Complex Analysis

552 Pages·2005·5.14 MB·English

by  Professor Dr. Eberhard Freitag

Eberhard Freitag
Rolf Busam
Complex Analysis
ABC
ProfessorDr.EberhardFreitag Translator
Dr.RolfBusam Dr.DanFulea
FacultyofMathematics FacultyofMathematics
InstituteofMathematics InstituteofMathematics
UniversityofHeidelberg UniversityofHeidelberg
ImNeuenheimerFeld288 ImNeuenheimerFeld288
69120Heidelberg 69120Heidelberg
Germany Germany
E-mail:[email protected] E-mail:[email protected]
[email protected]
MathematicsSubjectClassification(2000):30-01,11-01,11F11,11F66,11M45,11N05,
30B50,33E05
LibraryofCongressControlNumber:2005930226
ISBN-10 3-540-25724-1SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-25724-0SpringerBerlinHeidelbergNewYork
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liableforprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
springeronline.com
(cid:1)c Springer-VerlagBerlinHeidelberg2005
PrintedinTheNetherlands
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Typesetting:bytheauthorsandTechBooksusingaSpringerLATEXmacropackage
Coverdesign:design&productionGmbH,Heidelberg
Printedonacid-freepaper SPIN:11396024 40/TechBooks 543210
In Memoriam
Hans Maaß
(1911–1992)
Preface to the English Edition
This book is a translation of the forthcoming fourth edition of our German
book “Funktionentheorie I” (Springer 2005). The translation and the LATEX
files have been produced by Dan Fulea. He also made a lot of suggestions for
improvementwhichinfluencedtheEnglishversionofthebook.Itisapleasure
for us to express to him our thanks. We also want to thank our colleagues
DiarmuidCrowley,WinfriedKohnenandJo¨rgSixtforusefulsuggestionscon-
cerning the translation.
Over the years, a great number of students, friends, and colleagues have con-
tributed many suggestions and have helped to detect errors and to clear the
text.
The many new applications and exercises were completed in the last decade
to also allow a partial parallel approach using computer algebra systems and
graphictools,whichmayhaveafruitful,powerfulimpactespeciallyincomplex
analysis.
Last but not least, we are indebted to Clemens Heine (Springer, Heidelberg),
who revived our translation project initially started by Springer, New York,
and brought it to its final stage.
Heidelberg, Easter 2005 Eberhard Freitag
Rolf Busam
Contents
I Differential Calculus in the Complex Plane C ............ 9
I.1 Complex Numbers ................................... 9
I.2 Convergent Sequences and Series....................... 24
I.3 Continuity .......................................... 36
I.4 Complex Derivatives ................................. 42
I.5 The Cauchy–Riemann Differential Equations .......... 48
II Integral Calculus in the Complex Plane C................ 71
II.1 Complex Line Integrals ............................... 72
II.2 The Cauchy Integral Theorem ........................ 79
II.3 The Cauchy Integral Formulas........................ 94
III Sequences and Series of Analytic Functions, the Residue
Theorem.................................................105
III.1 Uniform Approximation ..............................106
III.2 Power Series.........................................111
III.3 Mapping Properties for Analytic Functions..............126
III.4 Singularities of Analytic Functions .....................136
III.5 Laurent Decomposition .............................145
A Appendix to III.4 and III.5............................158
III.6 The Residue Theorem ................................165
III.7 Applications of the Residue Theorem ...................174
IV Construction of Analytic Functions ......................195
IV.1 The Gamma Function ................................196
IV.2 The Weierstrass Product Formula ...................214
IV.3 The Mittag–Leffler Partial Fraction Decomposition ...223
IV.4 The Riemann Mapping Theorem ......................228
A Appendix : The Homotopical Version of the Cauchy
Integral Theorem ....................................239
B Appendix : The Homological Version of the Cauchy
Integral Theorem ....................................244
X Contents
C Appendix : Characterizations of Elementary Domains ....249
V Elliptic Functions ........................................257
V.1 The Liouville Theorems.............................258
A Appendix to the Definition of the Periods Lattice ........265
V.2 The Weierstrass ℘-function .........................267
V.3 The Field of Elliptic Functions.........................274
A Appendix to Sect. V.3 : The Torus as an Algebraic Curve .279
V.4 The Addition Theorem ...............................287
V.5 Elliptic Integrals .....................................292
V.6 Abel’s Theorem.....................................299
V.7 The Elliptic Modular Group...........................310
V.8 The Modular Function j ..............................319
VI Elliptic Modular Forms ..................................327
VI.1 The Modular Group and Its Fundamental Region ........328
VI.2 The k/12-formula and the Injectivity
of the j-function .....................................335
VI.3 The Algebra of Modular Forms ........................345
VI.4 Modular Forms and Theta Series.......................348
VI.5 Modular Forms for Congruence Groups .................362
A Appendix to VI.5 : The Theta Group...................374
VI.6 A Ring of Theta Functions ............................381
VII Analytic Number Theory ................................391
VII.1 Sums of Four and Eight Squares .......................392
VII.2 Dirichlet Series ....................................409
VII.3 Dirichlet Series with Functional Equations ............418
VII.4 The Riemann ζ-function and Prime Numbers ...........431
VII.5 The Analytic Continuation of the ζ-function.............439
VII.6 A Tauberian Theorem ...............................446
VIII Solutions to the Exercises................................459
