Complex Variables
Joseph L. Taylor
University of Utah
Contents
Preface
v
Chapter 1.
The Complex Numbers
1
1.1.
Definition and Simple Properties
1
1.2.
Convergence in C
8
1.3.
The Exponential Function
14
1.4.
Polar Form for Complex Numbers
19
Chapter 2.
Analytic Functions
27
2.1.
Continuous Functions
27
2.2.
The Complex Derivative
34
2.3.
Contour Integrals
41
2.4.
Properties of Contour Integrals
47
2.5.
Cauchy’s Integral Theorem for a Triangle
53
2.6.
Cauchy’s Theorem for a Convex Set
60
2.7.
Properties of the Index Function
66
Chapter 3.
Power Series Expansions
75
3.1.
Uniform Convergence
75
3.2.
Power Series Expansions
82
3.3.
Liouville’s Theorem
89
3.4.
Zeroes and Singularities
95
3.5.
The Maximum Modulus Principal
Chapter 4.
The General Cauchy Theorems
102
109
4.1.
Chains and Cycles
110
4.2.
Cauchy’s Theorems
117
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iv
Contents
4.3.
4.4.
4.5.
4.6.
Laurent Series
The Residue Theorem
Rouché’s Theorem and Inverse Functions
Homotopy
124
130
136
141
Chapter 5. Residue Theory
5.1. Computing Residues
5.2. Evaluating Integrals Using Residues
5.3. Fourier and Mellin Transforms
5.4. Summing Infinite Series
153
153
158
166
174
Chapter 6. Conformal Mappings
6.1. Definition and Examples
6.2. The Riemann Sphere
6.3. Linear Fractional Transformations
6.4. The Riemann Mapping Theorem
6.5. The Poisson Integral
6.6. The Dirichlet Problem
179
179
184
190
195
200
206
Chapter 7. Analytic Continuation
7.1. The Schwarz Reflection Principle
7.2. Continuation Along a Curve
7.3. Analytic Covering Maps
7.4. The Picard Theorems
213
213
218
224
228
Chapter 8. Infinite Products
8.1. Convergence of Infinite Products
8.2. Weierstrass Products
8.3. Entire Functions of Finite Order
8.4. Hadamard’s Factorization Theorem
235
235
241
248
252
Chapter 9. The Gamma and Zeta Functions
9.1. Euler’s Gamma Function
9.2. The Riemann Zeta Function
9.3. Properties of ζ
9.4. The Riemann Hypothesis and Prime Numbers
9.5. A Proof of the Prime Number Theorem
259
260
265
271
277
281
Bibliography
287
Preface
Basic complex variables is a very popular subject among mathematics faculty and
students. There is one big theorem (the Cauchy Integral Theorem) with a somewhat
difficult proof, but this is followed by a host of easy to prove consequences that are
both surprising and powerful. Furthermore, for undergraduates, the subject is
refreshingly new and different from the analysis courses which precede it.
Undergraduate complex variables is our favorite course to teach and we teach
it as often as we can. Over a period of years we developed notes for use in such a
course. The course is a one semester undergraduate course on complex variables at
the University of Utah. It is designed for junior level students who have completed
three semesters of calculus and have also had some linear algebra and at least one
semester of foundations of analysis.
Over the course of several years, these notes have been expanded and modified
to serve other audiences as well. We have used the expanded notes to teach a course
in applied complex variables for engineering students as well a course at the first
year graduate level for mathematics graduate students. The topics covered, the
emphasis given to topics, and the choice of exercises were different for each of these
audiences.
When taught as a one-semester junior level course for mathematics students,
the course is a transitional course between freshman and sophomore level calculus,
linear algebra, and differential equations and the much more sophisticated senior
level mathematics courses taught at Utah. The students are expected to understand
definitions and proofs and the exercises assigned will include proofs as well as
computations. The course moves at a leisurely pace and the material covered
includes only chapters 1, 2, and 3 and selected sections from chapters 4, 5, and 6.
A full year course could easily cover the entire text.
When we teach a one semester undergraduate course for engineers using these
notes, we cover essentially the same material as in the course for mathematics
majors, but not all the proofs are done in detail, and there is more emphasis on
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vi
Preface
computational examples and exercises. The applications in chapter 6 are given
special emphasis.
When taught as a one semester graduate course, we assume students have
some knowledge of complex numbers and so Chapter 1 is given just a brief review.
Chapters 2, 3, and 4 are covered completely, parts of Chapters 5 and 6 are covered,
along with Chapter 7. If time allows, Chapters 8 and 9 are summarized at the end
of the course. In the homework, the emphasis is on the theoretical exercises.
