
,.,"
'.,
. . . . . . . . .'...,... ,:, .
',,:'._., .."--' ",,","-,'- :,.,_.. '_'::.:'.'_'_"":"._. ___ ,._H , .. ..; .. ,.,\,._ .. ,,~, ...
C;O...LElI ANAL YS ..
Theorem 5. In R" the components of any opIln Mare opIln.
This is a CODBequence of the fact that the ~neighborhoods in R· are
connected. Consider a ! C(a) C E. If E is open it contains B(a,8)
and bec&l1!!R B(a,8) is connected B(a,&) C C(a). Hence C(a) is open. A
little more gene~aIly the
is true for any space S which is looally
connected. By this we mean that any neighborhood of a point a contains
a connected neighborhood of a. The proof is left to the reader.
In the case of R" we can conclude, furthermore, that the number of
components is countable. To see this we observe that every open set
must contain a point with rational coordinates. The set of points with
rational coordinates is countable, and may thWl be exprewted as a sequence
(p.l. For each component C(a), determine the smallest 1c such that
p. E O(a). To difierent component.s correspond difierent 1c. We con-
clude that the componentS are in one-to-one correspondence with a
subset of the natural numbers, and consequently the set of components is
COlmtable.
For instance, ~ opIln 3Ubaet uJ R ia a countable "nion oj dia,ioint
interval,.
Again, it is possible to analyze the proof and thereby arrive at a
more general result. We shall say that a set E is denae in S if E- = 8,
and we sball say that a metric space is separabk if there exists a countable
s..bset which is dense in S. We are led to the following result:
In a locally
8epaf'abk spaC6 we'/I opIln &Ill is a CQUfIIabk union
oj di8joint regions.
EXEaCISES
L If xes, show that the relatively open (closed) subsets of X are
precisely those sets that can be expressed as the intersection of X with an
open (closed) snbset of S.
2.. Show t bat the union of two regions is a region if and only if they have a common point.
S. Prove that the closure of a connected set is connected.
4.. Let A be the set of points (x,1/) E R" with x = 0, 1111 ~ I, and let B he the set with x> 0, 1/ = Bin l/x. Is A V B connected?
s. Let Ebe the set of points (x,/I) E Rlsuch that 0 ~ x ~. 1 and either
y = 0 or y = l/n for some positive ~teger n. What are the com-
ponents of E? Are they all dosed? Are they relatively open? Verify that E is not locally connected.
'" Prove that the components of a dosed set are closed (use Ex. 3). 7. A set i8 said to be diM:rele if all its Points are isolated. Sbo" that a
set in a separable metric space is countable.
ANAL YTIC FUNCTIONS AS IIAPPINGS 511
1.4. Compactne... The notions of convergent sequences and Cauchy
sequences are obviously meaningful in any metric space. Indeed, we
would say that x...... x if d(x.,x) ..... 0, and we would say that {x. I is a
Cauchy sequence if d(x.,x..) ..... 0 as n and m tend to 00. It is clear that every convergent sequence is a Cauchy sequence. For Band C we have
proved the converse, namely that every Cauchy sequence is convergent
(Chap. 2, See. 2.1), and it is not hard to see that this property carries over
to any B'. In view of its importance the property deserves a special name.
Ddinition 5. A metric space i8 said wbe comp/.u 'f wery Cauchy sequence
iB convergent.
A subset i8 complete if it is complete when regarded as a subspace. The reader will'find no difficulty in proving that a compleU BUb.., of a
mell ie space iB closed, and that a tWsed BUb.oet of a compleU 8fHJI'Il iB compkle.
We 8haJI now introduce the stronger concept of compacl'IIU'. It is stronger than completeness in the sense that every compact space Of set
is complete, but not conversely, As a matter of fact it will turn out that
the compact subsets of Band C are the closed bounded sets. In view of
this result it would be possible to dispense with the notion of compactness,
at least for the purposes of this book, but this would be unwise, for it would mean shutting our eye8 to the most striking property of bounded and closed sets of real or complex numbers. The outcome would he that
we would have to repeat essentially the 8ame proof in many different
connections. There are several equivalent characterizations of compactness, and it
is a matter of taste which one to choose as definition. Whatever we do the
uninitiated reader will feel somewhat hewildered, for he will not be able to
di.seern the purpose of the definition, This is not surprising, for it took
a whole generation of mathematicians to sgr<le on the hest approsch,
The c(>nsensus of present opinion is that it is hest to focus the attention on the different ways in which a given set can he covered by open sets.
Let us say that a collection of open sets is an open COVer1'11f/ of a set X if
X is contained in the union of the open sets. A IlUbcoveri'llf/ is a 8ubcolIection with the same property, and a finite covering is one that consists of a
finite number of sets. The definition of compactness reads:
Definition 6. A Bel X iB compacl 'if and only if every Opll/l COIJer1'11f/ of X
conl4im a finite BUbcovering.
. '.-
• ..... '.- . ..J. ',' .. ' ..... :-
..
.. ' , ' . . - .-"., .', ,.: :
.. , ,-.' '
. . ':. ',,'
-,
.. ".'.~."). '•-.·,1.>-,
. ..
: ",'- ::;'-'. " ,','"
IG
COMPLEX ANALYSIS
and the covering is by open sets of S. .But if U is an open set in S, then U (\ X is an open subset of X (a relatively open set), and conversely every open subset of X can be expressed in this fonn (Sec. 1.3, Ex. 1). For this reason it makes no difference whether we formulate the definition for a full SpllOO or for a subset.
The property in the definition is frequently referred to 88 the HeifUlBorel properly. Its importance lies in the fact that many proofs become
particularly simple when formulated in terms of open coverings. We prove first that every compact SpllOO is complete. Suppose that
X is compact, and let Ix.) be a Cauchy sequence in X. If y is not the
limit of Ix.l there exists an , > 0 such that d(x.,y) > 2, for infinitely many ... Determine ... suchthatd(x..,x.) < dorm,n ~ no. Wechoose a fixed n ~ n.forwhichd(x.,y) > 2.. Thcnd(x..,y) ~ d(x.,y) - d(x..,x.) > • for all m ~ no. It follows that the .-neighborhoOd H(y,e) contains
only finitely many x. (better: contains x. only for finitely many n).
Consider now the collection of aU open sets U which contain only finitely many x.. If Ix.l is not convergent, it follows by the preceding reasoning that this collection is an open covering of X. Therefore it muet contain a finite 8ubcovering, formed by U ... ,UN. But that is
clearly impossible, for since each U, contains" only finitely many x. it
would follow that the given sequence is finite. Secondly, a compact set is necessarily bounded (a metric space is
bounded if all distances lie under a finite bound). To see this, choose a point Xo and consider all balls B(x.,r}. They fonll an open covering of X, and if X is compact, it contains It finite subcovering; in other words,
X C B(xo,r,) U ... U B(x.,r..), which means the same as XC H(xo,r)
+ with r = max (r" , .•, r..). For any X,y E X it follows that d(x,y) ;:;;
d(x,x.) d(y,x.) < 2r, and we have proved· that X is bounded.
But boundedn".,.. is not all we can prove. It is convenient to define a stronger property called total boundedne88:
..
D,.6nitlon 7. A sel X is totally bounded iI, I'" every e > 0, X can be OOfI61'ed by finitely """,y ball.! 01 radim e.
This is certainly true of any compact set. For the collection of all
balls of radius e is an open covering, and the compactness implies that
we can Belect finitely many that cover X. We observe that a. totslly
bounded set is necessarily bounded, for i(X C B(xl, e) U ... U B(x.., e),
+ then any two poin~ of X have a distance <2. max d(x"xj). (The
preceding proof that any compact set is bounded becomes redundant.) We have already proved one part of the following theorem:
Theorem 6. A BIll ia cumpacl if and tmly if it ia complete and IotIJIly
botmded.
"N"LYTIC FUNCTIONS " • •"I'PINOS
To prove the other part, 888lU1le that the metric space S is complete
and totally bounded. Suppose that there exists an open covering which
does not contain any finite subcovering. Write t. = 2-. We know
that S can be covered by finitely many B(x,o,). If each had a finite subeovering, the same would be true of S; hence there exists a B(x".,) which does not admit a finite subeovering. Because B(x.,E,) is itself
totally bounded we can find an x. E B(x,,!!) such that B(x"e.) has no
finite subcovering.t It is clear how to continue the construction: we
obtain a sequence x. with the property that B(x., E.) has no finite sub-
covering and X.+1 E B(x., ••). The second property implies d(x.,x....,) < ••
and hence d(x.,x.+p ) < o. + .....' + ... + .""0-1 < 2-·+1. It follows
that x. is a Cauchy sequence. It converges to a limit 1/, and this 1/ belongs to one of the open sets U in the given covering. Because U is open, it
contains a ball B(I/,8). Choose n 80 large that d(x..,I/) < 8/2 and •• < &/2.
