Table Of ContentS. Ponnusamy
Herb Silverman
Complex Variables
with Applications
Birkha¨user
Boston • Basel • Berlin
S.Ponnusamy HerbSilverman
IndianInstituteofTechnology,Madras CollegeofCharleston
DepartmentofMathematics DepartmentofMathematics
Chennai,600036 Charleston,SC29424
India U.S.A.
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To my father, Saminathan Pillai
—S. Ponnusamy
To my wife, Sharon Fratepietro
—Herb Silverman
Preface
The student, who seems to be engulfed in our culture of specialization, too
quicklyfeelsthenecessitytoestablishan“area”ofspecialinterest.Inkeeping
with this spirit, academic bureaucracy has often forced us into a compart-
mentalization of courses, which pretend that linear algebra is disjoint from
modern algebra, that probability and statistics can easily be separated, and
even that advanced calculus does not build from elementary calculus.
This book is written from the point of view that there is an interdepen-
dence between real and complex variables that should be explored at ev-
ery opportunity. Sometimes we will discuss a concept in real variables and
then generalize to one in complex variables. Other times we will begin with
a problem in complex variables and reduce it to one in real variables. Both
methods—generalization and specialization—are worthy of careful considera-
tion.
We expect “complex” numbers to be difficult to comprehend and “imag-
inary” units to be shrouded in mystery. Hopefully, by staying close to the
realfield,weshallovercomethisregrettableterminologythathasbeenthrust
upon us. The authors wish to create a spiraling effect that will first enable
thereadertodrawfromhisorherknowledgeofadvancedcalculusinorderto
demystifycomplexvariables,andthenusethisnewlyacquiredunderstanding
of complex variables to master some of the elements of advanced calculus.
Wewillalsocompare,wheneverpossible,theanalyticandgeometricchar-
acter of a concept. This naturally leads us to a discussion of “rigor”. The
current trend seems to be that anything analytic is rigorous and anything
geometric is not. This dichotomy moves some authors to strive for “rigor” at
the expense of rich geometric meaning, and other authors to endeavor to be
“intuitive” by discussing a concept geometrically without shedding any ana-
lyticlightonit.Rigor,astheauthorsseeit,isusefulonlyinsofarasitclarifies
rather than confounds. For this reason, geometry will be utilized to illustrate
analyticconcepts,andanalysiswillbeemployedtounravelgeometricnotions,
without regard to which approach is the more rigorous.
viii Preface
Sometimes, in an attempt to motivate, a discussion precedes a theorem.
Sometimes,inanattempttoilluminate,remarksaboutkeystepsandpossible
implicationsfollowatheorem.Noapologiesaremadeforthislackofterseness
surroundingdifficulttheorems.Whilebrevitymaybethesoulofwit,itisnot
the soul of insight into delicate mathematical concepts. In recognition of the
primary importance of observing relationships between different approaches,
some theorems are proved in several different ways. In this book, traveling
quickly to the frontiers of mathematical knowledge plays a secondary role to
the careful examination of the road taken and alternative routes that lead to
the same destination.
Awordshouldbesaidaboutthequestionsattheendofeachsection.The
authors feel deeply that mathematics should be questioned—not only for its
internal logic and consistency, but for the reasons we are led where we are.
Doestheconclusionseem“reasonable”?Didweexpectit?Didthestepsseem
naturalorartificial?Canwere-provetheresultadifferentway?Canwestate
intuitively what we have proved? Can we draw a picture?1
“Questions”, as used at the end of each section, cannot easily be catego-
rized. Some questions are simple and some are quite challenging; some are
specific and some are vague; some have one possible answer and some have
many; some are concerned with what has been proved and some foreshadow
what will be proved. Do all these questions have anything in common? Yes.
They are all meant to help the student think, understand, create, and ques-
tion. It is hoped that the questions will also be helpful to the teacher, who
may want to incorporate some of them into his or her lectures.
Less need be said about the exercises at the end of each section because
exercises have always received more favorable publicity than have questions.
Very often the difference between a question and an exercise is a matter of
terminology. The abundance of exercises should help to give the student a
good indication of how well the material in the section has been understood.
The prerequisite is at least a shaky knowledge of advanced calculus. The
first nine chapters present a solid foundation for an introduction to complex
variables. The last four chapters go into more advanced topics in some detail,
inordertoprovidethegroundworknecessaryforstudentswhowishtopursue
further the general theory of complex analysis.
