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A First Course in

Complex

Analysis

with Applications

Dennis G. Zill

Loyola Marymount University

Patrick D. Shanahan

Loyola Marymount University

World Headquarters

Jones and Bartlett Publishers

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Copyright © 2003 by Jones and Bartlett Publishers, Inc.

Library of Congress Cataloging-in-Publication Data

Zill, Dennis G., 1940A first course in complex analysis with applications / Dennis G. Zill, Patrick D. Shanahan.

p. cm.

Includes indexes.

ISBN 0-7637-1437-2

1. Functions of complex variables. I. Shanahan, Patrick, 1931- II. Title.

QA331.7 .Z55 2003

515’.9—dc21

2002034160

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Printed in the United States of America

06 05 04 03 02

10 9 8 7 6 5 4 3 2 1

For Dana, Kasey, and Cody

Contents

7.1

Contents

Preface

ix

Chapter 1.

Complex Numbers and the Complex Plane 1

1.1

Complex Numbers and Their Properties 2

1.2

Complex Plane 10

1.3

Polar Form of Complex Numbers 16

1.4

Powers and Roots 23

1.5

Sets of Points in the Complex Plane 29

1.6

Applications 36

Chapter 1 Review Quiz 45

Chapter 2.

Complex Functions and Mappings 49

2.1

Complex Functions 50

2.2

Complex Functions as Mappings 58

2.3

Linear Mappings 68

2.4

Special Power Functions 80

2.4.1 The Power Function z n 81

2.4.2 The Power Function z 1/n 86

2.5

Reciprocal Function 100

2.6

Limits and Continuity 110

2.6.1 Limits 110

2.6.2 Continuity 119

2.7

Applications 132

Chapter 2 Review Quiz 138

Chapter 3.

Analytic Functions 141

3.1

Diﬀerentiability and Analyticity 142

3.2

Cauchy-Riemann Equations 152

3.3

Harmonic Functions 159

3.4

Applications 164

Chapter 3 Review Quiz 172

v

vi

Contents

Chapter 4.

Elementary Functions 175

4.1

Exponential and Logarithmic Functions 176

4.1.1 Complex Exponential Function 176

4.1.2 Complex Logarithmic Function 182

4.2

Complex Powers 194

4.3

Trigonometric and Hyperbolic Functions 200

4.3.1 Complex Trigonometric Functions 200

4.3.2 Complex Hyperbolic Functions 209

4.4

Inverse Trigonometric and Hyperbolic

Functions 214

4.5

Applications 222

Chapter 4 Review Quiz 232

Chapter 5.

Integration in the Complex Plane 235

5.1

Real Integrals 236

5.2

Complex Integrals 245

5.3

Cauchy-Goursat Theorem 256

5.4

Independence of Path 264

5.5

Cauchy’s Integral Formulas and Their

Consequences 272

5.5.1 Cauchy’s Two Integral Formulas 273

5.5.2 Some Consequences of the Integral

Formulas 277

5.6

Applications 284

Chapter 5 Review Quiz 297

Chapter 6.

Series

6.1

6.2

6.3

6.4

6.5

6.6

6.7

and Residues 301

Sequences and Series 302

Taylor Series 313

Laurent Series 324

Zeros and Poles 335

Residues and Residue Theorem 342

Some Consequences of the Residue

Theorem 352

6.6.1 Evaluation of Real Trigonometric

Integrals 352

6.6.2 Evaluation of Real Improper

Integrals 354

6.6.3 Integration along a Branch Cut 361

6.6.4 The Argument Principle and Rouch´e’s

Theorem 363

6.6.5 Summing Inﬁnite Series 367

Applications 374

Chapter 6 Review Quiz 386

Contents

vii

Chapter 7.

Conformal Mappings 389

7.1

Conformal Mapping 390

7.2

Linear Fractional Transformations 399

7.3

Schwarz-Christoﬀel Transformations 410

7.4

Poisson Integral Formulas 420

7.5

Applications 429

7.5.1 Boundary-Value Problems 429

7.5.2 Fluid Flow 437

Chapter 7 Review Quiz 448

Appendixes: I

II

III

Proof of Theorem 2.1 APP-2

Proof of the Cauchy-Goursat Theorem

Table of Conformal Mappings APP-9

Answers for Selected Odd-Numbered Problems

Index

IND-1

ANS-1

APP-4

Preface

7.2

Preface

Philosophy This text grew out of chapters 17-20 in Advanced Engineering Mathematics, Second Edition (Jones and Bartlett Publishers), by Dennis

G. Zill and the late Michael R. Cullen. This present work represents an expansion and revision of that original material and is intended for use in either

a one-semester or a one-quarter course. Its aim is to introduce the basic principles and applications of complex analysis to undergraduates who have no

prior knowledge of this subject.