Then B(z., ••) C Bty,&), for d(x,x.) < e. implies d(x,l/) ;:;; d(x,z.) +
d(x.,I/) < 3. Therefore B(x.,e.) admits a finite subeovering, namely by
the single set U. This is a contradiction, and we conclude that S has
the Heine-Borel property.
Coroll ery. A IlUb.eI of R or C is compact if and onll/ if it is closed and
bounMa.
We have already mentioned this particular consequence. In One
direction the conclusion is immediate: We know that a compact set is
bounded and complete; but R and C are complete, and complete subsets
of a complete space are closed. For the opposite conclusion we need to
show that every bounded set in R or C is totally bounded. Let I1B take
the case of C. If X is bounded it is contained in a disk, and hence in a
square. The I!quare can be subdivided into a finite number of I!quares
with arbitrarily small side, aud the squares can in turn be covered by disks
with arbitrarily small radius. This proves that X is totally bounded,
except for a small point that should not be glossed over. When Definition
x e s 7 is applied to a subset
it is slightly IUDbiguous, for it is not clear
whether the ...neighborhoods should be with respect to X or with respect
to S; that is, it is not clear whether we reqnire their centers to lie on X.
It happens that this is of no avail. In fact, suppose that we have covered
X by ...neighborhoods whoee centers do not nece:marily lie on X. If such
a neighborhood does not meet X it is superfluous, and can be dropped. If
it does
a point from X, then we caD replace it by a 2a-Deighborhood
around that point, and ...e obtain a finite covering by 2..neighborhoods
with centers on X. F!lrthi8 reason the ambiguity is only apparent, and
our
proof
that
boundedqbeet/l , ,"
of. C
.
.are
totally
o
o.
u
n
d
'
e
d
is
valid.
t Here we eM n,hl tbe INt that any subset of a totally bonodtd set is totally
bounded. , The ieael .. ~ PO" Ihia.
... . ,
-.-
'""," '-." ',',"'''I',- ':'-'.,i_.-''.;".";,
"
""
. " :,:~
",' ,.
·"·'t·",,"., -,",",
:
'.
""'
~'.'-"
,~.;:."_':..-,..:
'
.
,"
,'-,' ,
," '
.
''-.
.
:-',
-
'''.:
.
:
'-,.,,,",.,-.,'.,
.
,.' -;,
.
,.-
.. ~;, . ''-- - .. . .:,:""
,..:.
"
'
- .-, -
,. ~~
12
COMPLEX ANALYSIS
There is a third characterization of oompact sets. It deals with the
notion of limit paint (sometimes called c/usler value): We say that 11 is a
limit point of the sequence Ix.l if there exists a subsequence (x.. l that
converges to 11. A limit point is almost the Bame as an accumulation point
of the set formed by the points x., except that a sequence permits repeti-
tions of the ~ame point. If 11 is a limit point, every neighborhood of 11
contains infinitely many x.. The converse is also true. Indeed, suppose
that n -+ O. If every B(y, £.) oontains infinitely many x. we can choose
subscripts no, by induction, in such a way that x •• E B(y, e.) and n",., > no.
It is clear that 1z..1 oonverges to y.
Theorem 7. A flIIJtrU! &pIJCe Ur co,npad if IJtId only if every infinite
aeq~ Iuu a limit point.
This theorem is usually referred to as
The original formulation WIU! that every bounded sequence of oomplex
numbers has a convergent subsequence. It came to be recognized 8B an
important theorem precisely becal'''e of the role it plays in the theory of
analytic functions.
The first part of the proof is a repetition of an earlier argument. If
11 is not a limit point of Ix.1 it has a neighborhood which contains only
finitely mAny x. (abbreviated version of the conect phrase). If there were
nO limit points the open sets containing only finitely many x. would form
an open covering. In the compact ease we could select a finite subcover-
ing, and it would follow that the sequence is finite. The previous time we
used this reasoning was to prove that a compact space is complete. We
showed in
that every sequence has a limit point, and then we
observed that a Oauchy sequence With a limit point is necessarily con-
vergent. For strict economy of thought it would thus have been better to
prove Theorem 7 before Theorem 6, but we prefened to emphasize the
importance of total boundedness as early as possible.
It remains to prove the converse. In the first place it is clear that the
Bolzano-WeieistnL"8 property implies colDpletene88. Indeed, we just
pointed out that a Cauchy sequence with a limit point must be convergent.
Suppose now that the space is not totally bounded. Then there exists an
• > 0 such that the space cannot be covered·by finitely many ""neighbor-
hoods. We construct a sequence (x.1 ""follows: x. is arbitrary, and when
,x. X" ••• have been selected we cho,Qse X.+. 80 that it does Iiot lie in
B(x.,e) V . . . V B(x.,e). This is always pollSible because these neigh-
borh(lods do not cQver the whole space. But it is clear that (x.I has no convergent subsequence, for d(x..,x.) > • for all m and fl. We conclude
that the Bolzano-Weierstrass property implies total boundedness. In
view of Theorem 6 that is what we had to prove.
..
. .
.._-.,~ '."- .
"""LYTIC FUIICTIOII. " • •"""IIIGB 13
The reader should re8eet on the fact that we have exhibited three ch8l'acteruations of compactne:-'" whose logical equiv&lence is not at all trivial. It should be clear that results of this kind are particularly v&luable for the PUIpOse of presenting proofs as concisely as possible.
.
EXERCISES
1. Give an alternate proof of the fact that every bounded sequence of
complex numbers has a convergent subsequence (for instance by use of the
limes inferior). 2. Show that the Heine-Borel property can &leo be expretl8ed in the
following manner: Every collection of closed sets with an empty intersection contains a finite 8ubcollection with empty intenlection.
.. Use compactness to prove that a closed bounded set of real nllmben! has a maximum.
... If E, ::> E. ::> E.:J • . . is a decreasing sequence of nonempty
•
compact sets, then the interse.tion r\ E. is not I'mpty (Cantor's lemma).
I
Show by exl'mple that this nlled not be true if the sets are merely closed. 50 Let S be the set of all sequences % = (%.I of real num bers such
that only a finite number ofthe %. are ... O. Defined(%,y) = max 1%. - y.l.
Ia the space complete? Show that the ~neigbhorhoods are not totally
bounded.
1.ti. Continrcou.o Functro.... We shall consider functions/which are
defined on a metric apace S and have values in anotber metric space S'.
Functions are also referred to as mappinga: we say that / ma}l8 B into B',
and we write/:B .... B'. Naturally, we shall be mainly concerned with
real or complex-valued functions; occasicoually the latter are allowed
to take values in the extended complex plane, ordinary distance being
replaced by distAnce on the Riemann spbere.
The space S is the dl>lll4in of.the function. We are of
free to
consider functions / whose domain is only a subset of B, in which ease the
domain is regarded as a subspace. In moat
it is safe to slur over the
distiDction: a function on B and its restriction to a subset are usually
denoted by tbe same symbol. If XC Bthe set of all values/(z) for %f B
is called the itrUlf18 of X under I, and it is denoted by I(X). The inverse
itrUlf18rl{X') oC X' C B' conoistl!of all %e Bsuch that/(%) eX'. Observe
that/U-,{X'» C X', and rIU(X» ::> X.
The definition of a continuous function needs praetically no modifica.-
~on: / is continooWl at a if to every & > 0 there exists a > 0 such that ~(z,a) < a implies d'U(z),/(a» <.. We are mainly concelD~ with
rnnetiollS thllt'arecontiDuoua at all points in the 'domain of definition.
'.
...... '--, ..... "'._"
-,.' "
",
. .-'.,,"-
" • . ..:",·"·,,',·'.~'·,,~-.i·,-"
" ' . , ' . , ;,,-·.,!:·,'-~'-":"'-:1'_,·'
. . ' .
-'
'•__.I-.,"--!:'
. -..'f..:.,,-1"-":".'".
, ,, ' ,
.
,
-
.
'.....
. ,
'..".-.
.
.-.:
.
:...
:
-
.
_'.
," .••. ':- , ","'.' , '--,:,."r,...:. "::i ::~,,;:~~~> .(;~::-~,:..~:_:".,:.:):~~'.:.._~-:.:..,;..(';:.<''<t...-.;-~~~.:~~:"j ~:;': ':'..,,:,,,,,, :'.',
,-----
COMPLEX ANALYSIS·
The following characterizations are immediate consequences of the
definition:
'sA lundion ia contmUOUB il and tmly il tIuJ inve1"1lIJ image 01 every open
861 open.
A lunction i8 continuous il and only il tIuJ invm-se image 01 every closed
set ia closed.
IfI is not defined on ell of 8, the words "open" and "Closed," when
referring to the
image, should of course be interpreted relatively
to the domain of I. It is very important to observe that these properties
hold only for the
image, not for the direct image. For instance
the mapping I(x) =- x'/(1 + z') of R into R has the image f(R) =
{y; 0 ;:> y < I} which is neither open nor closed. In tills eXll.lllple I(R)
fails to be closed heeaWle R is not compact. In fact. the following is
true:
TheorelD 8. Untkr a comintuma fJUlpping tIuJ image 01 every compact set
.,
and consequently clOMl.
