Complex variables

213 Pages·2004·1.199 MB·English

by  Ash R.B., Novinger W.P.

Complex Variables
by R. B. Ash and W.P. Novinger
Preface
This book represents a substantial revision of the first edition which was published in
1971. Most of the topics of the original edition have been retained, but in a number of
instances the material has been reworked so as to incorporate alternative approaches to
these topics that have appeared in the mathematical literature in recent years.
Thebookisintendedasatext,appropriateforusebyadvancedundergraduatesorgradu-
atestudentswhohavetakenacourseinintroductoryrealanalysis,orasitisoftencalled,
advancedcalculus. Nobackgroundincomplexvariablesisassumed,thusmakingthetext
suitable for those encountering the subject for the first time. It should be possible to
cover the entire book in two semesters.
Thelistbelowenumeratesmanyofthemajorchangesand/oradditionstothefirstedition.
1. The relationship between real-differentiability and the Cauchy-Riemann equations.
2. J.D. Dixon’s proof of the homology version of Cauchy’s theorem.
3. The use of hexagons in tiling the plane, instead of squares, to characterize simple
connectedness in terms of winding numbers of cycles. This avoids troublesome details
that appear in the proofs where the tiling is done with squares.
4. Sandy Grabiner’s simplified proof of Runge’s theorem.
5. A self-contained approach to the problem of extending Riemann maps of the unit disk
to the boundary. In particular, no use is made of the Jordan curve theorem, a difficult
theorem which we believe to be peripheral to a course in complex analysis. Several
applications of the result on extending maps are given.
6. D.J. Newman’s proof of the prime number theorem, as modified by J. Korevaar, is
presentedinthelastchapterasameansofcollectingandapplyingmanyoftheideasand
results appearing in earlier chapters, while at the same time providing an introduction to
several topics from analytic number theory.
For the most part, each section is dependent on the previous ones, and we recommend
that the material be covered in the order in which it appears. Problem sets follow most
sections, with solutions provided (in a separate section).
1
2
We have attempted to provide careful and complete explanations of the material, while
at the same time maintaining a writing style which is succinct and to the point.
(cid:1)c Copyright 2004 by R.B. Ash and W.P. Novinger. Paper or electronic copies for non-
commercial use may be made freely without explicit permission of the authors. All other
rights are reserved.
Complex Variables
by Robert B. Ash and W.P. Novinger
Table Of Contents
Chapter 1: Introduction
1.1 Basic Definitions
1.2 Further Topology of the Plane
1.3 Analytic Functions
1.4 Real-Differentiability and the Cauchy-Riemann Equations
1.5 The Exponential Function
1.6 Harmonic Functions
Chapter 2: The Elementary Theory
2.1 Integration on Paths
2.2 Power Series
2.3 The Exponential Function and the Complex Trigonometric Functions
2.4 Further Applications
Chapter 3: The General Cauchy Theorem
3.1 Logarithms and Arguments
3.2 The Index of a Point with Respect to a Closed Curve
3.3 Cauchy’s Theorem
3.4 Another Version of Cauchy’s Theorem
Chapter 4: Applications of the Cauchy Theory
4.1 Singularities
4.2 Residue Theory
4.3 The Open mapping Theorem for Analytic Functions
4.4 Linear Fractional Transformations
4.5 Conformal Mapping
4.6 Analytic Mappings of One Disk to Another
1
2
4.7 The Poisson Integral formula and its Applications
4.8 The Jensen and Poisson-Jensen Formulas
4.9 Analytic Continuation
Chapter 5: Families of Analytic Functions
5.1 The Spaces A(Ω) and C(Ω)
5.2 The Riemann Mapping Theorem
5.3 Extending Conformal Maps to the Boundary
Chapter 6: Factorization of Analytic Functions
6.1 Infinite Products
6.2 Weierstrass Products
6.3 Mittag-Leffler’s Theorem and Applications
The Prime Number Theorem
7.1 The Riemann Zeta Function
7.2 An Equivalent Version of the Prime Number Theorem
7.3 Proof of the Prime Number Theorem
Chapter 1
Introduction
The reader is assumed to be familiar with the complex plane C to the extent found in
most college algebra texts, and to have had the equivalent of a standard introductory
course in real analysis (advanced calculus). Such a course normally includes a discussion
of continuity, differentiation, and Riemann-Stieltjes integration of functions from the real
line to itself. In addition, there is usually an introductory study of metric spaces and the
associated ideas of open and closed sets, connectedness, convergence, compactness, and
continuity of functions from one metric space to another. For the purpose of review and
to establish notation, some of these concepts are discussed in the following sections.