The motivation to adapt the material from Advanced Engineering Mathematics into a stand-alone text sprang from our dissatisfaction with the succession of textbooks that we have used over the years in our departmental

undergraduate course oﬀering in complex analysis. It has been our experience

that books claiming to be accessible to undergraduates were often written at a

level that was too advanced for our audience. The “audience” for our juniorlevel course consists of some majors in mathematics, some majors in physics,

but mostly majors from electrical engineering and computer science. At our

institution, a typical student majoring in science or engineering does not take

theory-oriented mathematics courses in methods of proof, linear algebra, abstract algebra, advanced calculus, or introductory real analysis. Moreover,

the only prerequisite for our undergraduate course in complex variables is

the completion of the third semester of the calculus sequence. For the most

part, then, calculus is all that we assume by way of preparation for a student

to use this text, although some working knowledge of diﬀerential equations

would be helpful in the sections devoted to applications. We have kept the

theory in this introductory text to what we hope is a manageable level, concentrating only on what we feel is necessary. Many concepts are conveyed

in an informal and conceptual style and driven by examples, rather than the

formal deﬁnition/theorem/proof. We think it would be fair to characterize

this text as a continuation of the study of calculus, but also the study of the

calculus of functions of a complex variable. Do not misinterpret the preceding

words; we have not abandoned theory in favor of “cookbook recipes”; proofs

of major results are presented and much of the standard terminology is used.

Indeed, there are many problems in the exercise sets in which a student is

asked to prove something. We freely admit that any student—not just majors in mathematics—can gain some mathematical maturity and insight by

attempting a proof. But we know, too, that most students have no idea how

to start a proof. Thus, in some of our “proof” problems, either the reader

ix

x

Preface

is guided through the starting steps or a strong hint on how to proceed is

provided.

The writing herein is straightforward and reﬂects the no-nonsense style

of Advanced Engineering Mathematics.

Content We have purposely limited the number of chapters in this text

to seven. This was done for two “reasons”: to provide an appropriate quantity

of material so that most of it can reasonably be covered in a one-term course,

and at the same time to keep the cost of the text within reason.

Here is a brief description of the topics covered in the seven chapters.

• Chapter 1 The complex number system and the complex plane are

examined in detail.

• Chapter 2 Functions of a complex variable, limits, continuity, and

mappings are introduced.

• Chapter 3 The all-important concepts of the derivative of a complex

function and analyticity of a function are presented.

• Chapter 4 The trigonometric, exponential, hyperbolic, and logarithmic functions are covered. The subtle notions of multiple-valued functions and branches are also discussed.

• Chapter 5 The chapter begins with a review of real integrals (including line integrals). The deﬁnitions of real line integrals are used to

motivate the deﬁnition of the complex integral. The famous CauchyGoursat theorem and the Cauchy integral formulas are introduced in

this chapter. Although we use Green’s theorem to prove Cauchy’s theorem, a sketch of the proof of Goursat’s version of this same theorem is

given in an appendix.

• Chapter 6 This chapter introduces the concepts of complex sequences

and inﬁnite series. The focus of the chapter is on Laurent series, residues,

and the residue theorem. Evaluation of complex as well as real integrals,

summation of inﬁnite series, and calculation of inverse Laplace and inverse Fourier transforms are some of the applications of residue theory

that are covered.

• Chapter 7 Complex mappings that are conformal are deﬁned and

used to solve certain problems involving Laplace’s partial diﬀerential

equation.

Features Each chapter begins with its own opening page that includes a

table of contents and a brief introduction describing the material to be covered

in the chapter. Moreover, each section in a chapter starts with introductory comments on the speciﬁcs covered in that section. Almost every section

ends with a feature called Remarks in which we talk to the students about

areas where real and complex calculus diﬀer or discuss additional interesting

topics (such as the Riemann sphere and Riemann surfaces) that are related

Preface

xi

to, but not formally covered in, the section. Several of the longer sections,

although uniﬁed by subject matter, have been partitioned into subsections;

this was done to facilitate covering the material over several class periods.

The corresponding exercise sets were divided in the same manner in order to

make the assignment of homework easier. Comments, clariﬁcations, and some

warnings are liberally scattered throughout the text by means of annotations

in the left margin marked by the symbol ☞.

There are a lot of examples and we have tried very hard to supply all

pertinent details in the solutions of the examples. Because applications of

complex variables are often compiled into a single chapter placed at the end

of the text, instructors are sometimes hard pressed to cover any applications

in the course. Complex analysis is a powerful tool in applied mathematics. So

to facilitate covering this beautiful aspect of the subject, we have chosen to

end each chapter with a separate section on applications. The exercise sets are

constructed in a pyramidal fashion and each set has at least two parts. The

ﬁrst part of an exercise set is a generous supply of routine drill-type problems;

the second part consists of conceptual word and geometrical problems. In

many exercise sets, there is a third part devoted to the use of technology.

Since the default operational mode of all computer algebra systems is complex

variables, we have placed an emphasis on that type of software. Although we

have discussed the use of Mathematica in the text proper, the problems are

generic in nature. Answers to selected odd-numbered problems are given in

the back of the text. Since the conceptual problems could also be used as

topics for classroom discussion, we decided not to include their answers. Each

chapter ends with a Chapter Review Quiz. We thought that something more

conceptual would be a bit more interesting than the rehashing of the same

old problems given in the traditional Chapter Review Exercises. Lastly, to

illustrate the subtleties of the action of complex mappings, we have used two

colors.