If this book is to be used as a one-semester course, Chapters 5, 6, 7,
8, and 9 should constitute the core. Chapter 1 can be covered rapidly, and
the concepts in Chapter 2 need be introduced only when applicable in latter
chapters. Chapter 3 may be omitted entirely, and the mapping properties in
Chapter 4 may be omitted.
We wanted to write a mathematics book that omitted the word “trivial”.
Unfortunately, the Riemann hypothesis, stated on the last page of the text,
1 For an excellent little book elaborating on the relationship between questioning
and creative thinking, see G. Polya, How to Solve It, second edition, Princeton
University press, Princeton, New Jersey, 1957.
Preface ix
could not have been mentioned without invoking the standard terminology
dealing with the trivial zeros of the Riemann zeta function. But the spirit, if
nottheletter,ofthisdesirehasbeenfulfilled.Detailedexplanations,remarks,
worked-outexamplesandinsightsareplentiful.Theteachershouldbeableto
leavesectionsforthestudenttoreadonhis/herown;infact,thisbookmight
serve as a self-study text.
A teacher’s manual containing more detailed hints and solutions to ques-
tions and exercises is available. The interested teacher may contact us by
e-mail and receive a pdf version.
We wish to express our thanks to the Center for Continuing Education
at the Indian Institute of Technology Madras, India, for its support in the
preparation of the manuscript.
Finally, we thank Ann Kostant, Executive Editor, Birkha¨user, who has
been most helpful to the authors through her quick and efficient responses
throughout the preparation of this manuscript.
S. Ponnusamy
IIT Madras, India
Herb Silverman
June 2005 College of Charleston, USA
Contents
Preface ........................................................ vii
1 Algebraic and Geometric Preliminaries .................... 1
1.1 The Complex Field ...................................... 1
1.2 Rectangular Representation............................... 5
1.3 Polar Representation..................................... 15
2 Topological and Analytic Preliminaries .................... 25
2.1 Point Sets in the Plane................................... 25
2.2 Sequences .............................................. 32
2.3 Compactness............................................ 39
2.4 Stereographic Projection ................................. 44
2.5 Continuity.............................................. 48
3 Bilinear Transformations and Mappings ................... 61
3.1 Basic Mappings ......................................... 61
3.2 Linear Fractional Transformations ......................... 66
3.3 Other Mappings......................................... 85
4 Elementary Functions ..................................... 91
4.1 The Exponential Function ................................ 91
4.2 Mapping Properties......................................100
4.3 The Logarithmic Function ................................108
4.4 Complex Exponents .....................................114
5 Analytic Functions ........................................121
5.1 Cauchy–Riemann Equation ...............................121
5.2 Analyticity .............................................130
5.3 Harmonic Functions .....................................141
xii Contents
6 Power Series...............................................153
6.1 Sequences Revisited......................................153
6.2 Uniform Convergence ....................................164
6.3 Maclaurin and Taylor Series ..............................173
6.4 Operations on Power Series ...............................186
7 Complex Integration and Cauchy’s Theorem...............195
7.1 Curves .................................................195
7.2 Parameterizations .......................................207
7.3 Line Integrals ...........................................217
7.4 Cauchy’s Theorem.......................................226
8 Applications of Cauchy’s Theorem.........................243
8.1 Cauchy’s Integral Formula................................243
8.2 Cauchy’s Inequality and Applications ......................263
8.3 Maximum Modulus Theorem .............................275
9 Laurent Series and the Residue Theorem ..................285
9.1 Laurent Series ..........................................285
9.2 Classification of Singularities..............................293
9.3 Evaluation of Real Integrals ..............................308
9.4 Argument Principle......................................331
10 Harmonic Functions .......................................349
10.1 Comparison with Analytic Functions.......................349
10.2 Poisson Integral Formula .................................358
10.3 Positive Harmonic Functions..............................371
11 Conformal Mapping and the Riemann Mapping Theorem..379
11.1 Conformal Mappings.....................................379
11.2 Normal Families.........................................390
11.3 Riemann Mapping Theorem ..............................395
11.4 The Class S ............................................405
12 Entire and Meromorphic Functions ........................411
12.1 Infinite Products ........................................411
12.2 Weierstrass’ Product Theorem ............................422
12.3 Mittag-Leffler Theorem ..................................437
13 Analytic Continuation .....................................445
13.1 Basic Concepts..........................................445
13.2 Special Functions........................................458
References and Further Reading...............................473
Index of Special Notations.....................................475
Contents xiii
Index..........................................................479
Hints for Selected Questions and Exercises ....................485
