Complex Analysis: In the Spirit of Lipman Bers

229 Pages·2007·3.299 MB·English

by  Jane P. Gilman, Irwin Kra, Rubí E. Rodríguez

245
Graduate Texts in Mathematics
Editorial Board
S. Axler
K.A. Ribet
Graduate Texts in Mathematics
1 Takeuti/Zaring. Introduction to Axiomatic 39 Arveson. An Invitation to C-Algebras.
Set Theory. 2nd ed. 40 Kemeny/Snell/Knapp. Denumerable
2 Oxtoby. Measure and Category. 2nd ed. Markov Chains. 2nd ed.
3 Schaefer. Topological Vector Spaces. 41 Apostol. Modular Functions and Dirichlet
2nd ed. Series in Number Theory. 2nd ed.
4 Hilton/Stammbach. A Course in 42 J.-P. Serre. Linear Representations of Finite
Homological Algebra. 2nd ed. Groups.
5 Mac Lane. Categories for the Working 43 Gillman/Jerison. Rings of Continuous
Mathematician. 2nd ed. Functions.
6 Hughes/Piper. Projective Planes. 44 Kendig. Elementary Algebraic Geometry.
7 J.-P. Serre. A Course in Arithmetic. 45 Loève. Probability Theory I. 4th ed.
8 Takeuti/Zaring. Axiomatic Set Theory. 46 Loève. Probability Theory II. 4th ed.
9 Humphreys. Introduction to Lie Algebras 47 Moise. Geometric Topology in Dimensions
and Representation Theory. 2 and 3.
10 Cohen. A Course in Simple Homotopy 48 Sachs/Wu. General Relativity for
Theory. Mathematicians.
11 Conway. Functions of One Complex 49 Gruenberg/Weir. Linear Geometry.
Variable I. 2nd ed. 2nd ed.
12 Beals. Advanced Mathematical Analysis. 50 Edwards. Fermat’s Last Theorem.
13 Anderson/Fuller. Rings and Categories 51 Klingenberg. A Course in Differential
of Modules. 2nd ed. Geometry.
14 Golubitsky/Guillemin. Stable Mappings 52 Hartshorne. Algebraic Geometry.
and Their Singularities. 53 Manin. A Course in Mathematical Logic.
15 Berberian. Lectures in Functional Analysis 54 Graver/Watkins. Combinatorics with
and Operator Theory. Emphasis on the Theory of Graphs.
16 Winter. The Structure of Fields. 55 Brown/Pearcy. Introduction to Operator
17 Rosenblatt. Random Processes. 2nd ed. Theory I: Elements of Functional Analysis.
18 Halmos. Measure Theory. 56 Massey. Algebraic Topology: An
19 Halmos. A Hilbert Space Problem Book. Introduction.
2nd ed. 57 Crowell/Fox. Introduction to Knot Theory.
20 Husemoller. Fibre Bundles. 3rd ed. 58 Koblitz. p-adic Numbers, p-adic Analysis,
21 Humphreys. Linear Algebraic Groups. and Zeta-Functions. 2nd ed.
22 Barnes/Mack. An Algebraic Introduction 59 Lang. Cyclotomic Fields.
to Mathematical Logic. 60 Arnold. Mathematical Methods in
23 Greub. Linear Algebra. 4th ed. Classical Mechanics. 2nd ed.
24 Holmes. Geometric Functional Analysis 61 Whitehead. Elements of Homotopy
and Its Applications. Theory.
25 Hewitt/Stromberg. Real and Abstract 62 Kargapolov/Merizjakov. Fundamentals
Analysis. of the Theory of Groups.
26 Manes. Algebraic Theories. 63 Bollobas. Graph Theory.
27 Kelley. General Topology. 64 Edwards. Fourier Series. Vol. I. 2nd ed.
28 Zariski/Samuel. Commutative Algebra. Vol. I. 65 Wells. Differential Analysis on Complex
29 Zariski/Samuel. Commutative Algebra. Vol. II. Manifolds. 2nd ed.
30 Jacobson. Lectures in Abstract Algebra I. 66 Waterhouse. Introduction to Affi ne Group
Basic Concepts. Schemes.