Acknowledgments We would like to express our appreciation to our

colleague at Loyola Marymount University, Lily Khadjavi, for volunteering to

use a preliminary version of this text. We greatly appreciate her careful reading of the manuscript. We also wish to acknowledge the valuable input from

students who used this book, in particular: Patrick Cahalan, Willa Crosby,

Kellie Dyerly, Sarah Howard, and Matt Kursar. A deeply felt “thank you”

goes to the following reviewers for their words of encouragement, criticisms,

and thoughtful suggestions:

Nicolae H. Pavel, Ohio University

Marcos Jardim, University of Pennsylvania

Ilia A. Binder, Harvard University

Finally, we thank the editorial and production staﬀ at Jones and Bartlett,

especially our production manager, Amy Rose, for their many contributions

and cooperation in the making of this text.

✐

✐

xii

Preface

A Request Although the preliminary versions of this book were class

tested for several semesters, experience has taught us that errors—typos or

just plain mistakes—seem to be an inescapable by-product of the textbookwriting endeavor. We apologize in advance for any errors that you may ﬁnd

and urge you to bring them to our attention.

Dennis G. Zill

Patrick D. Shanahan

Los Angeles, CA

✐

✐

20

pa

1

Complex Numbers

and the

Complex Plane

3π

2π

π

0

–π

–2π

–3π

1

–1

0

0

1 –1

Riemann surface for arg( z ). See

page 97.

1.1

Complex Numbers and Their Properties

1.2

Complex Plane

1.3

Polar Form of Complex Numbers

1.4

Powers and Roots

1.5

Sets of Points in the Complex Plane

1.6

Applications

Chapter 1 Review Quiz

Introduction In elementary courses you learned

about the existence, and some of the properties,

of complex numbers. But in courses in calculus,

it is most likely that you did not even see a complex number. In this text we study nothing but

complex numbers and the calculus of functions

of a complex variable.

We begin with an in-depth examination of

the arithmetic and algebra of complex numbers.

1

2

Chapter 1 Complex Numbers and the Complex Plane

1.1

Complex Numbers and Their Properties

No one person 1.1

“invented” complex numbers, but controversies surrounding the use of these

numbers existed in the sixteenth century. In their quest to solve polynomial equations by

formulas involving radicals, early dabblers in mathematics were forced to admit that there

were other kinds of numbers besides positive integers. Equations such as x2 + 2x + 2 = 0

√

√

√

and x3 = 6x + 4 that yielded “solutions” 1 + −1 and 3 2 + −2 + 3 2 − −2 caused

particular consternation within the community of √

ﬂedgling mathematical

scholars because

√

everyone knew that there are no numbers such as −1 and −2, numbers whose square is

negative. Such “numbers” exist only in one’s imagination, or as one philosopher opined, “the

imaginary, (the) bosom child of complex mysticism.” Over time these “imaginary numbers”

did not go away, mainly because mathematicians as a group are tenacious and some are even

practical. A famous mathematician held that even though “they exist in our imagination

. . . nothing prevents us from . . . employing them in calculations.” Mathematicians

also hate to throw anything away. After all, a memory still lingered that negative numbers

at ﬁrst were branded “ﬁctitious.” The concept of number evolved over centuries; gradually

the set of numbers grew from just positive integers to include rational numbers, negative

numbers, and irrational numbers. But in the eighteenth century the number concept took a

gigantic evolutionary step forward when the German mathematician Carl Friedrich Gauss

put the so-called imaginary numbers—or complex numbers, as they were now beginning to

be called—on a logical and consistent footing by treating them as an extension of the real

number system.

Our goal in this ﬁrst section is to examine some basic deﬁnitions and the arithmetic of

complex numbers.

The Imaginary Unit Even after gaining wide respectability, through

the seminal works of Karl Friedrich Gauss and the French mathematician Augustin Louis Cauchy, the unfortunate name “imaginary” has survived down

the centuries. The

√ symbol i was originally used as a disguise for the embarrassing symbol −1. We now say that i is the imaginary unit and deﬁne

it by the property i2 = –1. Using the imaginary unit, we build a general

complex number out of two real numbers.

Definition 1.1

Complex Number

A complex number is any number of the form z = a + ib where a and

b are real numbers and i is the imaginary unit.

Note: The imaginary part of

z = 4 − 9i is −9 not −9i.

☞

Terminology The notations a + ib and a + bi are used interchangeably.

The real number a in z = a+ ib is called the real part of z; the real number b

is called the imaginary part of z. The real and imaginary parts of a complex

number z are abbreviated Re(z) and Im(z), respectively. For example, if

z = 4 − 9i, then Re(z) = 4 and Im(z) = −9. A real constant multiple

of the imaginary unit is called a pure imaginary number. For example,

z = 6i is a pure imaginary number. Two complex numbers are equal if their

1.1 Complex Numbers and Their Properties

3

corresponding real and imaginary parts are equal. Since this simple concept

is sometimes useful, we formalize the last statement in the next deﬁnition.

