Chen W. - Introduction to Complex Analysis Lecture Notes.pdf - Analysis - Kuya

INTRODUCTION TO COMPLEX ANALYSIS

WWLCHEN

WWLChen, 1996, 2003.

This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.

It is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gains,

and may be downloaded and/or photocopied, with or without permission from the author.

However, this document may not be kept on any information storage and retrieval system without permission

from the author, unless such system is not accessible to any individuals other than its owners.

Chapter 1

COMPLEX NUMBERS

1.1. Arithmetic and Conjugates

The purpose of this chapter is to give a review of various properties of the complex numbers that we shall

need in the discussion of complex analysis. As the reader is expected to be familiar with the material,

all proofs have been omitted.

.To“solve” this equation, we have to introduce extra

numbers into our number system. To do this, we deﬁne the number i by i 2 +1 =0,and then extend the

ﬁeld of all real numbers by adjoining the number i, which is then combined with the real numbers by the

operations addition and multiplication in accordance with the Field axioms of the real number system.

The numbers a +i b , where a, b

∈ R

,ofthe extended ﬁeld are then added and multiplied in accordance

with the Field axioms, suitably extended, and the restriction i 2 +1=0. Note that the number a + 0i,

where a

∈ R

,behaves like the real number a .

What we have said in the last paragraph basically amounts to the following. Consider two complex

numbers a +i b and c +i d , where a, b, c, d

∈ R

.Wehave the addition and multiplication rules

( a +i b )+( c +i d )=( a + c )+i( b + d )

and

( a +i b )( c +i d )=( ac

−

bd )+i( ad + bc ) .

These lead to the subtraction rule

( a +i b )

−

( c +i d )=( a

−

c )+i( b

−

d ) ,

and the division rule, that if c +i d

=0,then

a +i b

c +i d

= ac + bd

c 2 + d 2

+i bc

c 2 + d 2 .

−

The equation x 2 +1 =0has no solution x

1–2

WWLChen : Introduction to Complex Analysis

Note the special case a =1and b =0.

. The real number x is called the real part of z , and denoted

by x = Re z . The real number y is called the imaginary part of z , and denoted by y = I m z . The set

Suppose that z = x +i y , where x, y

∈ R

{

z = x +i y : x, y

∈ R }

is called the set of all complex numbers. The complex number z = x

−

i y is

called the conjugate of z .

It is easy to see that for every z

∈ C

,wehave

Re z = z + z

and

Im z = z

−

Furthermore, if w

∈ C

, then

z + w = z + w

and

zw = z w.

1.2. Polar Coordinates

Suppose that z = x +i y , where x, y

∈ R

. The real number

r = x 2 + y 2

is called the modulus of z , and denoted by

.Onthe other hand, if z

=0,then any number θ

∈ R

satisfying the equations

(1)

x = r cos θ

and

y = r sin θ

is called an argument of z , and denoted by arg z . Hence we can write z in polar form

z = r (cos θ +isin θ ) .

, arg z is not unique. Clearly we can add any integer multiple of

2 π to θ without aﬀecting (1). We sometimes call a real number θ

∈ C

∈ R

the principal argument of z if θ

satisﬁes the equations (1) and

−

π<θ

≤

π . The principal argument of z is usually denoted by Arg z .

It is easy to see that for every z

∈ C

,wehave

2 = zz . Also, if w

∈ C

, then

and

z + w

|≤|

Furthermore, if

z = r (cos θ +isin θ )

and

w = s (cos φ +isin φ ) ,

where r, s, θ, φ

∈ R

and r, s > 0, then

zw = rs (cos( θ + φ )+isin( θ + φ ))

and

= r

s (cos( θ

−

φ )+isin( θ

−

φ )) .

1.3. Rational Powers

De Moivre’s theorem, that

(2)

cos nθ +isin nθ =(cos θ +isin θ ) n

for every n

∈ N

and θ

∈ R

Note, however, that for a given z

Chapter 1 : Complex Numbers

1–3

is useful in ﬁnding n -th roots of complex numbers.

Suppose that c = R (cos α +isin α ), where R, α

∈ R

and R> 0. Then the solutions of the equation

z n = c are given by

z =

√ R cos α +2 kπ

+isin α +2 kπ

where k =0 , 1 ,...,n

−

1 .

Finally, we can deﬁne c b

for any b

∈ Q

and non-zero c

∈ C

as follows. The rational number b can

have no prime factors in common.

Then there are exactly q distinct numbers z satisfying z q = c .Wenow deﬁne c b = z p , noting that the

expression (2) can easily be extended to all n

∈ Z

and q

∈ N

∈ Z

.Itisnot too dicult to show that there are q distinct

values for the rational power c b .