1.1 Basic Definitions
The complex plane C is the set of all ordered pairs (a,b) of real numbers, with addition
and multiplication defined by
(a,b)+(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac−bd,ad+bc).
If i = (0,1) and the real number a is identified with (a,0), then (a,b) = a+bi. The
expression a+bi can be manipulated as if it were an ordinary binomial expression of real
numbers, subject to the relation i2 = −1. With the above definitions of addition and
multiplication, C is a field.
If z = a+bi, then a is called the real part of z, written a = Rez, and b is called the
imaginary part of z, written b = Imz. The absolute value or magnitude or modulus of z
is defined as (a2+b2)1/2. A complex number with magnitude 1 is said to be unimodular.
Anargument ofz (writtenargz)isdefinedastheanglewhichthelinesegmentfrom(0,0)
to(a,b)makeswiththepositiverealaxis. Theargumentisnotunique,butisdetermined
up to a multiple of 2π.
If r is the magnitude of z and θ is an argument of z, we may write
z =r(cosθ+isinθ)
and it follows from trigonometric identities that
|z z |=|z ||z | and arg(z z )=argz −argz
1 2 1 2 1 2 1 2
1
2 CHAPTER 1. INTRODUCTION
(that is, if θ is an argument of z ,k = 1,2, then θ +θ is an argument of z z ). If
k k 1 2 1 2
z (cid:3)=0, then arg(z /z )=arg(z )+arg(z ). If z =a+bi, the conjugate of z is defined as
2 1 2 1 2
z =a−bi, and we have the following properties:
|z|=|z|, argz =−argz, z +z =z +z , z −z =z −z ,
1 2 1 2 1 2 1 2
z z =z z , Rez =(z+z)/2, Imz =(z−z)/2i, zz =|z|2.
1 2 1 2
The distance between two complex numbers z and z is defined as d(z ,z )=|z −z |.
1 2 1 2 1 2
So d(z ,z ) is simply the Euclidean distance between z and z regarded as points in
1 2 1 2
the plane. Thus d defines a metric on C, and furthermore, d is complete, that is, every
Cauchy sequence converges. If z ,z ,... is sequence of complex numbers, then z →z if
1 2 n
and only if Rez →Rez and Imz →Imz. We say that z →∞ if the sequence of real
n n n
numbers |z | approaches +∞.
n
Manyoftheaboveresultsareillustratedinthefollowinganalyticalproofofthetriangle
inequality:
|z +z |≤|z |+|z | for all z ,z ∈C.
1 2 1 2 1 2
The geometric interpretation is that the length of a side of a triangle cannot exceed the
sum of the lengths of the other two sides. See Figure 1.1.1, which illustrates the familiar
representation of complex numbers as vectors in the plane.
(cid:2)(cid:2)(cid:2)z(cid:2)1(cid:2)+(cid:2)z(cid:2)(cid:2)(cid:2)2(cid:2)(cid:3)(cid:2)(cid:3)(cid:3)(cid:2)(cid:3)(cid:2)(cid:3)z(cid:2)(cid:3)2(cid:2)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)
z1
Figure 1.1.1
The proof is as follows:
|z +z |2 =(z +z )(z +z )=|z |2+|z |2+z z +z z
1 2 1 2 1 2 1 2 1 2 1 2
=|z |2+|z |2+z z +z z =|z |2+|z |2+2Re(z z )
1 2 1 2 1 2 1 2 1 2
≤|z |2+|z |2+2|z z |=(|z |+|z |)2.