Definition 1.2 Equality

Complex numbers z1 = a1 + ib 1 and z2 = a2 + ib 2 are equal, z1 = z2 , if

a1 = a2 and b1 = b2 .

In terms of the symbols Re(z) and Im(z), Deﬁnition 1.2 states that z1 = z2 if

Re(z1 ) = Re(z2 ) and Im(z1 ) = Im(z2 ).

The totality of complex numbers or the set of complex numbers is usually

denoted by the symbol C. Because any real number a can be written as

z = a + 0i, we see that the set R of real numbers is a subset of C.

Arithmetic Operations Complex numbers can be added, subtracted,

multiplied, and divided. If z1 = a1 + ib 1 and z2 = a2 + ib 2 , these operations

are deﬁned as follows.

Addition:

z1 + z2 = (a1 + ib1 ) + (a2 + ib2 ) = (a1 + a2 ) + i(b1 + b2 )

Subtraction:

z1 −z2 = (a1 + ib1 ) − (a2 + ib2 ) = (a1 − a2 ) + i(b1 − b2 )

Multiplication:

z1 · z2 = (a1 + ib1 )(a2 + ib2 )

= a1 a2 − b1 b2 + i(b1 a2 + a1 b2 )

Division:

z1

a1 + ib1

=

, a2 = 0, or b2 = 0

z2

a2 + ib2

b1 a2 − a1 b2

a1 a2 + b1 b2

+i

=

2

2

a2 + b2

a22 + b22

The familiar commutative, associative, and distributive laws hold for complex numbers:

z +z = z +z

1

2

2

1

Commutative laws:

z z =z z

1 2

2 1

Associative laws:

z + (z + z ) = (z + z ) + z

1

2

3

1

2

3

z (z z ) = (z z )z

1

Distributive law:

2 3

1 2

3

z1 (z2 + z3 ) = z1 z2 + z1 z3

In view of these laws, there is no need to memorize the deﬁnitions of

addition, subtraction, and multiplication.

4

Chapter 1 Complex Numbers and the Complex Plane

Addition, Subtraction, and Multiplication

(i ) To add (subtract ) two complex numbers, simply add (subtract ) the

corresponding real and imaginary parts.

(ii ) To multiply two complex numbers, use the distributive law and the

fact that i2 = −1.

The deﬁnition of division deserves further elaboration, and so we will discuss

that operation in more detail shortly.

EXAMPLE 1

Addition and Multiplication

If z1 = 2 + 4i and z2 = −3 + 8i, ﬁnd (a) z1 + z2 and (b) z1 z2 .

Solution (a) By adding real and imaginary parts, the sum of the two complex

numbers z1 and z2 is

z1 + z2 = (2 + 4i) + (−3 + 8i) = (2 − 3) + (4 + 8)i = −1 + 12i.

(b) By the distributive law and i2 = −1, the product of z1 and z2 is

z1 z2 = (2 + 4i) (−3 + 8i) = (2 + 4i) (−3) + (2 + 4i) (8i)

= −6 − 12i + 16i + 32i2

= (−6 − 32) + (16 − 12)i = −38 + 4i.

Zero and Unity The zero in the complex number system is the number 0 + 0i and the unity is 1 + 0i. The zero and unity are denoted by 0 and

1, respectively. The zero is the additive identity in the complex number

system since, for any complex number z = a + ib, we have z + 0 = z. To see

this, we use the deﬁnition of addition:

z + 0 = (a + ib) + (0 + 0i) = a + 0 + i(b + 0) = a + ib = z.

Similarly, the unity is the multiplicative identity of the system since, for

any complex number z, we have z · 1 = z · (1 + 0i) = z.

There is also no need to memorize the deﬁnition of division, but before

discussing why this is so, we need to introduce another concept.

Conjugate If z is a complex number, the number obtained by changing

the sign of its imaginary part is called the complex conjugate, or simply

conjugate, of z and is denoted by the symbol z¯. In other words, if z = a + ib,

1.1 Complex Numbers and Their Properties

5

then its conjugate is z¯ = a − ib. For example, if z = 6 + 3i, then z¯ = 6 − 3i;

if z = −5 − i, then z¯ = −5 + i. If z is a real number, say, z = 7, then

z¯ = 7. From the deﬁnitions of addition and subtraction of complex numbers,

it is readily shown that the conjugate of a sum and diﬀerence of two complex

numbers is the sum and diﬀerence of the conjugates:

z1 − z2 = z¯1 − z¯2 .

z1 + z2 = z¯1 + z¯2 ,

(1)

Moreover, we have the following three additional properties:

z1 z2 = z¯1 z¯2 ,

z1

z2

=

z¯1

,

z¯2

z¯ = z.

(2)

Of course, the conjugate of any ﬁnite sum (product) of complex numbers is

the sum (product) of the conjugates.

The deﬁnitions of addition and multiplication show that the sum and

product of a complex number z with its conjugate z¯ is a real number:

z + z¯ = (a + ib) + (a − ib) = 2a

(3)

z z¯ = (a + ib)(a − ib) = a − i b = a + b .