Suppose that I is defined and continuous on the compact set X.
Consider a covering of I(x) by open sets U. The inverse images I-'(U) are open and form a covering of X. Because X is compact we "an select a finite sUbcovering: X C I-'(U,) U ... U I-I(U.). It follows that f(X) CU, V • . • V U.., and we have proved that I(x) is compact.
Corollary. A contin1WllB real-valued function on a compad set hall a marl-
mum and a minimum.
. .
..
The image is a closed bounded subset of R. The existence of a
maximum and a minimum follows by Theorem 2.
TheoreJD 9. Under a continooUB mapping the imnge 01 anll connected 8el ia connected.
We may assume that I is defined and continuous on the whole space
8, and that f(8) is ell of 8'. Suppose that 8' = A V B where A a.nd B
are open and disjoint. Then S = f-I(A) U f-'(B) is a
of
S as a union of disjoint open sets. If 8 is connected either I-I(A) = 0 or
l-I (B) = 0, and hence A=-O or B.~ O. We conclude that 8' is
connected.
A typical application is the assertion that a real-valued function
which is continuous and never zero on·a connected set is either always
positive or always negative. In fact, the image is connected, and hence
an interval. But an interval which cont.ains positive and negative .011111-
ANALYTIC fUNCTIONS A. MAPPINGS
bers also contains zero.
A mapping f:8 ..... S' is said to be one to one if I(x) = fey) only for
x = Yi it is said to be onto if f(8) = 8'.t A mapping with both these
properties has an inverse j'-', defined on 8'i it eatisfiesf-'(f(x» .. xand
f(j'-'(x'» - x'. In this situation, if f and f-1 are both continuous we
say that f is a topoWgiool mapping or a 1wmeOlllorphi8m. A property of a
set which is shared by all topological images is called a topological properly. For instance, we have proved that compactneJ!S and connecrednesa are
topological properties (Theorems 8 and 9). In this connection it is per-
haps u-eful to point out that the property of being an open subset is not
topological. If X C 8 and Y C 8' and if X is homeomorphic to Y there is no resson why X and Y should he simultaneously open. It happens to be true if 8 .. 8' = R' (invarionce of the regirm), but this is a deep theorem
that We shall not need.
The notion of unilornt crmtinud1l will be in constant use. Quite
generally, a condition is said to hold uniformly with respect to a parameter
if it can he expwllJed by inequalities which do not involve the parameter.
Accordingly, a function f is said to be uniforlll11l continuous on X if, to
every • > 0, there exists a 3 > 0 such that d'(/(xl),J(z.» < • for all
pairs (Xl,X.) with d(x,,x.) < 3. The emphasis is on the fact that ais not
allowed to depend on x,.
Theorem 10. On 0 ro.npact BBl every ronlinuotU function is uniforml1l
contimroUB.
The proof is typical of the way the Heine-Borel property can he used.
Suppose thatf is continuous on II> compact set X. For every y • X there is II> baIl B(y,p) such that d'(f(x)J(1I» < ./2 for x • B(1I,p) i here p may depend
on 11. Consider the covering of X by. the smal.ler balls B(1I,P/2). There exists a finite subcovering: X C B(1IbP,/2) V ... V B(1I.,p../2). Let 6
be the smallest of the numbers Pl/2, .•• , p../2, and suppose that d(x"z,) < I. There is a y. with d(Xl,II') < p./2, and we obtain d(x',II') < p./2 +
3 ;:iii 3.. Hence d'(/(z,).!(1I.» < ./2 and d'(f(x,),f(1I.» < ./2 80 that
d'(f(xl),f(xi) < 1&8 desired.
On sets which are not compact 80me continuous functions are uni-
formly continuous and otbers are not. For instance, the function z is
uniformly continuous on the whole complex plane, but the func\ion .' is not.
t TheM IiDauiatico.ll7 c1um.,. ~ c." be ...pleced b7 .Jfi.. (for on. to one) and Iwjsc'" (lor onto). A-.;.q·Wn& with both paoperi.tM ia..ned. wjn'":"
H
COlfPLEX ANALYSIS
EXlRella
1. Construct a topological mapping of the open diu Izi < 1 onto the
whole pl&ne. 2. Prove that a subset of the real line which is topologically equiva-
lent to an open interval is an open interval. (Consider the effect of removing a point.)
S. Prove that every continuous one-to-one mapping of a compact space is topological. (Show that closed sets are mapped on closed sets.)
4. Let X and Y be compact sets in a complete metric space. Prove
that there exist x E X, 1/ E Y such that d(x,y) is a minimum.
s. Which of the foUowing functions are uniformly ,continuous on the
whole real line: sin x, x ain x, x sin (x'), Ixlt sin x?
1.6. Topofogialf SPOC6IJ. It is not necessary, and not always con-
venient, to express nearness in terUlsof distance. The observant reader will have noticed that most results in the preceding sections were fOrInulated in terms of open sets. True enough, we used distances to define open sets, but there is really no strong reason to do this. If we decide to consider the open sets as the primary obieets we must postulate axioms that they have to satisfy. The following axioms lead to the commonly accepted definition of a topologirol space:
Definition 8.,' A topologirol space ill a set T togotlter with a colled:ian of its
Btlb8et8, called open sel8. The JoUlllJ!ing coMtlilmB have 10 be fuljUled:
(i) The emply set {lJ and !he whole apace Tare open w.
(ii) The inUrsectWn of Imy lIDO upM Btll. ill an open set. "
(iii) The union of on arbitrary co/led.ion oJ open set. ill an open set.
We recognize at once that this terminology is consistent with our earlier definition of an open subset of a metric space. Indeed, properties (ii) and (iii) were strongly emphasized, and (i) is trivial.
Closed sets are the complements of open sets, ....d it is immediately clear how to define interior, closure, boundary, and 80 on. Neigbbor-
hoods could be avoided, but they are rather convenient: N is aneighbor-
hood of x if there exists an open set U such that x E U and U C N. Connectedness was defined purely by means of open sets. ' Hence the
definition carries over to topological spaces, and the theoreJfu! remain true. The Heine-Borel property is alai> one that deals only with open sets. Therefore it,makes perfect sense to speak of a compact topological space. However, Theorem 6 become. meaningless, and Theorem 7 becomes false:
As a matter of fact, the first serious difficulty we encounter iz, with
ANALYTIC fUNCTIONS AI MAPPINGS .7
say z convergent sequences. The definition is clear: we that %. -> if
every neighborhood of % contains all but a finite number of the %.. But
if %. -> % and %. -> Y we are not able to prove that % = y. This awkward
situation is remedied by introducing a new axiom which characteri~e8 the
topological space as a HausdtJrfl opace:
Definition 9. A topological opace ill called a HautJorfl space if any two
distinct points are contmned in di8;joint open seta.
In other words, if % "" Y we require the existence of open sets U, V
such that % f U, 1/ f V and U f\ ·V = /If. In the presence of this condition
it is obvious that the limit of a convergent sequence is unique. We shaD
never in this book have occasion to consider a space that is not a HaU8dorfr
space.
This is not the place to give examples of topologies that cannot be derived from a distance function. Such ell'ampies would necessarily be
very complicated and would not fUrther the purposes of this book. The
point is that it may be unnatural to introduce a distance in situations when one is not really need.ed. The reasOn for including this section has been to alert the reader that distances are dispensable.
2. CON FORMALITY
We now return to our original setting where all functions and variables are restricted to real or complex numbers. The role of metric spaces will seem disproportionately small: all we actually need are some simple
applications of connectedn""" and compactness. The whole section is mainly descriptive. It centers on the geometric
consequences of the existence of a derivative.
%.1. ,4,."" and ClDaed Cur...... The equation of an are "f in the plane
is most conveniently given in parametric form % = %(t), y = yet) where t
+ runs through an interval .. :l!! t :;; P and :ret), 1/(t) are continuous func-
tions. We can also use the complex notation. - !let) = :r(1) iy(t)
which has several advantages. It is also customary to identify the are "f
with the continuous mapping of [..,P]. When followiug this custom it is
preferable to denote the mapping by. = "f(t).
.
Considered as a point set an are is the imagll of a closed finite interval
under a continuous mapping. As such it is compact and connected. How-
ever; an are is not merely a set of points, but very Lssentially also a 8UC-
cession of points, ordered by inczening values of the parameter. Jf a
n
I
on
::Ii
d/J'._,
eMing
-.
then ".
function t
=
a
( ,-"
. "
,
(
T
)
~ .--
,
. maps
·the' ,III'
."
-
an
me
siuno,tcee-. nesaiol n...i:c:ifli-p.o.'.m:. :It,istJa',
on s"
to
~ '
.. :i
a(t) .