31 Jacobson. Lectures in Abstract Algebra II. 67 Serre. Local Fields.
Linear Algebra. 68 Weidmann. Linear Operators in Hilbert
32 Jacobson. Lectures in Abstract Algebra III. Spaces.
Theory of Fields and Galois Theory. 69 Lang. Cyclotomic Fields II.
33 Hirsch. Differential Topology. 70 Massey. Singular Homology Theory.
34 Spitzer. Principles of Random Walk. 2nd ed. 71 Farkas/Kra. Riemann Surfaces. 2nd ed.
35 Alexander/Wermer. Several Complex 72 Stillwell. Classical Topology and
Variables and Banach Algebras. 3rd ed. Combinatorial Group Theory. 2nd ed.
36 Kelley/Namioka et al. Linear Topological 73 Hungerford. Algebra.
Spaces. 74 Davenport. Multiplicative Number Theory.
37 Monk. Mathematical Logic. 3rd ed.
38 Grauert/Fritzsche. Several Complex 75 Hochschild. Basic Theory of Algebraic
Variables. Groups and Lie Algebras.
(continued after index)
Jane P. Gilman
Irwin Kra
Rubí E. Rodríguez
Complex Analysis
In the Spirit of Lipman Bers
Jane P. Gilman Irwin Kra Rubí E. Rodríguez
Department of Math for America Pontifi cia Universidad
Mathematics and 50 Broadway, 23 Floor Católica de Chile
Computer Science New York, NY 10004 Santiago
Rutgers University and Chile
Newark, NJ 07102 Department of Mathematics [email protected]
USA State University of New York
[email protected] at Stony Brook
Stony Brook, NY 11794
USA
[email protected]
Editorial Board
S. Axler K.A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]
ISBN: 978-0-387-74714-9 e-ISBN: 978-0-387-74715-6
Library of Congress Control Number: 2007940048
Mathematics Subject Classifi cation (2000): 30-xx 32-xx
© 2007 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written
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Printed on acid-free paper.
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To the memory of
Mary and Lipman Bers
Preface
This book presents fundamental material that should be part of the
education of every practicing mathematician. This material will also
be of interest to computer scientists, physicists, and engineers.
Complex analysis is also known as function theory. In this text
we address the theory of complex-valued functions of a single complex
variable. This is a prerequisite for the study of many current and
rapidlydeveloping areasofmathematics, includingthetheoryofseveral
andinfinitely many complex variables, the theory of groups, hyperbolic
geometry and three-manifolds, and number theory. Complex analysis
has connections and applications to many other many other subjects
in mathematics, and also to other sciences as an area where the classic
and the modern techniques meet and benefit from each other. We will
try to illustrate this in the applications we give.
Because function theory has been used by generations of practicing
mathematicians working in a number of different fields, the basic re-
sults have been developed and redeveloped from a number of different
perspectives. We are not wedded to any one viewpoint. Rather we
will try to exploit the richness of the subject and explain and interpret
standard definitions and results using the most convenient tools from
analysis, geometry, and algebra.
The key first step in the theory is to extend the concept of dif-
ferentiability from real-valued functions of a real variable to complex-
valued functions of a complex variable. Although the definition of com-
plex differentiability resembles the definition of real differentiability, its
consequences are profoundly different. A complex-valued function of a
complex variable that is differentiable iscalled holomorphic oranalytic,
and the first part of this book is a study of the many equivalent ways of
understanding the concept of analyticity. Many of the equivalent ways
of formulating the concept of an analytic function are summarized in
what we term the Fundamental Theorem for functions of a complex
variable. Chapter 1 begins with two motivating examples, followed by
the statement of the Fundamental Theorem, an outline of the plan for
vii
viii PREFACE
proving it, and a description of the text contents: the plan for the rest
of the book.