1 2 1 2 1 2
The proof is completed by taking the square root of both sides.
If a and b are complex numbers, [a,b] denotes the closed line segment with endpoints
a and b. If t and t are arbitrary real numbers with t <t , then we may write
1 2 1 2
t−t
[a,b]={a+ 1 (b−a):t ≤t≤t }.
t −t 1 2
2 1
The notation is extended as follows. If a ,a ,... ,a are points in C, a polygon from
1 2 n+1
a to a (or a polygon joining a to a ) is defined as
1 n+1 1 n+1
(cid:2)n
[a ,a ],
j j+1
j=1
often abbreviated as [a ,... ,a ].
1 n+1
1.2. FURTHER TOPOLOGY OF THE PLANE 3
1.2 Further Topology of the Plane
Recall that two subsets S and S of a metric space are separated if there are open sets
1 2
G ⊇ S and G ⊇ S such that G ∩ S = G ∩ S = ∅, the empty set. A set is
1 1 2 2 1 2 2 1
connected iff it cannot be written as the union of two nonempty separated sets. An open
(respectively closed) set is connected iff it is not the union of two nonempty disjoint open
(respectively closed) sets.
1.2.1 Definition
A set S ⊆C is said to be polygonally connected if each pair of points in S can be joined
by a polygon that lies in S.
Polygonal connectedness is a special case of path (or arcwise) connectedness, and it
follows that a polygonally connected set, in particular a polygon itself, is connected. We
will prove in Theorem 1.2.3 that any open connected set is polygonally connected.
1.2.2 Definitions
If a ∈ C and r > 0, then D(a,r) is the open disk with center a and radius r; thus
D(a,r) = {z : |z−a| < r}. The closed disk {z : |z−a| ≤ r} is denoted by D(a,r), and
C(a,r) is the circle with center a and radius r.
1.2.3 Theorem
If Ω is an open subset of C, then Ω is connected iff Ω is polygonally connected.
Proof. If Ω is connected and a ∈ Ω, let Ω be the set of all z in Ω such that there is a
1
polygoninΩfromatoz,andletΩ =Ω\Ω . Ifz ∈Ω ,thereisanopendiskD(z,r)⊆Ω
2 1 1
(because Ω is open). If w ∈ D(z,r), a polygon from a to z can be extended to w, and
it follows that D(z,r) ⊆ Ω , proving that Ω is open. Similarly, Ω is open. (Suppose
1 1 2
z ∈Ω , and choose D(z,r)⊆Ω. Then D(z,r)⊆Ω as before.)
2 2
Thus Ω and Ω are disjoint open sets, and Ω (cid:3)= ∅ because a ∈ Ω . Since Ω is
1 2 1 1
connected we must have Ω = ∅, so that Ω = Ω. Therefore Ω is polygonally connected.
2 1
The converse assertion follows because any polygonally connected set is connected. ♣
1.2.4 Definitions
A region in C is an open connected subset of C. A set E ⊆ C is convex if for each pair
of points a,b ∈ E, we have [a,b] ⊆ E; E is starlike if there is a point a ∈ E (called a
star center) such that [a,z] ⊆ E for each z ∈ E. Note that any nonempty convex set is
starlike and that starlike sets are polygonally connected.