._. . ._. ," ,", ,,-' .....-......,". -, -;'_":, '_' ", • ,j:j• '-',"-'-.
, .•:- .. -.- - .,,' : "-," -.. -.. __. __ . _'_: :'
-"-- '-•. "-. ... --,~,>.,~~", " " , . , . . -
·.',,'.,.-~._..:.w.f,.~.l'",·>.'".,-~.".f.'"'_:,_•'.
~--.,-.~-.,.
'•,
_,"--_n,..,."."'..~,'_~_,I'
"' £. '.."..'...
-"~,':~'..,....,...:,.....,"F._.,~•'•,.......:...."-..'..-'.'..'.,.~'"-~I
•.".-..',~.~-"--c·"-"';'':-"''I"n,,-:';.
.
.
',''
II
COIIPLEX ANALYIIS
We say that the first equation arises from the second by a change 01 parame-
ter. The change is rever8ihle if and only if (1)(,,) is strictly increasing. For
+ instance, the equation 11 = t' it", 0 ~ t ;;; 1 arises by a reversible
of parameter from the equation z = t + 11.', 0 ;;; t :iii 1. A change of
pa.rametric interval (a,{J) can always be brought about by a l i _
+ of parameter, which is one of the fonn t = 111' b, a > O.
Logically, the simplest course is to consider two &l'CS as difierent as
soon as they are given by diilerent equations,
of whether one
equation may arise from the other by a change of parameter. In follow-
ing this course, as we will, it is important to show that cerf.ain properties of
arcs are invariant under a change of parameter. For instance, the initial
IUld tel ",ina! point of an arc remain the same after a cbal;lge of parameter.
+ If the derivative t(t) - Z/(!) iy'(t) exists and is ~O, the arc "I has
a tangent whose direction is determined by arg t(l). We shall say that
the arc is differentiable if t(l) exists and is continuous (the term con-
tinuously differentiable is too 1lDwieldy); if, in addition, z'(t) ~ 0 the arc
iR said to be regular. An arc is piecetl1ise differentiable or ~ regular
if the same conditions hold except for a finite number of values t; at these
points z(t) shall still be continuous with left and right derivatives which are equal to the left and right limits of z' (t) and, in the case of a piecewise
regular arc, ¢O.
The diilerentiable or regular character of an arc is invariant under the
change of parameter t = 'P(T) provided that 'P'(T) is continuous and, for
regularity, ¢O. When this is the
we speak of a difierentiable or
regular change of parameter.
An arc is mnple, or a Jordan arc, if z(t,) = z(/.) only for"''''' t.. An
arc is a closed CUf'!I6 if the end points coincide: z(a) = z(ft). For closed
curves a shift of the parameter is defined as follows: If the origin•.! equa-
tion is z = z(t), a ~ t ;;; /J, we choose a point t. from the interval (a,(:I) and
define a new closed curve whose equation is 11 = ,et) for I. ~ t ~ fJ and
+ 2 = z(t - fJ + a) fol' (J :; t ;;; t. {J - a. The purpose of the shift is to
get rid of the distinguished position of the initial point. The correct
definitions of a differentiable or regular cloiIed curve and of a Bimple clo8ed
CUnJS (or Jordan cunJs) are obvious.
The opposite arc of 2 = z(I), a ;;; t ~ /J, is the arc z ~ z( -t), - II ;:ii
t ;;; -a. Opposite arcs are sometimes denoted by "I and -"I, sometimes
by 'Y and "1-1, depending on the connection. A constant function z(t)
defines a point CW'II6.
'
+ A circle C, originally defined as a locus 12 - "I = r, can be considered
as a closed curve with the equation z = a rsu, 0 :; t :; 2r. We will
lise this standard parametrization whenever a circle is introduced.
This convention saves us from writing down the eqll8tion each time it is
ANALYTIC FUNCTIONS AS MAPPINGS ..
needed; also, and this is it. most important purpose, it serve. as a definite
rule to distinguish between C and - C.
2.2. Analytic Functions In Regions. When we consider the derivative
f'(e) _ lim !(z + h) - !(!)
.....
h
of a complex-valued function, defined on a set A in the complex plane, it is of course understood that Z E A and that the limit is with respect to values
+ h such that z h EA. The existence of the derivative will therefore
have a different meaning depending on whether. is an interior point or a
boundary point of A. The way to avoid this is to insist that all analytic
functions be defined on open sets.
We give a formal statement of the definition:
Definition 10. A complez,..valued !utICtion f(.), deji.1It!d on an open Bel n, is 8fJid to be analy/i£ in n if it 00s a t1.erioolive at each point of O.
Sometimes one says more explicitly that /(z) is complex analytic. A commonly used synonym is holmnorphic.
It is important to stress that the open set n is part of the definition.
As a rule one should avoid speaking of an analytic function f(z) without
referring to a specific open set n on which it is defined, but the rule can
be broken if it is clear from the context what the set is. Observe that f
must first of all be a fundion, and hence ringk-flalued. If 0' is an open
subset of n, and if f(.) is analytic in Il, then the restriction of f to II' is
analytic in 0'; it is customary to denote the restriction by tbe 88me letterf.
In particular, since the components of an open set are open, it is no loss
of generality to consider only the case where a is connected, that is to say
a reg1•on.
For greater flexibility of the l8ngll.g~ it is desirable to introduce the
following complement to Definition 10:
Definition ll. A function f(.) i. analytic on an arbitrary point let A if
it is 1M reslridion to A of a function which is analytic in _ open 86t conlaining A.
The last definition is merely an
to use a convenient te....j-
nology. This i8 a ca"" in which the set Q need not be explicitly men-
tioned, for the specific choice of Q is usually immaterial as long as it contain.
A. Another instance in which the mention of a can be 8Uppreased i.e the
phrase: "Let I{z) be
at z.." It mean8 that .. f.mction f(z) is
defined and hae ..
of 2<>
.
<-
,.--"---
COIIPLEX ANALYSIS
Although our definition requires all analytic functions to be single-
valued. it is possible to consider such multiple-valued functions 88 VZ.
log z, or e.rc cos z, provided that they e.re restricted to a definite region
in which it is possible to select a single-valued and analytic branch of the
function.
For instance, we may choose for n the complement of the negative
vz real axis .0 ~ 0; this set is indeed open and connected. In 0 one and
only one of the values of has a positive real part. With this choice
to ~ VZ becomes a single-valued function in 0; let us prove that it is
continuous. Choose two points z,• .., E 0 and denote the corresponding
+ values of w by w• ., 11. iv,. w, = u, + iv, with u,. u, > O. Then
1.0. -- .0.1 = Iw1 - to;1 = Iw, - to,l . Iw. + w.1
and Iw. + to.1 !l:; u, + u. > 11,. Hence
Iw, -
w. I
<
.1-.'-0-.'---,---.0-.'1" It.
VZ and it follows that w = is continuous at .0,. Once the continuity is
established the analyticity follows by deriV&tion of the inverse function
• = w'. Indeed, with the notations used in calculus .<loa --t 0 implies aw -+ O. Therefore.
aw.= ato AI..tm~o Ilz
II' m-,-
.4111' .0 .4z
and we obtain .
dw 1 1
1
-
dz=dz=2w=2VZ
dw
with the same branch of VZ.
In the case of log .0 we can use the same region n, obtained by exclud-
ing the negative real axis. and define the printipal branch of the logarithm
by the condition 11m log zl < T. Again,. the continuity must be proved.
but this time we have no algebraic identity at our disposal. and we are
forced to use a more general reasoning. Denote the principal branch by
+ w = 11 iv - log .o. For a given point w, = + 11, iv,• Iv.1 <: .... and a
given • > O. consider the set A in the w-plane which is defined by the
inequalities Iw - w,1 iii:; ", Ivl ;!!!! T, lu .,.. u,l ~ log 2. This set is closed
and bounded, and for sufficiently small. it is not empty. The continu-
OUS function Ie" - e"> I has consequently a minimum p on A (Theorem 8,
Corollary). This minimum is positive, for A does not contain any point
+ w, n . 2wi. Choose 5 ,,;. min (P,~"). and aBBume that
IZI - ...1= Ie"> - e"'1 < i. -
ANALYTIC FUNCTION. AS MAPPINGS
71
Then w. cannot lie in A, for this would make Ie... - e"\ ~ p ~ B.
+ Neither is it possible that Us < fl, - log 2 or u. > u, log 2; in the
former case we would obtain \e'" - 6".\ ~ eV' - 6" > jeW, ~ 4, and in
the latter
Ie'" - e"'\ ~ eO, - e"' > eO, > B. Hence w. mWJt lie
in the disk ltD - tD.\ < c, and we have proved that w is a continuous
function of z. From the continuity we conclude as above that the
derivative exists and equals l/z.