We have tried to present this material in a fashion which is both rigorous
and concise, with simple, straightforward explanations. We feel that the modern
tendency to expand textbooks with ever more material, excessively verbose explanations, and more and more bells and whistles, simply gets in the way of the
student’s understanding of the material.
The exercises differ widely in level of abstraction and level of difficulty. They
vary from the simple to the quite difficult and from the computational to the theoretical. There are exercises that ask students to prove something or to construct
an example with certain properties. There are exercises that ask students to apply
theoretical material to help do a computation or to solve a practical problem. Each
section contains a number of examples designed to illustrate the material of the
section and to teach students how to approach the exercises for that section.
This text, in its various incarnations, has been used by the author and his
colleagues for several years at the University of Utah. Each use has led to improvements, additions, and corrections.
The text begins, in Chapter 1, with a discussion of the fact that the real number
system is insufficient for some purposes (solving polynomial equations). The system
of complex numbers is developed in an attempt to remedy this problem. We then
study basic arithmetic of complex numbers, convergence of sequences and series of
complex numbers, power series, the exponential function, polar form for complex
numbers and the complex logarithm.
The core material of a complex variables course is the material covered here
in Chapters 2 and 3. Analytic functions are introduced in Chapter 2 as functions
which have a complex derivative. This leads to a discussion of the Cauchy-Riemann
equations and harmonic functions. We then introduce contour integrals and the
index function (winding number) for closed paths. We prove the Cauchy-integral
theorem for triangles and then for convex sets. We believe that this approach leads
to the simplest and quickest rigorous proof of a form of Cauchy’s theorem that can
be used to prove the existence of power series expansions for analytic functions.
The proof that analytic functions have power series expansions occurs early in
Chapter 3. This is followed by a wide range of powerful applications with simple
proofs – Morera’s Theorem, Liouville’s Theorem, The Fundamental Theorem of
Algebra, the characterization of zeroes and singularities of analytic functions, the
Maximum Modulus Principal, and Schwarz’s Lemma.
Chapter 4 begins with a proof of the general form of Cauchy’s Theorem and
Cauchy’s Formula. These theorems involve functions which are analytic on a general
open set. The integration takes place around a cycle (a generalization of a closed
path) which is required to have index zero about any point not in the set. We go on
Preface
vii
to study Laurent series, the Residue Theorem, Rouché’s Theorem, inverse functions
and the Open Mapping Theorem. The chapter ends with a discussion of homotopy
and its relationship to integrals around closed paths. We normally don’t cover this
last section in our undergraduate course, but it would certainly be appropriate for
a graduate course using this text.
Residue theory is covered in Chapter 5. We discuss techniques for computing
residues as well as a wide variety of applications of residue theory to problems
involving the calculation of integrals. We normally don’t cover all of this in our
undergraduate course. We typically skip the last two sections of this chapter in
favor of covering most of Chapter 6.
Chapter 6 deals with conformal maps. We prove the Riemann Mapping Theorem and show how it can be used to transform problems involving analytic or
harmonic functions on a simply connected subset of the plane to the analogous
problems on the unit disc. We use this technique to study the Dirichlet problem
on open, proper, simply connected subsets of C. We discuss applications to heat
flow, electrostatics, and hydrodynamics. This material is of particular interest to
students in a course on complex variables for engineering students.
The goal of Chapter 7 on analytic continuation is to prove the Picard theorems
concerning meromorphic functions at essential singularities. Along the way, we
prove the Schwarz Reflection Principle and the Monodromy Theorem, and discuss
lifting analytic functions through a covering map.
Chapters 8 and 9 are normally not covered in our one semester undergraduate
complex variables course. Some of the topics in these chapters are, however, often covered in the graduate course. In Chapter 8 we use Weierstrass products to
construct an analytic function with a given discrete set of zeroes, with prescribed
multiplicities. This is the Weierstrass Theorem. It leads directly to the Weierstrass
Factorization Theorem for entire functions. We also prove the Mittag-Leffler Theorem – which gives the existence of a meromorphic function with a prescribed set of
poles and principle parts. The final result of the chapter is the proof of Hadamard’s
Theorem characterizing entire functions of finite order. This is a key ingredient in
the proof of the Prime Number Theorem in Chapter 9.
In Chapter 9 we introduce the gamma and zeta functions, develop their basic
properties, discuss the Riemann Hypothesis, and prove the Prime Number Theorem. Course notes by our colleague Dragan Milićić provided the original basis
for this material. It then went through several years of expansion and refinement
before reaching its present form. There is a lot of technical calculation in this material and we do not cover it in our undergraduate course. However, a few lectures
summarizing this material has proved to be a popular way to end the graduate
course.
Several standard texts in complex variables and related topics were useful guides
in preparing this text. These may also be of interest to the student who wishes to
learn more about the subject. They are listed in a short bibliography at the end of
the text.