2

2 2

2

2

(4)

The diﬀerence of a complex number z with its conjugate z¯ is a pure imaginary

number:

z − z¯ = (a + ib) − (a − ib) = 2ib.

(5)

Since a = Re(z) and b = Im(z), (3) and (5) yield two useful formulas:

Re(z) =

z + z¯

2

and

Im(z) =

z − z¯

.

2i

(6)

However, (4) is the important relationship in this discussion because it enables

us to approach division in a practical manner.

Division

To divide z 1 by z 2 , multiply the numerator and denominator of z 1 /z2 by

the conjugate of z 2 . That is,

z1 z¯2

z1 z¯2

z1

=

·

=

z2

z2 z¯2

z2 z¯2

(7)

and then use the fact that z2 z¯2 is the sum of the squares of the real and

imaginary parts of z 2 .

The procedure described in (7) is illustrated in the next example.

EXAMPLE 2

Division

If z1 = 2 − 3i and z2 = 4 + 6i, ﬁnd z1 /z2 .

6

Chapter 1 Complex Numbers and the Complex Plane

Solution We multiply numerator and denominator by the conjugate

z¯2 = 4 − 6i of the denominator z2 = 4 + 6i and then use (4):

2 − 3i

−10 − 24i

2 − 3i 4 − 6i

8 − 12i − 12i + 18i2

z1

=

=

=

=

.

z2

4 + 6i

4 + 6i 4 − 6i

42 + 62

52

Because we want an answer in the form a + bi, we rewrite the last result by

dividing the real and imaginary parts of the numerator −10 − 24i by 52 and

reducing to lowest terms:

5

6

10 24

z1

= − − i = − − i.

z2

52 52

26 13

Inverses In the complex number system, every number z has a unique

additive inverse. As in the real number system, the additive inverse of

z = a + ib is its negative, −z, where −z = −a − ib. For any complex number

z, we have z + (−z) = 0. Similarly, every nonzero complex number z has a

multiplicative inverse. In symbols, for z = 0 there exists one and only one

nonzero complex number z −1 such that zz −1 = 1. The multiplicative inverse

z −1 is the same as the reciprocal 1/z.

EXAMPLE 3

Reciprocal

Find the reciprocal of z = 2 − 3i.

Solution By the deﬁnition of division we obtain

1

1

1

2 + 3i

2 + 3i

2 + 3i

=

=

=

=

.

z

2 − 3i

2 − 3i 2 + 3i

4+9

13

Answer should be in the form a + ib.

☞

That is,

3

2

1

= z −1 =

+ i.

z

13 13

You should take a few seconds to verify the multiplication

zz −1 = (2 − 3i)

2

13

+

3

13 i

= 1.

Remarks Comparison with Real Analysis

(i ) Many of the properties of the real number system R hold in the

complex number system C, but there are some truly remarkable

diﬀerences as well. For example, the concept of order in the

real number system does not carry over to the complex number

system. In other words, we cannot compare two complex numbers

z1 = a1 + ib1 , b1 = 0, and z2 = a2 + ib2 , b2 = 0, by means of

1.1 Complex Numbers and Their Properties

7

inequalities. Statements such as z1 < z2

meaning in C except in the special case

bers z1 and z2 are real.

See Problem

Therefore, if you see a statement such as

it is implicit from the use of the inequality

bol α represents a real number.

or z2 ≥ z1 have no

when the two num55 in Exercises 1.1.

z1 = αz2 , α > 0,

α > 0 that the sym-

(ii ) Some things that we take for granted as impossible in real analysis,

such as ex = −2 and sin x = 5 when x is a real variable, are perfectly correct and ordinary in complex analysis when the symbol x

is interpreted as a complex variable. See Example 3 in Section 4.1

and Example 2 in Section 4.3.

We will continue to point out other diﬀerences between real analysis and

complex analysis throughout the remainder of the text.

EXERCISES 1.1

Answers to selected odd-numbered problems begin on page ANS-2.

1. Evaluate the following powers of i.

(a) i8

(b) i11

(c) i42

(d) i105

2. Write the given number in the form a + ib.

(a) 2i3 − 3i2 + 5i

(c)

(b) 3i5 − i4 + 7i3 − 10i2 − 9

20

2

5

+ 3 − 18

i

i

i

2

−i

(d) 2i6 +

3

+ 5i−5 − 12i

In Problems 3–20, write the given number in the form a + ib.

3. (5 − 9i) + (2 − 4i)

4. 3(4 − i) − 3(5 + 2i)

5. i(5 + 7i)

6. i(4 − i) + 4i(1 + 2i)

7. (2 − 3i)(4 + i)

8.

9. 3i +

1

2−i

1

2

− 14 i

10.

i

1+i

2

3

+ 53 i

11.

2 − 4i

3 + 5i

12.

10 − 5i

6 + 2i

13.

(3 − i)(2 + 3i)

1+i

14.

(1 + i)(1 − 2i)

(2 + i)(4 − 3i)

15.

(5 − 4i) − (3 + 7i)

(4 + 2i) + (2 − 3i)

16.