The infinitely m8.llY values of arc cos z are the same 88 the values of
+ i log (z y z· - 1). In this ease we restrict z to the complement (I' of
the half lines z ~ -1, 11 = 0 and z ;;:; 1, 11 - O. Since 1 - 2' is never
+ real and ~ 0 in 0', we ean define y1 - z· 88 in the first example and then
set y,,' - 1 ~ iyl - %t. Moreover, % YZ' - 1 is never real in 0',
for z + Y zt - 1 and z - y z' - 1 are reeiprooals and hence real only if z
and yz' - 1 are both real; this happens only when z lies on the excluded
parts of the real axis. Because 0' is connected, it follows that all values
of. z + Y zt - i in 0' are on the same side of the leal axis, and since i is
such a value they are all in the upper half plane. We can therefore define
an analytic branch of log (z + yz' - 1) whose imaginary part lies between
oand.... In this way we obtain a: single-valued analytic function
arc COBS = ilog (.. + yz· - 1)
in (I' whoBe derivative is
+. . + D
arc
cos
%
.
-.
z
1
1
yZ'-1
yz'Z -1
where yi - zi has a positive real part.
There is nothing unique about the way in which the region and the single-valued branches have becn choBen in theBe examples. Therefore,
each time we consider a function euch as log z the choice of the branch has to be specified. It is a fundamental fact that it is impo8lible to define a single-valued and analytic branch of log z in certain regions.
This will be proved in the chapter on integration.
All the results of Chap. II, Sec. 1.2 remain valid for functions which
o.re analytic on. an open set. In partiClllar, the real and imaginary parts )f an analytic function in (I satisfy the Cauchy-Riemann equations
-ia/xu= -aiIyv,
Jonversely, if II and v satisfy theBe equations in. (I, and if the partial
lerivativee are continuous, then u + iii lsan BD&Iytic (nnctijl/1 in O.
An analytic f,",-ction in O. .. . . ... it.educee to • e6_n~ l"M!
:
-
-.
.,', '. . .
• _
. ,-.'
_.:,-.:;"'0;;;'
_.;:,~
,
'":;:.
.·:',:' ""_;""';-.:,·:,"'~';h.-.""~-"-:':~'~_."~".',:':;':.:";":~'"-"'·.;<·;"":-:",':.;l-.'"::';·~<~~'.\;.o~_:i":,:'•,.·;;·-"..-.:,,:.,.
..•.;..
·:,.~:"
72
COMPLEX ANALYSIS
the following theorem we shall list BOrne simple conditions which have this
consequence:
Theorem ll. An analytic fUndWn in a region 0 whose dMivative van-
w.a identicaUII IIlmt re8t1C1l to a eonatam. The 3IJm6 Ut truei! eilker the
real pa1'~, the imaginary part, the flWdulm, or the argument Ut constant.
The vanishing of the derivative implies that au/in, au/ay, ltv/in, ltv/ayare all zero. It follows that u and v are constant on any line seg-
ment in Il which is parallel tc one of the coordinate axes. In Sec. 1.3 we
remarked, in connection with Theorem 3, that any two pohits in a region
can be joined within the region by a polygon whose sides are parallel 10
+ the axes. We conclude that u ill is constant.
If u or II is constant,
1'(z) --
au ax
-
~. aauy-_a-ayll + J .alt- v - x_ 0'
and hence I(z) must be constant. If u· + 112 is constant, we obtain
uaa-3u+ : v-ilJt= 3v: O
and
uaa-uy
+
v
ltv ay
=
-uiJa-xv +11aa3u-: =
O.
•
These equations pCI'II1it the conclusion au/ax = ltvja:r: = 0 uDless the
+ + determinant u' v' vanishes. But if u' v' = 0 at a single point it is
constantly zero and I(z) vanishes identically. Hence I(z) is in any case
a constant.
Finally, if arg I(z) is constant, we can set u = ku with constant k
+ (uDless II is identically zero). But u - kv is the real part of (1 ik)!,
and we conclude again that I must reduce tc a constant.
Note that for this theorem it is essential that 0 is a region. If not,
we can oDly
that 1(%) is constant on each component of O.
EXERCISES
+ + Vi L Give a precise definition of .. mngle-valued branch of
II
VI - % in .. suitable region, and prove that it is analytic.
2. Same problem for log log z.
I. Suppose that !(z) is analytic and satisfies the condition \/(z)' - 1\
< 1 in a region o. Show that either Re I(z) > 0 or Re I(z) < 0 throughout o.
ANALYTIC FUNCTIONS AS MAPPINGS
2.3. Conformal Mapping. Suppose that an arc 'Y with the equation
z = z(t), a :;; t ;;;; fl, is contained in a region n, and let I(z) be defined
and continuous in ll. Then the equation w ~ w(t) = f(.(t)) defines an
arc .,' in the w-plane which may he called the image of .,.
Consider the case of an I(z) which is analytic in ll. If .'(t) exists,
we find that ""(t) aiM exists and is determined by
(1)
w'(t) = 1'(.(I»z'(I).
,
We will investigate the meaning of this equation at a point Zo = z(I.)
with z'(t.) ;!If- 0 andr(z.) ;!If- O.
The first eonclusion is that w'(Ie) ;!If- O. Hence 'Y' has a tangent at
w. = fC••), and its direction is determined by
(2)
+ arg w'(t.) = argl'(z.) atg %'(1.).
This relation
that the angle between the directed tangents to 'Y
at z. and to 'Y' at w. is equal to arg f(z.). It is hence independent of
the curve .,. For this reason curves through ". which are tangent to
each other are mapped onto curves with a eornmon tangent at w..
Moreover, two curves which form an angle at z. are mapped upon curves
forming the same angle, in sense 88 well as in size. In view of this
property the mapping by w = f(.) is said to be conformal at all points
with fCz) ;!If- O.
A related property of the mapping is derived by consideration of the
modulus 1/'(4)1. We have
lim If~z) - 1(4)1 = If(z.)I,
--.. I· - ••1
and this means that any small line segment with one end point at ZD is,
in the limit, contracted or expanded in the ratio If('o)l. In other words,
the linear change of scale at 4, effected by the tranzfomlation w = f('), is independent of the direction. In general this change of scale will vary
from point to point.
Conversely, it is clear that both kinds of eonformaIity together imply
the existence of f (ZD). It is less obvious that each kind will separately
imply the same result, at least under additional regularity assumptions.
To he more precise, let us asswne that the' partial derivatives iJflax and iJlllIy are continuous. Under this condition the derivative of 1I>(t) = f(lI(t» can he expressed in the form
! . "+., w'(l.) - X' (I.)
1I'(to)
. .'
.
where the partial dari~ _fUkM at .0;' In 1enne of %'(to)"ftllB
" . _ ->.' . , :'."" .. . ,- '-'''-:'~ -. ,,".: .. ;~ --' • ..'. .....•. ......-.. ....•.. ,....••_..c.'.:··.;':••••·· ....•.,.:,.'..
74
COIIPL•• ANALYSIS
be rewritten as
If angles are preserved, arg [w'(to)/z'(to») must be independent of
arg z'(to). The expression
(3)
must therefore have a colll!tant argument. AIl arg z'(to) is allowed to
vary, the point represented by (3) describes a circle baving the radius
ll(allib:) + i(fJllay)\· The argument eannot be constant on this eircle
unlees its radius vanishes, and hence we must have
(4)
-aaxl= - t.aa-yl
which is the complex form of the Cauchy-Riemann equations. Quite similarly, the condition that the ehange .of sceJe sheJl be the
same in all directions implies that the expression (3) has a constant
modulus. On a circle the modulUJl is constant only if the radius vanishes or if the center lies at the origin. In the fim case we obtain (4), and in the second case
,,_. -al . al
ax ay :;::::::.
The last equation expresses the facit that I(z) is analytic. A mapping
by the conjugate of an analytie fnnction with a nonvanisbing derivative
is said to be indirectly wnlllN'ilal. It evidently preserves the size but
reverses the sense of angles.
If the mapping of (l by '" = I(z) is topological, then tbe inverse fnnc-
tion z - I-I(w) is also analytic. This follows easily if f'(z) '" 0, for then
the derivative of the inverse function must be equal to III' (z) at the point
z ..; I-'(w). We shall prove later that I'(z) can never vanish in the case
of a topo\ogiceJ mapping by an analytic function.
The knowledge thatf'(zo) '" 0 is sufficient to conclude that the ma~
ping is topological if it is restricted to a sufficiently small neighborhood of
Zo. This follows by the theorem on implicit functions known from the oal-
culus, for the Jacobian of the functions u = u(x,y), v .. v(x,y) at the point
Zo i81/'(zo)\' and hence '" O. Later we shall present a simpler proof of this
important theorem.
..
But even if I'(z) '" 0 throughout the region 0, we cannot
that
the mapping of the whole region is necesse.rily topological. Toillust.mte
ANALYTIC FUNCTIONS AS MAPPINGS l'
what may happen we refer to Fig. 3-1. Here the mappings of the sub-
nI'
FIG.lt Doubly oovered region.
regions (1, and Il, are one to one, but the images overlap. It is helpful to think of the image of the whole region &8 & trlUlsparent film which partly
covers itself. This is the llimple IUld fruitful idea used by Riemann when he introduced the genera.1ized regions now known &8 Riemann IIUrlaus.