Problems for Chapter 1

1. Suppose that z 0 ∈ C

is ﬁxed. A polynomial P ( z )issaid to be divisible by z

−

z 0 if there is another

polynomial Q ( z ) such that P ( z )=( z

−

z 0 ) Q ( z ).

z 0 .

b) Consider the polynomial P ( z )= a 0 + a 1 z + a 2 z 2 + ... + a n z n , where a 0 ,a 1 ,a 2 ,...,a n

∈ C

and k

∈ N

, the polynomial c ( z k

−

z 0 )isdivisible by z

−

∈ C

are

arbitrary. Show that the polynomial P ( z )

−

P ( z 0 )isdivisible by z

−

z 0 .

z 0 if P ( z 0 )=0.

d) Suppose that a polynomial P ( z )ofdegree n vanishes at n distinct values z 1 ,z 2 ,...,z n ∈ C

−

,so

that P ( z 1 )= P ( z 2 )= ... = P ( z n )=0. Show that P ( z )= c ( z

−

z 1 )( z

−

z 2 ) ... ( z

−

z n ), where

is a constant.

e) Suppose that a polynomial P ( z )ofdegree n vanishes at more than n distinct values. Show

that P ( z )=0identically.

∈ C

2. Suppose that α

∈ C

is ﬁxed and

< 1. Show that

|≤

1ifand only if

−

≤

−

αz

3. Suppose that z = x +i y , where x, y

∈ R

. Express each of the following in terms of x and y :

−

z +1

z +i

−

i z

=0.

a) Show that αz + αz + c = 0 is the e qua tion of a straight line on the plane.

b) What does the equation zz + αz + αz + c =0represent if

∈ R

a nd α

∈ C

with α

≥

c ?

5. Suppose that z, w

∈ C

. Show that

z + w

2 +

−

2 =2(

2 +

2 ).

6. Find all the roots of the equation ( z 8

−

1)( z 3 +8)=0.

7. For each of the following, compute all the values and plot them on the plane:

a) (1 + i) − 1 / 2

b) (

−

4) 3 / 4

c) (1

−

i) 3 / 8

be written uniquely in the form b = p/q , where p

a) Show that for every c

c) Deduce that P ( z )isdivisible by z

4. Suppose that c

INTRODUCTION TO COMPLEX ANALYSIS

WWLCHEN

WWLChen, 1996, 2003.

This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.

It is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gains,

and may be downloaded and/or photocopied, with or without permission from the author.

However, this document may not be kept on any information storage and retrieval system without permission

from the author, unless such system is not accessible to any individuals other than its owners.

Chapter 2

FOUNDATIONS OF COMPLEX ANALYSIS

2.1. Three Approaches

We start by remarking that analysis is sometimes known as the study of the four C’s: convergence,

continuity, compactness and connectedness. In real analysis, we have studied convergence and continuity

to some depth, but the other two concepts have been somewhat disguised. In this course, we shall try

to illustrate these two latter concepts a little bit more, particularly connectedness.

Complex analysis is the study of complex valued functions of complex variables. Here we shall

restrict the number of variables to one, and study complex valued functions of one complex variable.

Unless otherwise stated, all functions in these notes are of the form f : S

→ C

, where S is a set in

We shall study the behaviour of such functions using three diﬀerent approaches. The ﬁrst of these,

discussed in Chapter 3 and usually attributed to Riemann, is based on diﬀerentiation and involves pairs

of partial diﬀerential equations called the Cauchy-Riemann equations. The second approach, discussed in

Chapters 4–11 and usually attributed to Cauchy, is based on integration and depends on a fundamental

theorem known nowadays as Cauchy’s integral theorem. The third approach, discussed in Chapter 16

and usually attributed to Weierstrass, is based on the theory of power series.

2.2. Point Sets in the Complex Plane

.Inmost situations, various

properties of the point sets S play a crucial role in our study. We therefore begin by discussing various

typesofpoint sets in the complex plane.

→ C

, where S is a set in

Before making any deﬁnitions, let us consider a few examples of sets which frequently occur in our

subsequent discussion.

We shall study functions of the form f : S

2–2

WWLChen : Introduction to Complex Analysis

Example 2.2.1.

Suppose that z 0 ∈ C

, r, R

∈ R

and 0 <r<R . The set

{

∈ C

−

z 0 |

represents a disc, with centre z 0 and radius R , and the set

{

∈ C

: r<

−

z 0 |

}

represents an

annulus, with centre z 0 , inner radius r and outer radius R .

z 0

Example 2.2.2.

Suppose that A, B

∈ R

and A<B . The set

{

z = x +i y

∈ C

: x, y

∈ R

and x>A

}

represents a half-plane, and the set

{

z = x +i y

∈ C

: x, y

∈ R

and A<x<B

}

represents a strip.

Example 2.2.3.

Suppose that α, β

∈ R

and 0

≤

α<β< 2 π . The set

{

z = r (cos θ +isin θ )

∈ C

: r, θ

∈ R

and r> 0 and α<θ<β

}

represents a sector.

We now make a number of important deﬁnitions. The reader may subsequently need to return to

these deﬁnitions.

}

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