(4 + 5i) + 2i3

(2 + i)2

17. i(1 − i)(2 − i)(2 + 6i)

19. (3 + 6i) + (4 − i)(3 + 5i) +

18. (1 + i)2 (1 − i)3

1

2−i

20. (2 + 3i)

2−i

1 + 2i

2

8

Chapter 1 Complex Numbers and the Complex Plane

In Problems 21–24, use the binomial theorem∗

n(n − 1) n−2 2

n n−1

B+

B + ···

A

A

1!

2!

n(n − 1)(n − 2) · · · (n − k + 1) n−k k

B + · · · + Bn,

A

+

k!

where n = 1, 2, 3, . . . , to write the given number in the form a + ib.

(A + B)n = An +

1 − 12 i

21. (2 + 3i)2

22.

23. (−2 + 2i)5

24. (1 + i)8

3

In Problems 25 and 26, ﬁnd Re(z) and Im(z).

1

i

1

26. z =

25. z =

3−i

2 + 3i

(1 + i)(1 − 2i)(1 + 3i)

In Problems 27–30, let z = x + iy. Express the given quantity in terms of x and y.

27. Re(1/z)

28. Re(z 2 )

29. Im(2z + 4¯

z − 4i)

30. Im(¯

z2 + z2 )

In Problems 31–34, let z = x + iy. Express the given quantity in terms of the

symbols Re(z) and Im(z).

31. Re(iz)

32. Im(iz)

33. Im((1 + i)z)

34. Re(z 2 )

In Problems 35 and 36, show that the indicated numbers satisfy the given equation.

In each case explain why additional solutions can be found.

√

√

2

2

+

i. Find an additional solution, z2 .

35. z 2 + i = 0, z1 = −

2

2

36. z 4 = −4; z1 = 1 + i, z2 = −1 + i. Find two additional solutions, z3 and z4 .

In Problems 37–42, use Deﬁnition 1.2 to solve each equation for z = a + ib.

37. 2z = i(2 + 9i)

2

39. z = i

41. z + 2¯

z=

2−i

1 + 3i

38. z − 2¯

z + 7 − 6i = 0

40. z¯2 = 4z

z

42.

= 3 + 4i

1 + z¯

In Problems 43 and 44, solve the given system of equations for z1 and z2 .

43.

iz1 −

iz2 = 2 + 10i

−z1 + (1 − i)z2 = 3 − 5i

44.

iz1 + (1 + i)z2 = 1 + 2i

(2 − i)z1 +

2iz2 = 4i

Focus on Concepts

45. What can be said about the complex number z if z = z¯? If (z)2 = (¯

z )2 ?

46. Think of an alternative solution to Problem 24. Then without doing any signiﬁcant work, evaluate (1 + i)5404 .

∗ Recall that the coeﬃcients in the expansions of (A + B)2 , (A + B)3 , and so on, can

also be obtained using Pascal’s triangle.

1.1 Complex Numbers and Their Properties

9

47. For n a nonnegative integer, in can be one of four values: 1, i, −1, and −i. In

each of the following four cases, express the integer exponent n in terms of the

symbol k, where k = 0, 1, 2, . . . .

(a) in = 1

(b) in = i

(c) in = −1

(d) in = −i

48. There is an alternative to the procedure given in (7). For example, the quotient

(5 + 6i)/(1 + i) must be expressible in the form a + ib:

5 + 6i

= a + ib.

1+i

Therefore, 5 + 6i = (1 + i)(a + ib). Use this last result to ﬁnd the given quotient.

Use this method to ﬁnd the reciprocal of 3 − 4i.

√

49. Assume for the moment that 1 + i makes sense in the complex number system.

How would you then demonstrate the validity of the equality

√

1+i=

1

2

+

1

2

√

2+i

− 12 +

1

2

√

2?

50. Suppose z1 and z2 are complex numbers. What can be said about z1 or z2 if

z1 z2 = 0?

51. Suppose the product z1 z2 of two complex numbers is a nonzero real constant.

Show that z2 = k¯

z1 , where k is a real number.

52. Without doing any signiﬁcant work, explain why it follows immediately from

(2) and (3) that z1 z¯2 + z¯1 z2 = 2Re(z1 z¯2 ).

53. Mathematicians like to prove that certain “things” within a mathematical system are unique. For example, a proof of a proposition such as “The unity in

the complex number system is unique” usually starts out with the assumption

that there exist two diﬀerent unities, say, 11 and 12 , and then proceeds to show

that this assumption leads to some contradiction. Give one contradiction if it

is assumed that two diﬀerent unities exist.

54. Follow the procedure outlined in Problem 53 to prove the proposition “The zero

in the complex number system is unique.”

55. A number system is said to be an ordered system provided it contains a

subset P with the following two properties:

First, for any nonzero number x in the system, either x or −x is (but not both)

in P.

Second, if x and y are numbers in P, then both xy and x + y are in P.

In the real number system the set P is the set of positive numbers. In the real

number system we say x is greater than y, written x > y, if and only if x − y

is in P . Discuss why the complex number system has no such subset P . [Hint:

Consider i and −i.]