.
.
11.4. Length and Ar..... We have found that under a COnfOl'lllal mapping
I(z) the length of an infinitesimal line segment at the point z is multiplied
by the factor If'(,,) I. Because the distortion is the same in all directions, infinitesimal areas will clearly be multiplied by If'(o) I'.
Let us put this on a rigorous basis. We know from calcul\18 that the
length of a differentiable are 'Y with the equation. = o(t) = x(l) + iy(t),
a ;:I t ~ b, is given by
J' J' L('Y) = y'",'(O· + y'(t)1 de = 1.'(01 dt.
•
•
The image curve 'Y' is determined by tD = w(t) = I(.(t» with the derivative
w(O = f(.(I»"(I). Its length is th\18
I: L(..,') =
1f'(0(1)) 11.'(1) I tit.
It is customary to use the ahorter notations
(5)
f f L('Y) = Idol, L(..,') = If'(z) lldol·
T
T
Observe that in complex notation the calculus eymbol d3 for integration
with respect to.arc length is replaced by ldo I.
Now let E be a point set in the pilUle whose area
/fdz A(E)=
dll
•
. - '-
,- . .'-,
..•.,.,",
.
.'
"
- '.-". ",' -.;' "-" ':.:'.',., " '.' -"
'.::.'-- ".\ ',.,~'
COMPLEX ANALYSIS
+ ran be evaluated as a double Riemann integral. If f(.) - u(x,y)
iv(z,y) is a bijective differentiable mapping, then by the rule for changing
integration variables tbe area of the image E' = feE) is given by
A(E') = fflu... - ..,.v.1 d% dy.
"
But if f(.) is a confol'lMI mapping of an open set containing E, then
..... - ..,.v. - 11'(.) I' by virtue of the Cauchy-Riemann equatiollll, and
we obtain
(6)
ff A(E') = 11'(.)I' d% dy.
B
The (ol'mnl"" (5) and (6) bave important applications in the part of
complex analyais that is frequently referred to 88 geometric function
theory.
I. LINEAR TRANSFORMATIONS
Of all analytic functions the firslr<lrder rational functions have the simplest mapping properties, for they define mappings of the extended plane onto itself which are at the same time conformal and topological. The linear transformations have also very remarkable geometric properties, and for that reason their importance goes far beyond serving as simple examples of conformal mappings. The reader will do well to pay particular attention
to this geometric aspect, for it will equip him with simple but very valua-
ble techniques.
3.1. The Unear Group. We have already remarked in Chap. 2, Sec.
1.4 that a limar fradionol tt-am!ormtJIion
(7)
+ to = 8(.) = -lcI-zZ;-+--d-b,
with ad - be ¢ 0 has an inve.se
z = 8-'(10) = '-d=w +- ba.'
The special values 8( co) = ale and 8,( -die) = co can be introduced
either by convention or as limits for t ..... co and z ..... -die. With the
latter interpretation it becomes obvious that 8 is a topological mapping of
the extended plane onto itself, the topology being defined by distances on the Riemann sphere.
For linear transformations we shall usually replace the notation S(.)
, ,
ANALYTIC FUNCTIONS AS MAPPINGS 77
by Sz. The representatjon (7) is aaid to be norma1;zed if ad - bc = 1.
It is .11'.8.1' that every linear transformsoon has two normalized represen-
taoons, obtained from each other by changing the signs of the coefficients.
A convenient way to expre88 a linear transformation is by use of
homogeneous coordinates. If we write z = 1<,/Z2,W = w,/w, we find that
w=Szif
(8)
or, in matrix notation,
w, = oz, + bz,
+ 10. = "'" dz.
w,
w,
" a b
c d z, •
The main advantage of this notation is that it leads to a simple determina.-
8.s... tion of a composite trlUlsformation w =
If we use BUbseripts to
distinguish between the matrices that collellD"nd to S" S, it is immediate
that S.s, belongs to the matri~ product
+ + a. b, _ atilt b,e, a,b, b,d.
+ e, d,
c,a, d,c, e ,b, + d,d, •
All linear transformations form a group. Indeed, the associative law (S,S.)S. = S,(S,sj) holds for arbitrary transformatjons, the identity
w = z is a linear transformation, and the inverse of a linear transformation
is linear. The ratios %, :z. ;0< 0:0 are the points of the complex projective
line, and (8) identifies the group of linear transformations with the one-
dimensional projective group over the complex numbers, UBUa11y denoted
by P(l,C). If we UIIe only normalimd representations, we can aI"" identify it with the group of two-by-two matrices with determinant 1 (denoted
SL(2,C», except that there are two opposite matrices conesponding to the
same linear transformation. We shall make no further use of the matrix notation, except for
remarking that the simplest linear transformations belong to matrices of
the form
1 a Ie 0 0 1
01'01'10·
+ The first of these, to = " ot, i8 called a parallel b"amlation. The second,
to = lez, is a rotation if llel = 1 and a hbmothel~ trantlfONtl4tion if Ie > o.
For arbitrary complel< Ie ". 0 We can set Ie = 1.1:1 • k/lkl, and hence w = kz
can be represented &8 the lCSalt of a hornothetic transformation followed
by a rotation. The third transformation, to = l/z, is called an inversion.
If c ". 0 we can write
,
COMPLEX ANALYSIS
and this decomposition shows that the most general linear transform&ion is composed by a translation, an inversion, a rotation, and a homo-
thetic tl'ansformation followed by another translation. If c == 0, the
inversion falls out and the last tr8 muation is not needed.
aXERCISES
L Prove that the reflection z ..... i is not a linear transformation.
1. If
z+2
T,z = z +3'
T,z = z +z l'
find T,T,z, T.T,z and T;:'TtZ,
L Prove that the most general transforulation which leaves the origin fixed and preserves all distances is either a rotation or a rotation followed by reftexion in the real axis.
... Show that any linear transformation which transiOIlIll! the real axis into itself can be written with real coefficients.
8.2. The CrOSlt Ratio. Given three distinct points 1$0, z., z. in the
extended plane, there exists a linear tra.nsforulation S which carries them
into 1, 0, co in this order. If none of the points is co, 8 will be given by
(9)
Bz = :I - : :Is %1 - %1.
z - %" %2 - z .
If %"~' or 2. == co the transformation reduces to
, Z - Za
Z - Zt
, Z2 - Zf,
Z - z..
respectively. If T were another linear tl.LDsiormation with the same property,
then ST-' would leave 1, 0, co invariant.. Direct calculation shows that
this is true only for the identity trausf4)nnation, and we would have
8 - T. We conclude \hat S is uniquely deWlilined.
Definition 12. The eros. roIw (z"z.,.,.,.,.) iB the im4(J' of z, umkr the
linear traM/oltlilll'Wn which carriB8 z"z.,~. into 1, 0, co.
The definition is meaningful only if Z"Z,,%, are distinct. A conven-
tional value can be introduced 88 llOon 88 any three of the points are
distinct, but this iaunimport9u t.
ANALYTIC FUNCTlON8 108 MAPPINGS
71
The cross ratio is invariant under linear tran.'ormatiOIl8. In more precise formulation:
Theoreln 12. 1/Z" Zt, Za, ii, are dulind point. in the alMidod plIJne and T any linear tTam/ormaiian, then (Tz"Tzt,Tz"Tz.) ~ (ZI,Z.,Z,,2.).
The proof is immediate, for if. Bz = (z,zt,z,,z,), thenBT-' canies
Tzt, Tz" Tz. into 1, 0, 00. By definition webave hence
(Tzl, Tz., Tz" Tz.) = BT-'(Tz,) = 8Z1 = (z,,.f.,.,,,,,.).
With the help of this property we can immediately write down the linear trMsfQrmation which carries three given P<lintli Zl, ." Z, to pre
scribed P<lsitions W" to" W" The corxe8P<lndence must indeed be given by
In general it is of course necessary to solve this equation with
toto .
.
Theorem IS. The or08' ratio (z"Z.,%I,...) i& real if and only if tM f<YUr
points lie an a circle or an a 8!1'aight line.
This is evident by elementary geometry, for we obtain
and if the P<lints lie on a circle this diJJerence of angles is either 0 or ±..,
depending on the relative location.
For an analytic proof we need only show that the image of the real
axis under any linear transformation is either a circle or a straight line. Indeed, Tz - (z,,",,z.,z.) is real. on the image of the real. axis under the
transfotmation T-' and nowhere else.
The val.ues of to = T-'z for z satisfy the equation Tto = l'to.
Explicitly, this eondition is nf the fom,
aw+b
+ C1D d
=
dCttliO++iiil.
By C1'OM multiplication we .obtain
(at - ca)lwtl + (ad.:'" ct.)w + (be - d4)1li + bd - db = o.