In devoting the first part of this book to the precise goal of stating
andproving the Fundamental Theorem, we follow a path charted for us
by Lipman Bers, from whom we learned the subject. In his teaching,
expository, andresearchwritingheoftenstartedbyintroducingamain,
often technical, result and then proceeded to derive its important and
seemingly surprising consequences. Some of the grace and elegance
of this subject will not emerge until a more technical framework has
been established. In the second part of the text, we proceed to the
leisurely exploration of interesting ramifications and applications of
the Fundamental Theorem.
We are grateful to Lipman Bers for introducing us to the beauty
of the subject. The book is an outgrowth of notes from Bers’s original
lectures. Versions of these notes have been used by us at our respective
home institutions, some for more than 20 years, as well as by others at
various universities. We are grateful to many colleagues and students
who read and commented on these notes. Our interaction with them
helpedshapethisbook. Wetriedtofollowalluseful adviceandcorrect,
of course, any mistakes or shortcomings they identified. Those that
remain are entirely our responsibility.
Jane Gilman
Irwin Kra
Rub´ı E. Rodr´ıguez
June 2007
Standard Notation and Commonly Used Symbols
A LIST OF SYMBOLS
TERM MEANING
Z integers
Q rationals
R reals
C complex numbers
(cid:2)
C C∪{∞}
ı a square root of −1
ıR the imaginary axis in C
(cid:4)z real part of z
(cid:5)z imaginary part of z
z = x+ıy x = (cid:4)z and y = (cid:5)z
z¯ conjugate of z
r = |z| absolute value of z
θ = arg z an argument of z
z = reıθ r = |z| and θ = arg z
u , u real partial derivatives
x y
u , u complex partial derivatives
z z¯
∂f partial derivative
∂x
∂R boundary of set R
|R| cardinality of set R
clR closure of set R
intR interior of set R
X the set of x ∈ X that satisfy condition
condition
ν (f) order of the function f at the point ζ
ζ
i(γ) interior of the Jordan curve γ
e(γ) exterior of the Jordan curve γ
f| restriction of the function f to the subset B
B
of its domain
U(z,r) = U (r) {ζ ∈ C; |ζ −z| < r}
z
D U(0,1)
H2 {z ∈ C; (cid:5)z > 0}
ix
x STANDARD NOTATION AND COMMONLY USED SYMBOLS
STANDARD TERMINOLOGY
TERM MEANING
LHS left-hand side
RHS right-hand side
deleted neighborhood of z neighborhood with z removed
CR Cauchy Riemann equations
⊂ proper subset
⊆ subset, may not be proper
d Euclidean distance on C
ρ hyperbolic distance on D
D
MMP Maximum Modulus Property
MVP Mean Value Property
Contents
Preface vii
Standard Notation and Commonly Used Symbols ix
Chapter 1. The Fundamental Theorem in Complex Function
Theory 1
1.1. Some motivation 1
1.2. The Fundamental Theorem 3
1.3. The plan for the proof 4
1.4. Outline of text 5
Chapter 2. Foundations 7
2.1. Introduction and preliminaries 7
2.2. Differentiability and holomorphic mappings 14
Exercises 18
Chapter 3. Power Series 23
3.1. Complex power series 24
3.2. More on power series 32
3.3. The exponential function, the logarithm function,
and some complex trigonometric functions 36
3.4. An identity principle 42
3.5. Zeros and poles 47
Exercises 52
Chapter 4. The Cauchy Theory–A Fundamental Theorem 59
4.1. Line integrals and differential forms 60
4.2. The precise difference between closed and exact forms 65
4.3. Integration of closed forms and the winding number 70
4.4. Homotopy and simple connectivity 72
4.5. Winding number 75
4.6. Cauchy Theory: initial version 78
Exercises 80
Chapter 5. The Cauchy Theory–Key Consequences 83
5.1. Consequences of the Cauchy Theory 83
xi