10

Chapter 1 Complex Numbers and the Complex Plane

1.2

Complex Plane

A complex number

1.2 z = x + iy is uniquely determined by an ordered pair

of real numbers

(x, y). The ﬁrst and second entries of the ordered pairs correspond, in turn, with the

real and imaginary parts of the complex number. For example, the ordered pair (2, −3)

corresponds to the complex number z = 2 − 3i. Conversely, z = 2 − 3i determines the

ordered pair (2, −3). The numbers 7, i, and −5i are equivalent to (7, 0), (0, 1), (0, −5),

respectively. In this manner we are able to associate a complex number z = x + iy with a

point (x, y) in a coordinate plane.

y-axis

or

imaginary axis

z = x + iy or

(x, y)

y

x

x-axis

or

real axis

Complex Plane

Because of the correspondence between a complex

number z = x + iy and one and only one point (x, y) in a coordinate plane,

we shall use the terms complex number and point interchangeably. The coordinate plane illustrated in Figure 1.1 is called the complex plane or simply

the z -plane. The horizontal or x-axis is called the real axis because each

point on that axis represents a real number. The vertical or y-axis is called

the imaginary axis because a point on that axis represents a pure imaginary

number.

Figure 1.1 z-plane

Vectors

In other courses you have undoubtedly seen that the numbers

in an ordered pair of real numbers can be interpreted as the components of

a vector. Thus, a complex number z = x + iy can also be viewed as a twodimensional position vector, that is, a vector whose initial point is the origin

and whose terminal point is the point (x, y). See Figure 1.2. This vector

interpretation prompts us to deﬁne the length of the vector z as the distance

x2 + y 2 from the origin to the point (x, y). This length is given a special

name.

y

z = x + iy

x

Figure 1.2 z as a position vector

Definition 1.3

Modulus

The modulus of a complex number z = x + iy, is the real number

|z| =

x2 + y 2 .

(1)

The modulus |z| of a complex number z is also called the absolute value

of z. We shall use both words modulus and absolute value throughout this

text.

EXAMPLE 1

Modulus of a Complex Number

If z = 2 − 3i, then √from (1) we ﬁnd the modulus of the number to be

|z| = 22 + (−3)2 = 13. If z = −9i, then (1) gives |−9i| = (−9)2 = 9.

1.2 Complex Plane

11

Properties Recall from (4) of Section 1.1 that for any complex number

z = x + iy the product z z¯ is a real number; speciﬁcally, z z¯ is the sum of the

squares of the real and imaginary parts of z: z z¯ = x2 + y 2 . Inspection of (1)

2

then shows that |z| = x2 + y 2 . The relations

2

|z| = z z¯ and |z| =

√

z z¯

(2)

deserve to be stored in memory. The modulus of a complex number z has the

additional properties.

|z1 z2 | = |z1 | |z2 |

and

|z1 |

z1

.

=

z2

|z2 |

(3)

Note that when z1 = z2 = z, the ﬁrst property in (3) shows that

2

z 2 = |z| .

(4)

The property |z1 z2 | = |z1 | |z2 | can be proved using (2) and is left as an

exercise. See Problem 49 in Exercises 1.2.

z1 + z2

or

(x1 + x2, y1 + y2)

y

z2

Distance Again The addition of complex numbers z1 = x1 + iy1 and

z2 = x2 + iy2 given in Section 1.1, when stated in terms of ordered pairs:

(x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 )

z1

is simply the component deﬁnition of vector addition. The vector interpretation of the sum z1 + z2 is the vector shown in Figure 1.3(a) as the main

diagonal of a parallelogram whose initial point is the origin and terminal point

is (x1 + x2 , y1 + y2 ). The diﬀerence z2 − z1 can be drawn either starting from

the terminal point of z1 and ending at the terminal point of z2 , or as a position

vector whose initial point is the origin and terminal point is (x2 − x1 , y2 − y1 ).

See Figure 1.3(b). In the case z = z2 −z1 , it follows from (1) and Figure 1.3(b)

that the distance between two points z1 = x1 + iy1 and z2 = x2 + iy2

in the complex plane is the same as the distance between the origin and the

point (x2 − x1 , y2 − y1 ); that is, |z| = |z2 − z1 | = |(x2 − x1 ) + i(y2 − y1 )| or

x

(a) Vector sum

y

z 2 – z1

or

(x2 – x1, y2 – y1)

z2

z2

–z

1

z1

|z2 − z1 | =

x

(b) Vector difference

Figure 1.3 Sum and diﬀerence

of vectors

(x2 − x1 )2 + (y2 − y1 )2 .

(5)

When z1 = 0, we see again that the modulus |z2 | represents the distance

between the origin and the point z2 .

EXAMPLE 2

Set of Points in the Complex Plane

Describe the set of points z in the complex plane that satisfy |z| = |z − i|.