4 CHAPTER 1. INTRODUCTION
1.3 Analytic Functions
1.3.1 Definition
Let f : Ω → C, where Ω is a subset of C. We say that f is complex-differentiable at the
point z ∈Ω if for some λ∈C we have
0
f(z +h)−f(z )
lim 0 0 =λ (1)
h→0 h
or equivalently,
f(z)−f(z )
lim 0 =λ. (2)
z→z0 z−z0
Conditions (3), (4) and (5) below are also equivalent to (1), and are sometimes easier to
apply.
f(z +h )−f(z )
lim 0 n 0 =λ (3)
n→∞ hn
for each sequence {h } such that z +h ∈Ω\{z } and h →0 as n→∞.
n 0 n 0 n
f(z )−f(z )
lim n 0 =λ (4)
n→∞ zn−z0
for each sequence {z } such that z ∈Ω\{z } and z →z as n→∞.
n n 0 n 0
f(z)=f(z )+(z−z )(λ+(cid:28)(z)) (5)
0 0
for all z ∈Ω, where (cid:28):Ω→C is continuous at z and (cid:28)(z )=0.
0 0
To show that (1) and (5) are equivalent, just note that (cid:28) may be written in terms of
f as follows:
(cid:3)
f(z)−f(z0) −λ if z (cid:3)=z
(cid:28)(z)= z−z0 0
0 if z =z .
0
The number λ is unique. It is usually written as f(cid:4)(z ), and is called the derivative of f
0
at z .
0
Iff iscomplex-differentiableateverypointofΩ,f issaidtobeanalyticorholomorphic
on Ω. Analytic functions are the basic objects of study in complex variables.
Analyticity on a nonopen set S ⊆ C means analyticity on an open set Ω ⊇ S. In
particular, f is analytic at a point z iff f is analytic on an open set Ω with z ∈Ω. If f
0 0 1
and f are analytic on Ω, so are f +f ,f −f ,kf for k ∈C,f f , and f /f (provided
2 1 2 1 2 1 1 2 1 2
that f is never 0 on Ω). Furthermore,
2
(f +f )(cid:4) =f(cid:4) +f(cid:4), (f −f )(cid:4) =f(cid:4) −f(cid:4), (kf )(cid:4) =kf(cid:4)
1 2 1 2 1 2 1 2 1 1
(cid:4) (cid:5)
f (cid:4) f f(cid:4) −f f(cid:4)
(f f )(cid:4) =f f(cid:4) +f(cid:4)f , 1 = 2 1 1 2.
1 2 1 2 1 2 f f2
2 2
1.4. REAL-DIFFERENTIABILITY AND THE CAUCHY-RIEMANN EQUATIONS 5
The proofs are identical to the corresponding proofs for functions from R to R.
Since d (z) = 1 by direct computation, we may use the rule for differentiating a
dz
product (just as in the real case) to obtain
d
(zn)=nzn−1, n=0,1,...
dz
This extends to n=−1,−2,... using the quotient rule.
If f is analytic on Ω and g is analytic on f(Ω)={f(z):z ∈Ω}, then the composition
g◦f is analytic on Ω and
d
(cid:4) (cid:4)
g(f(z))=g (f(z)f (z)
dz
just as in the real variable case.
As an example of the use of Condition (4) of (1.3.1), we now prove a result that will
be useful later in studying certain inverse functions.
1.3.2 Theorem
Let g be analytic on the open set Ω , and let f be a continuous complex-valued function
1
on the open set Ω. Assume
(i) f(Ω)⊆Ω ,
1
(ii) g(cid:4) is never 0,
(iii) g(f(z))=z for all z ∈Ω (thus f is 1-1).
Then f is analytic on Ω and f(cid:4) =1/(g(cid:4)◦f).