If ail - CC1 = 0 this is the equation of a straight line, for under this eon-
dition the coefficient ad -
: .. ,,' _-'r: ""'~"" :,
eli catinot also
"_:'''''''" .•. "
va-ni'sh.
If
.
of
-
' ..
CC111"
,,-
0....w' .e
can
, "-, '
- ', ... . C'"
.:
-
'
;
.'.-'
"
..
'
.
.',.,.'..'.'.:":'.:'-:-',-<--';."'.-',"'.--'.:;::'..:-: "-',-..' '..,~''.' ......"' '..••:.-., ......".:",._...:..........:.::.;.~-, . !.'>.:...~~:.:..,.•..,••,.. ,.~;. ,,',,'•.,.:.:.',,
eo
COMPLEX ANALYSIS
divide by this coefficient and complete the square. After a simple computation we obtain
ad-be
= lie-fa
which is the equation of a circle. The last resalt makes it clear th.e.t we should not, in the theory of
linear transformations, distinguish between circles and straight lines. A
further justification was found in the fact that bo~h correspond to circles on the Riemann sphere. Accordingly we sha.ll'e.gl ee to UBe the word
circle in this wider sense.t , The following is an immediate corollary of Theorems 12 and 13:
Tbeorem 14. A linear tranaformalian carrie. circle. iNo circle..
EXERCISES
L Find the linear transformation which carries 0, i, - i into 1, -1, O. Z. Express the cross ratios cOnesponding to the 24 permutations of four points in tern.s of " = (Z.,%2,%,,%,).
a. If the consecutive vertices %1, %t, Z" z. of a qlllulrilateral lie on a
circle, prove t.bat
\z, - z.\·\z, - %.\ = \It. - z,\ ·Iz. - z.\ + \z. - 1t.1 . lit. - z.1
and interpret the result geometrically.
... Show that four distinct points can be earried by a linear
transformation, to positions I, -1, Te, -Te, where the value of k depends on
the points. How many solutions are tbere, and how are they related?
3.3. Symmetry. The points z and j are symmetric with rupect to the real axis. A linear transformation with real coefficients carries the real axis into itself and z, Ii into points which- are again sym~tric. More generally, if a linear transformation T carries the real axis into a circle C, we shall say that the points 10 = Ta and 10* = Ti are symmetric with
re8pect to C. This is a relation between 10, 10* and C which does not
depend on T. For if S is another transformation which carries the real
axis into C, then S-'T is a real transfornlation, and hence S-'w - S-'Tz and 8-'111* = S-'TJ are also conjugate. Symmetry can thus be defined
in the following tenus:
,
t'I'hil eg'eement will be in foroe only when desliDC with linear t;ranefonnation .,
ANALYTIC FUNCTIONS AS MAPPINGS
II
Definition 13. The points It and.* an said 10 be qmmetric wiQl rupect
to the circle C thrMJ{/h z" Z" z, if mul onlll if (z*••••z.,z.) = (z.z.,z,.z.).
The points on C. and only those. are symmetric to themselves. The
mapping which carries z into •• is a one-to-one oonespondence and is
called reflection with respect to C. Two redections will evidently result
in a linear transformation.
.
We wish to investigate the geometric significance of symmetry. Sup-
pose first that C is a straight line. Then we can choose " = co and the
condition for symmetry becomes
(10)
Taking absolute values we obtain 1.* - ••1= I.r - 8.1. Here .. can be
any finite point on C, and we conclude that. and .. are
from all pointe on C. By (10) we have futther
1mz" · -., = -1m z- .. •
1:1-';.
*1-Z,
and hence 11 and z· are in dilferent half planes determined by C.t We
leave to the reader to prove that C is the bisecting normal of the segment betWW.1. and , •.
Consider now the case of a finite circle C of center a and radiu R.
Sy"'sllllclmatic use of the invariance of the cross ratio allows us to conclude
as foOows:
(z,z.,z,,z.) = (z - a,z. - a,z, - a,z. - il)
-
i-a_, R..'., R' • :CI-=t':,--
.11-4 &'s-a z,-a
=
R" ,z,- 0_1- Gs.-a
'-4
'
I
- ._R'4+ a,zI•••••• •
+ This equation shows that the symmetric point ohis.* - R'/(! - a) a
or that. and z· aatilsfy the relation
(11)
(,. - 4)(1 - a) = R·.
The produet 1.* - 01 . I_ - al of the distances to the center is hence R'.
Further, the ratio (II· - a}/(. - a) is positive. whioh
that z and
z· are
on the same half line from o. There is a simple geometric
coDlltruction for the 8ymmetrie point of " (Fig. 3-2). We note that the
eymmetric point of (I is ...
tUDlIiII
. ..... .. .. .... _,_ _ _. .
. .
,
':": ,.
' '
·r,.,·.',.·'
."",",',,
..
;,,::'", ,
"
,
',:,
'"
'~',,:'
.
,:
,,'"
.
.
.......
'
...'....,..._,' ...."'~'''''.',..,,,..:..'
.. : .. .':
'~'
.•'., .•...:..-........~.:•.... '<...;.,.',.'.....:.....,:.:',:.;.'..>, .:.
IZ
COMPLEX ANALYSIS
ZO
c
'"
Fl•. W. ReHection in 8 circle.
Theor.,.,. IS. (ThB symmetry priMiple.) If a linear Irana/ormation carries a circle C. into a circle C., then it trana/o""'8 any pair '" SY1n'IMtric
poinIB with """Peel to C. into a pmr of symmetric poinIB with respecl to C•.
Briefly, linear transfonnations preserve symmetry. If C, or C, is the
real axis, the principle follows from the definition of BY mmetry. In the
general the
follows by use of an intermedis.te transfOn.,ation
which carries C, into the real axis.
There are two ways in which the principle of symmetry can be used.
If the images of z and C under a certain linear transformation are known,
then the principle a.1lows us to find the ime,ge of ~*. On the other hand,
if the images of z and ~. are known, we conclude that the image of C
must be a line of symmetry of these images. While this is not enough
to determine the image of C, the infonnation we gain is nevertheless .
valuable.
The principle of symmetry is put to practical use in the problem of
finding the linear transformations which carry a circle C into a circle C'.
We can always determine the transformation by requiring that three
w., points 21" 21., z. on C go over into three points WI, W, on C'; the trans-
formation is then (w,w"w.,w.) = (z,z"z.,z.}. But the transformation is
also determined if we prescribe that a point z. on C sball couespond to
a point w. on C' and that a point z. not on C sha.1l be carried into a point
w: w. not on C'. We know then that z: (the symmetric point ,of z. with
respect to C) must correspond to (the symmetric point of Wt with
rwpect to C'). Hence the trllnsformation will be obtained from the
relation (w,w.,wo,w:) = (21,21,,210,21:).
EXERCISES
L Prove that every reflection carries circles into circles. •
ANALYTIC FUNCTION' AS MAPPINGS
2. Reflect the imaginary axis, the line :I: = 1/, and the circle Izi = 1
in the circle Iz - 21 = l.
I, Cany out the reflections in the preceding exercise by geometric construction.
4. Find the linear transform a.tion which ca.rries the circle Izi = 2 iuto
+ I. 11 - 1, the point - 2 into the origin, and the origin into i.
L Find the most general linear transforma.tion of the circle 1.1 = R
into itee\f. 8. Suppose that a linear transformation carries one pair of concentric
circles into another pair of concentric circles. Prove that the ratioe of the
radii must he the same.
7. Find a linear transformation which ca.rriea 1.1 - 1 and Iz - tl ~ {-
into concentric circles. What is the ratio of the radii?
L Same problem for Izi = 1 and z = 2.
.
11.4. Oriented C...",.... Because S(z) is analytic and
S'(.) = (aczd+-bde). ¢ 0
the mapping to .. S(.) is conformal fou ¢ -die and 00. ItIollows that
a pair of intersecting circles are ma.pped on circles that include the same
angle. In addition, the sense of an angle is preserved. From an intui-
tive point of view this means that right and left are preserved, but a more
precise formulation is desirable.
An orientation of a circle C is determined by an ordered triple of
points '.,Z.,ZI on C. With respect to this orientation a point" not on C
i8 nid to lie to the right of C if 1m (.".".".) > 0 and to the left of C is
1m (Il,","."') < 0 (this checks with everyday .IIIA beea".... (i,l,O, 00) = i).
It is eesential to show that there are only two different orientatioll8. By
this we mean that the distinction
left and right is the eame for all
triples, while the meaning may be revelsod. Since the Ct088 ratio is invari-
ant, it is sufficient to consider the ease where C is the real axis. Then
cz + /Ill +b
(z,...,.,,") =
d
coo be writtsn with real coefficients. and a simple calculation gives
•
1m (s,z.,'.,") = Ia..d+-bdel' 1m •.
We recognize tha~ the distinction between right and left is the same as
. half plene. Which is which
.:.;~..