Solution We can interpret the given equation as equality of distances: The

distance from a point z to the origin equals the distance from z to the point

12

Chapter 1 Complex Numbers and the Complex Plane

y

i. Geometrically, it seems plausible from Figure 1.4 that the set of points z

lie on a horizontal line. To establish this analytically, we use (1) and (5) to

write |z| = |z − i| as:

i

|z – i|

z

x2 + y 2 =

|z|

x2 + (y − 1)2

x2 + y 2 = x2 + (y − 1)2

x

x2 + y 2 = x2 + y 2 − 2y + 1.

Figure 1.4 Horizontal line is the set of

points satisfying |z| = |z − i|.

y

The last equation yields y = 12 . Since the equality is true for arbitrary x,

y = 12 is an equation of the horizontal line shown in color in Figure 1.4.

Complex numbers satisfying |z| = |z − i| can then be written as z = x + 12 i.

Inequalities In the Remarks at the end of the last section we pointed

out that no order relation can be deﬁned on the system of complex numbers.

However, since |z| is a real number, we can compare the absolute values of

two complex

numbers. For

√

√ example, if z1 = 3 + 4i and z2 = 5 − i, then

|z1 | = 25 = 5 and |z2 | = 26 and, consequently, |z1 | < |z2 |. In view of (1),

a geometric interpretation of the last inequality is simple: The point (3, 4) is

closer to the origin than the point (5, −1).

Now consider the triangle given in Figure 1.5 with vertices at the origin,

z1 , and z1 + z2 . We know from geometry that the length of the side of the

triangle corresponding to the vector z1 + z2 cannot be longer than the sum

of the lengths of the remaining two sides. In symbols we can express this

observation by the inequality

z1 + z2

|z1 + z2|

|z1|

|z2|

z1

x

Figure 1.5 Triangle with vector sides

This inequality can be derived using

the properties of complex numbers

in Section 1.1. See Problem 50 in

Exercises 1.2.

☞

|z1 + z2 | ≤ |z1 | + |z2 |.

(6)

The result in (6) is known as the triangle inequality. Now from the identity

z1 = z1 + z2 + (−z2 ), (6) gives

|z1 | = |z1 + z2 + (−z2 )| ≤ |z1 + z2 | + |−z2 | .

Since |z2 | = |−z2 | (see Problem 47 in Exercises 1.2), solving the last result for

|z1 + z2 | yields another important inequality:

|z1 + z2 | ≥ |z1 | − |z2 |.

(7)

But because z1 + z2 = z2 + z1 , (7) can be written in the alternative form

|z1 + z2 | = |z2 + z1 | ≥ |z2 | − |z1 | = − (|z1 | − |z2 |) and so combined with the

last result implies

|z1 + z2 | ≥ |z1 | − |z2 | .

(8)

It also follows from (6) by replacing z2 by −z2 that |z1 + (−z2 )| ≤ |z1 | +

|(−z2 )| = |z1 | + |z2 |. This result is the same as

|z1 − z2 | ≤ |z1 | + |z2 | .

(9)

1.2 Complex Plane

13

From (8) with z2 replaced by −z2 , we also ﬁnd

|z1 − z2 | ≥ |z1 | − |z2 | .

(10)

In conclusion, we note that the triangle inequality (6) extends to any ﬁnite

sum of complex numbers:

|z1 + z2 + z3 + · · · + zn | ≤ |z1 | + |z2 | + |z3 | + · · · + |zn | .

(11)

The inequalities (6)–(10) will be important when we work with integrals of a

function of a complex variable in Chapters 5 and 6.

EXAMPLE 3

An Upper Bound

−1

if |z| = 2.

Find an upper bound for 4

z − 5z + 1

Solution By the second result in (3), the absolute value of a quotient is the

quotient of the absolute values. Thus with |−1| = 1, we want to ﬁnd a positive

real number M such that

1

≤ M.

|z 4 − 5z + 1|

To accomplish this task we want the denominator as small as possible. By

(10) we can write

z 4 − 5z + 1 = z 4 − (5z − 1) ≥ z 4 − |5z − 1| .

(12)

But to make the diﬀerence in the last expression in (12) as small as possible, we

want to make |5z − 1| as large as possible. From (9), |5z − 1| ≤ |5z| + |−1| =

5|z| + 1. Using |z| = 2, (12) becomes

4

z 4 − 5z + 1 ≥ z 4 − |5z − 1| ≥ |z| − (5 |z| + 1)

4

= |z| − 5 |z| − 1 = |16 − 10 − 1| = 5.

Hence for |z| = 2 we have

1

1

≤ .

| z 4 − 5z + 1 |

5

Remarks

We have seen that the triangle inequality |z1 + z2 | ≤ |z1 | + |z2 | indicates

that the length of the vector z1 + z2 cannot exceed the sum of the lengths

of the individual vectors z1 and z2 . But the results given in (3) are

interesting. The product z1 z2 and quotient z1 /z2 , (z2 = 0), are complex

numbers and so are vectors in the complex plane. The equalities |z1 z2 | =

|z1 | |z2 | and |z1 /z 2 | = |z1 | / |z2 | indicate that the lengths of the vectors

z1 z2 and z1 /z2 are exactly equal to the product of the lengths and to the

quotient of the lengths, respectively, of the individual vectors z1 and z2 .