VIII.1 Solutions to the Exercises of Chapter I..................459
VIII.2 Solutions to the Exercises of Chapter II.................471
VIII.3 Solutions to the Exercises of Chapter III ................476
VIII.4 Solutions to the Exercises of Chapter IV ................488
VIII.5 Solutions to the Exercises of Chapter V.................496
VIII.6 Solutions to the Exercises of Chapter VI ................505
VIII.7 Solutions to the Exercises of Chapter VII ...............513
References.....................................................523
Symbolic Notations............................................533
Index..........................................................535
Introduction
The complex numbers have their historical origin in the 16th century when
theywerecreatedduringattemptstosolvealgebraic equations.G. C√ardano
(1545)hasalreadyintroducedformalexpressionsasforinstance5± −15,in
order to express solutions of quadratic and cubic equations. Around 1560 R.
Bombelli computed systematically using such expressions and found 4 as a
solution of the equation x3 =15x+4 in the disguised form
(cid:1) (cid:1)
√ √
4= 3 2+ −121+ 3 2− −121 .
AlsointheworkofG.W. Leibniz(1675)onecanfindequationsofthiskind,
e.g. (cid:1) (cid:1)
√ √ √
1+ −3+ 1− −3= 6 .
√
Intheyear1777L.Eulerintroducedthenotationi= −1fortheimaginary
unit.
The terminology “complex number” is due to C.F. Gauss (1831). The rig-
orous introduction of complex numbers as pairs of real numbers goes back to
W.R. Hamilton (1837).
Sometimes it is already advantageous to introduce and make use of complex
numbers in real analysis. One should for example think of the integration of
rational functions, which is based on the partial fraction decomposition, und
therefore on the Fundamental Theorem of Algebra:
Over the field of complex numbers
any polynomial decomposes as a product of linear factors.
AnotherexampleforthefruitfuluseofcomplexnumbersisrelatedtoFourier
series.FollowingEuler(1748)onecancombinetherealangularfunctionssine
and cosine, and obtain the “exponential function”
eix :=cosx+isinx .
Then the addition theorems for sine and cosine reduce to the simple formula
2 Introduction
ei(x+y) =eixeiy .
In particular, (cid:2) (cid:3)
eix n =einx holds for all integers n .
The Fourier series of a sufficiently smooth function f, defined on the real
line, with period 1, can be written in terms of such expressions as
(cid:4)∞
f(x)= a e2πinx .
n
n=−∞
Here it is irrelevant whether f is real or complex valued.
In these examples the complex numbers serve as useful, but ultimatively dis-
pensabletools.Newaspectscomeintoplaywhenweconsidercomplexvalued
functions depending on a complex variable, that is when we start to study
functionsf :D →Cwithtwo-dimensionaldomainsD systematically.Thedi-
mensiontwoisensuredwhenwerestricttoopendomainsofdefinition D ⊂C.
Analogouslytothesituationinrealanalysisoneintroducesthenotionofcom-
plex differentiability by requiring the existence of the limit
f(z)−f(a)
f(cid:3)(a):= lim
z→a z−a
z(cid:5)=a
foralla∈D.Itturnsoutthatthisnotionbehavesmuchmoredrasticallythen
realdifferentiability.Wewillshowforinstancethata(firstorder)complexdif-
ferentiable function is automatically arbitrarily often complex differentiable.
We will see more, namely that complex differentiable functions can always
be developed locally as power series. For this reason, complex differentiable
functions (defined on open domains) are also called analytic functions.
“Complex analysis” is the theory of such analytic functions.
Many classical functions from real analysis can be analytically extended to
complexanalysis.Itturnsoutthattheseextensionsareunique,asforinstance
in the case
ex+iy :=exeiy .
From the relation
e2πi =1
it follows that the complex exponential function is periodic with the purely
imaginaryperiod2πi.Thisobservationisfundamentalforthecomplexanaly-
sis. As a consequence one can observe further phenomena:
1. The complex logarithm cannot be introduced as the unique inverse
functionoftheexponentialfunctioninanaturalway.Itisa priorideter-
mined only up to a multiple of 2πi.
Introduction 3
2. The function 1/z (z (cid:5)= 0) does not have any primitive in the punc-
tured complex plane. A related fact is the following: the path integral
of 1/z with respect to a circle line centered in the origin and oriented
anticlockwise yields the non-zero value
(cid:5)
1
dz = 2πi (r >0) .
z
|z|=r
Centralresultsofcomplexanalysis,likee.g.theResidueTheorem,arenothing
but a highly generalized version of these statements.
Realfunctionsoftenshowtheirtruenaturefirstafterconsideringtheiranalytic
extensions. For instance, in the real theory it is not directly transparent why
the power series representation
1
=1−x2+x4−x6±···
1+x2
is valid only for |x| < 1. In the complex theory this phenomenon becomes
moreunderstandable,simplybecausetheconsideredfunctionhassingularities
in ±i. Then its power series representation is valid in the biggest open disk
excluding the singularities, namely the unit disk.
IntherealtheoryitisalsohardtounderstandwhytheTaylorseriesaround
0 of the C∞ function
(cid:6)
e−1/x2 , x(cid:5)=0 ,
f(x)=
0 , x=0 ,
convergesforallx∈R,butdoesnotrepresentthefunctioninanypointother
than zero. In the complex theory this phenomenon becomes understandable,
because the function e−1/z2 has an essential singularity in zero.
Lesstrivialexamplesaremoreimpressive.Here,oneshouldmentiontheRie-
mann ζ-function
(cid:4)∞
ζ(s)= n−s ,
n=1
whichwillbeextensivelystudiedinthelastchapterofthebookasafunctionof
thecomplex variabletraditionallydenotedbysusingthemethodsofcomplex
analysis, which will be presented throughout the preceeding chapters. From
the analytical properties of the ζ-function we will deduce the Prime Number
Theorem.
Riemann’s celebrated work on the ζ function [Ri2] is a brilliant example for
thethesishealreadypresentedeightyearsinadvanceinhisdissertation[Ri1]