Proof. Let z ∈Ω, and let {z } be a sequence in Ω\{z } with z →z . Then
0 n 0 n 0
(cid:6) (cid:7)
f(z )−f(z ) f(z )−f(z ) g(f(z ))−g(f(z )) −1
n 0 = n 0 = n 0 .
z −z g(f(z ))−g(f(z )) f(z )−f(z )
n 0 n 0 n 0
(Note that f(z ) (cid:3)= f(z ) since f is 1-1 and z (cid:3)= z .) By continuity of f at z , the
n 0 n 0 0
expression in brackets approaches g(cid:4)(f(z )) as n → ∞. Since g(cid:4)(f(z )) (cid:3)= 0, the result
0 0
follows. ♣
1.4 Real-Differentiability and the Cauchy-Riemann Equa-
tions
Let f :Ω→C, and set u=Ref,v =Imf. Then u and v are real-valued functions on Ω
and f = u+iv. In this section we are interested in the relation between f and its real
and imaginary parts u and v. For example, f is continuous at a point z iff both u and v
0
are continuous at z . Relations involving derivatives will be more significant for us, and
0
for this it is convenient to be able to express the idea of differentiability of real-valued
function of two real variables by means of a single formula, without having to consider
partial derivatives separately. We do this by means of a condition analogous to (5) of
(1.3.1).
6 CHAPTER 1. INTRODUCTION
Convention
From now on, Ω will denote an open subset of C, unless otherwise specified.
1.4.1 Definition
Let g : Ω → R. We say that g is real-differentiable at z = x +iy ∈ Ω if there exist
0 0 0
realnumbersAandB,andrealfunctions(cid:28) and(cid:28) definedonaneighborhoodof(x ,y ),
1 2 0 0
such that (cid:28) and (cid:28) are continuous at (x ,y ), (cid:28) (x ,y )=(cid:28) (x ,y )=0, and
1 2 0 0 1 0 0 2 0 0
g(x,y)=g(x ,y )+(x−x )[A+(cid:28) (x,y)]+(y−y )[B+(cid:28) (x,y)]
0 0 0 1 0 2
for all (x,y) in the above neighborhood of (x ,y ).
0 0
It follows from the definition that ifg is real-differentiable at (x ,y ), then the partial
0 0
derivatives of g exist at (x ,y ) and
0 0
∂g ∂g
(x ,y )=A, (x ,y )=B.
0 0 0 0
∂x ∂y
If,ontheotherhand, ∂g and ∂g existat(x ,y )andoneoftheseexistsinaneighborhood
∂x ∂y 0 0
of (x ,y ) and is continuous at (x ,y ), then g is real-differentiable at (x ,y ). To verify
0 0 0 0 0 0
this, assume that ∂g is continuous at (x ,y ), and write
∂x 0 0
g(x,y)−g(x ,y )=g(x,y)−g(x ,y)+g(x ,y)−g(x ,y ).
0 0 0 0 0 0
Now apply the mean value theorem and the definition of partial derivative respectively
(Problem 4).
1.4.2 Theorem
Let f : Ω → C,u = Ref,v = Imf. Then f is complex-differentiable at (x ,y ) iff u and
0 0
v are real-differentiable at (x ,y ) and the Cauchy-Riemann equations
0 0
∂u ∂v ∂v ∂v
= , =−
∂x ∂y ∂x ∂y
are satisfied at (x ,y ). Furthermore, if z =x +iy , we have
0 0 0 0 0
∂u ∂v ∂v ∂u
f(cid:4)(z )= (x ,y )+i (x ,y )= (x ,y )−i (x ,y ).
0 0 0 0 0 0 0 0 0
∂x ∂x ∂y ∂y
Proof. Assume f complex-differentiable at z , and let (cid:28) be the function supplied by (5)
0
of (1.3.1). Define (cid:28) (x,,y)=Re(cid:28)(x,y),(cid:28) (x,y)=Im(cid:28)(x,y). If we take real parts of both
1 2
sides of the equation
f(x)=f(z )+(z−z )(f(cid:4)(z )+(cid:28)(z)) (1)
0 0 0
we obtain
u(x,y)=u(x ,y )+(x−x )[Ref(cid:4)(z )+(cid:28) (x,y)]
0 0 0 0 1
+(y−y )[−Imf(cid:4)(z )−(cid:28) (x,y)].
0 0 2