..
',..-' «-" - '' -",.-'"'. '-'" ,... ..- .
.,' , :'. '.. -:< :.,-, ... - "':""",,'
'.
"
.-:.
,:
",:>
~-'~'=.i:~:,
";':J1.:".':~+&i[~~k:~cii':i'i~0~,,,,~'~
-~~;j.>~.~- ~,,
•
-.,:,·).~·:,.';i,:~~~~t:,:J.;,i1.~~ii.-:J.:'::.;:;;";;.~~:z,';{,L~
..COIIPLEX ANALYSIS
A linear transformation S carries the oriented circle C into a circle
which we orient through the triple SZl, &., Sz.. From the invariance of
the crOM ratio it follows that the left and right of C will be mapped on the
left and right of the image circle.
If two circles are tang<lnt to each other, their orientatioDJ! can be
compared. Indeed, we can use a linear transformation which throws their
common point to GO. The circles become parallel straight lines, and we
know how to compare the directions of parallel lines.
In the geometric
the orientation 11, Z., ZI can be indi-
cated by an arrow which points from 11 over z. to ZI. With the wrual
choice of the coordinate system left and right will have their customary
meaning with respect to this arrow.
When the finite plane is considered as part of the extended plane, the
point at infinity is distinguished. We can therefore define an absolute
positive orientation of all finite circles by the requirement that GO should
lie to the right of the oriented circles. The points to the left are said to
form the imitk of the circle and the points to the right form its outside.
EXERCISES
:a. If Z" eo, Z" z. are points on " circle, show that Z" Z" •• and ZI, ZI, t.
determine the same orientation if and only if (11,ZI,Z"I.) > o..
2. Prove ~hat a tangent to a circle is perpendicular to the radius
through the point of contact (in this connection a tangent should be defined
as a straight line with only one point in common with the circle).
.. Verily that the inside of the circle Iz - al - R is formed by all
points z with Iz - al < R.
.
4. The angle between two oriented circles at a point of inteIsection is
defined as the angle between the tangents at that point, equipped with the
orientation. Prove by analytic r8ll8ODing, rather thaD geometric
inspection, that the angles at the two points of intersection are opposite
to each other.
3.5. Familiett of arc".. A great deal be done toward the visual-·
ization of linear transformations by the introduction of certain families of circles which may be thought of as coordinate lines.. in a circular coordinate system.
Consider a Iioear transformation of the form
ID = k. I-a. z-b
Here z = a
= = to ID 0 and 1= b to ID GO. It follows that
the straight lines through the origin of the tD-p1ane are-images of the
ANALYTIC fUNCTIONS AS MAPPINGS
circles through a and b. On the other hand, the concentric circles about the origin, IIDI = p, correspond to circles with the equation
~-G
"z----"";:b
= pllll·
TheBe are the circle8 oj.ApolkmiWl with limit points a and b. By their equation they are the loci of points whoBe distances from a and b have
B constant ratio.
Denote by C, the cireles through A, b and by C. the cireles of Apol-
Ioniua with these limit points. The configuration (Fig. 3-3) formed by all
the circles C, and C. will be referred to 88 the circulaf- 11& or the 8teiner
circle8 determined by a and b. It has many interest.ing properties of
which we shalIli.t a few: 1. There is P.xa.,tly one C, and one C. through each point in the plane
with the exception of the limit pointe. 2. Every C, meets every C. under right angles. 3. Reflection in a C1 transforms every C. into itself and every C1 into
another C,. lUl1Iection in a C. transfomlS every C1 into itself and every C. into another C•.
4. The limit points are symmetric with respect to each C., but not
with l'68pect to any other cirele.
PIG. ..... Stain !I circlM.
.-, ,
.' .'.. . ". ., .
,- .. .
..... . "'- "" . ', '
"- '.' ,- -: '''--' ,-
•
COMPLEX ANALYSIS
These properties are all trivial when the limit points are 0 and GO I
a, a. i.e., when the are lines through the origin and the concentric
circles. Since the properties are invariant under linear transfonnations,
they must continue to hold in the general "&se.
lf a transfolmation UP = T. carries G, b into G', b' it can be written in
the f6rm
(12)
•
It iB clear that T transforms the circles C, and C. into circles C; and
a~ with the Ii mit point. a', b'.
The situation i8 particularly simple if G' = G, b' = b. Then a, b are
said to be fiud points of T, and it i8 convenient to represent" and Tz in
the· same plane. Under these circumstance. the whole circular net will
be mapped upon itself. The value of k serves to identify the image
a; a circles and c~. Indeed, with appropriate orientations 1 forms the
an«le arg k with its image a;, and the quotient of the oonBtant ratios
\. - al/lz - b\ on a; and a. iB Ik\.
The zpeeial c&ses in which all C1 or all C. are mAPped upon themselves
are particularly important. We have C: = C1 for all C1 if k > 0 (if
k < 0 the circles are still the same, but the orientation i. rev,,,seJ).
The transformation is then said to be hyperbolic. When k increases the
a points T" z ~ G, b, will flow along the circles 1 toward b. The con-
sideration of this flow provides a very elear picture of a hYPlrbolic
transfonnation.
,
a. The ~.,., C; = oecurs when Ik\ = 1. Transformations with thi8
property are called elliplic. When srgk varies, the points Tz move
along the eircles C.. The eorresponding flow circulates about G and b
in difierent directions.
The gtlnerallinear transformation with two fixed points is the product
of a hyperbolic and an elliptic transformation with the same fixed points.
The fixed. points of a linear transformation are found by solving the
equation
(13)
In gtlneral this is a quadratic equation with two roots; if ., = 0 one of the fixed points i. GO. It may happen, however, that the roots coincide. A linear tr&psformation with ooinciding fixed points i8 said to be parabolic.
The condition for this i. (a - ~)' = ~.,. If the equation (13) i8 found to have two distinct roots G and b. the
transformation can be written in the f o m . .
ANALYTIC FUNCTIONS AS IIApPING8 a7
v, -a =
v-b
k zz--ab.
We can then use the Steip.er circles determined by II, b to discuss the nature of the transformation. It is important to note, however, that the method is by DO means restricted to this CIlSe. We can write any linear transformation in the form (12) with arbitrary II, b and use the two circular nets to great advantage.
.For the discussion of parabolic transfOiDIatioDS it is desirable to intro-
duce still another type of circular net. Consider the transformation
.,
w = z-a +c.
It is evident that straight lines in the tD-plane conespond to eircles
through a; moreover, pareJIellines correspond to mutually tangent circles.
+ w In particular, if v = u
the lines u = constant and • - constant
correspond to two families of mutually tangent circles which intersect
at right angles (Fig. 3-4). This configuration can be considered 88 a
degenerate set of Steiner circles. It is determined by the Point II and
the tangent to one of the families of circles. We shall denote the images
of the lines v = constant by V .. the circles of the other family by V•.
Clearly, the line v ~ 1m c conesponds to the tangent of the circles 01;
its direction is given by arg <JI.
,.., ," ,
, . ..,'
J. ,~: ,.,~",;,
. .,
. .
"
.. . ...
•
COMPLEX ANALYSIS
Any tnmsfonnation which carries a into a' can be written in the form
'" =,.-,"--:a:.,
=
z
.,
-a
+
c.
It is clear that the circles C. end C. are carried into the circles C: and
C; determined by a' and ",'. We suppose now that a = a' is the only
fixed point. Then., = ",' and we can write
(14)
,.,-a - .. .. .. '+0. Z-II
.
By this transformation the configuration consisting of the circles C. and
C. is mapped upon itsell. In (14) a multiplicative factor is arbitrary, and we can hence suppose that 0 is real. Then every C. is mapped upon
itsell and the parabolic transfofillation can be comndered as a 80w along the circlesC . . - ·
A linear VaosfoIDlation that is neither hyperbolic, elliptic, nor
parabolic is to be
EX.RCISES
L Find the fixed points of the linear transformations
,., .. 2z - I'
2z ,., - 3z - l'
to
=
3z
z
-
-
14,
II!
to= 2-z.
Is any of these
elliptic, hYpc.'bolic, or parabolic?
:r. Suppose that the coefficients of the transfC1rmation
Sz.=aczz++bd
are normalised by ad. - be = 1. Show that S is elliptic if and only if
+ + + -2 < II d < 2, parabolic if a d ... ±2, hyperbolic if II d < -2
or >2.
L Show that a linear transformation which satisfies 8"1: .. z for
some integer n is
elliptic.
4. If S ill hyperbolic or loxodromic, show that Soz converges to a fixed
point as n"'" "', the Bame for all z, except, when z coincides with the other
fixed point. (The limit ill the aUractilJf1, the other the repellent fixed point.
What happens when n .... - "'? What happens in the parabolic case?)
L Find all linear transformations which
rotations of the
Riem"DD sphere.
L Find all circl!'8 which are orthogonal to \2\ -= 1 and \z - 1\ = ..