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COMPLEX VARIABLES

AND APPLICATIONS

SEVENTH EDITION

James Ward Brown Professor of Mathematics The University of Michigan-Dearborn

Rue1 V. Churchill Late Professor of Mathematics The University of Michigan

Higher Education

Boston Burr Ridge, tL Dubuque, lA Madison, WI New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

CONTENTS

Preface

• Sums and Products 1 Complex Numbers
• Basic Algebraic Properties
• Further Properties
• Moduli
• Complex Conjugates
• Exponential Fonn
• Products and Quotients in Exponential Form
• Roots of Complex Numbers
• Examples
• Regions in the Complex Plane
• Functions of a Complex Variable 2 Analytic Functions
• Mappings
• Mappings by the Exponential Function
• Limits
• Theorems on Limits
• Limits Involving the Point at Infinity
• Continuity
• Derivatives
• Differentiation Formulas
• Cauchy-Riemann Equations
  • Sufficient Conditions for Differentiability
  • Polar Coordinates
  • Analytic Functions
  • Examples
  • Harmonic Functions
  • Uniquely Determined Analytic Functions
  • Reflection Principle
  • The Exponential Function 3 Elementary Functions
  • The Logarithmic Function
  • Branches and Derivatives of Logarithms
  • Some Identities Involving Logarithms
  • Complex Exponents
  • Trigonometric Functions
  • Hyperbolic Functions
    • Inverse Trigonometric and Hyperbolic Functions
  • Derivatives of Functions w ( t ) 4 Integrals
    • Definite Integrals of Functions w ( t )
    • Contours
    • Contour Integrals
  • Examples
    • Upper Bounds for Moduli of Contour Integrals
    • Antiderivatives
    • Examples
    • Cauchy-Goursat Theorem
    • Proof of the Theorem
• , Simply and Multiply Connected Domains - Cauchy Integral Formula - Derivatives of Analytic Functions - Liouville's Theorem and the Fundamental Theorem of Algebra - Maximum Modulus Principle - Convergence of Sequences 5 Series - Convergence of Series - Taylor Series - Examples - Laurent Series - Examples - Absolute and Uniform Convergence of Power Series - Continuity of Sums of Power Series - Integration and Differentiation of Power Series - Uniqueness of Series Representations - Multiplication and Division of Power Series
  • Residues 6 Residues and Poles
  • Cauchy's Residue Theorem
  • Using a Single Residue
  • The Three Types of Isolated Singular Points
  • Residues at Poles
  • Examples
  • Zeros of Analytic Functions
  • Zeros and Poles
  • Behavior off Near Isolated Singular Points
• Evaluation of Improper Integrals 7 Applications of Residues
  • Example
  • Improper Integrals from Fourier Analysis
  • Jordan's Lemma
  • Indented Paths
  • An Indentation Around a Branch Point
  • Integration Along a Branch Cut
• Definite Integrals involving Sines and Cosines
  • Argument Principle
• Roucht's Theorem
• Inverse Laplace Transforms
  • Examples
• Linear Transformations 8 Mapping by Elementary Functions
• The Transformation w = l/z
• Mappings by llz
• Linear Fractional Transformations
• An Implicit Fonn
• Mappings of the Upper Half Plane
• The Transformation w = sin z
• Mappings by z2 and Branches of z'I2
• Square Roots of Polynomials
• Riemann Surfaces
  • Surfaces for Related Functions
• Preservation of Angles 9 Conformal Mapping
• Scale Factors
• Local Inverses
• Harmonic Conjugates
• Transformations of Harmonic Functions
• Transformations of Boundary Conditions
• Steady Temperatures 10 Applications of Conformal Mapping
• Steady Temperatures in a Half Plane
• A Related Problem
• Temperatures in a Quadrant
• Electrostatic Potential
• Potential in a Cylindrical Space
• Two-Dimensional Fluid How
• The Stream Function
• Flows Around a Comer and Around a Cylinder
• Mapping the Real Axis onto a Polygon 11 The Schwarz-Christoffel Transformation
• Schwarz-Christoffel Transformation
• Triangles and Rectangles
• Degenerate Polygons
• Fluid Flow in a Channel Through a Slit
• Flow in a Channel with an Offset
• Electrostatic Potential about an Edge of a Conducting Plate
• Poisson Integral Formula 12 Integral Formulas of the Poisson Type
• Dirichlet Problem for a Disk
• Related Boundary Value Problems
  • Schwarz Integral Formula
• Dirichlet Problem for a Half Plane
• Neumann Problems
• Bibliography Appendixes
• Table of Transformations of Regions

function; and the sections on trigonometric and hyberbolic functions are now closer to the ones on their inverses. Encouraged by comments from users of the book in the past several years, we have brought some important material out of the exercises and into the text. Examples of this are the treatment of isolated zeros of analytic functions in Chap. 6 and the discussion of integration along indented paths in Chap. 7. The Jirst objective of the book is to develop those parts of the theory which are prominent in applications of the subject. The second objective is to furnish an introduction to applications of residues and conformal mapping. Special emphasis is given to the use of conformal mapping in solving boundary value problems that arise in studies of heat conduction, electrostatic potential, and fluid flow. Hence the

book may be considered as a companion volume to the authors' "Fourier Series and

Boundary Value Problems" and Rue1 V. Churchill's "Operational Mathematics," where other classical methods for solving boundary value problems in partial differential equations are developed. The latter book also contains further applications of residues in connection with Laplace transforms.

This book has been used for many years in a three-hour course given each term at

The University of Michigan. The classes have consisted mainly of seniors and graduate

students majoring in mathematics, engineering, or one of the physical sciences. Before taking the course, the students have completed at least a three-term calculus sequence, a first course in ordinary differential equations, and sometimes a term of advanced

calculus. In order to accommodate as wide a range of readers as possible, there are

footnotes referring to texts that give proofs and discussions of the more delicate results from calculus that are occasionally needed. Some of the material in the book need not be covered in lectures and can be left for students to read on their own. If mapping by elementary functions and applications of conformal mapping are desired earlier in the course, one can skip to Chapters 8, 9, and 10 immediately after Chapter 3 on elementary functions.

Most of the basic results are stated as theorems or corollaries, followed by

examples and exercises illustrating those results. A bibliography of other books, many of which are more advanced, is provided in Appendix 1. A table of conformal transformations useful in applications appears in Appendix 2. In the preparation of this edition, continual interest and support has been provided

by a number of people, many of whom are family, colleagues, and students. They

include Jacqueline R. Brown, Ronald P. Morash, Margret H. Hoft, Sandra M. Weber,

Joyce A. Moss, as well as Robert E. Ross and Michelle D. Munn of the editorial staff

at McGraw-Hill Higher Education.

James Ward Brown

COMPLEX VARIABLES AND APPLICATIONS

C H A P T E R

COMPLEX NUMBERS

In this chapter, we survey the algebraic and geometric structure of the complex number system. We assume various corresponding properties of real numbers to be known.

1. SUMS AND PRODUCTS Complex numbers can be defined as ordered pairs (x, y) of real numbers that are to be interpreted as points in the complex plane, with rectangular coordinates x and y, just as real numbers x are thought of as points on the real line. When real numbers

x are displayed as points (x, 0) on the real axis, it is clear that the set of complex

numbers includes the real numbers as a subset. Complex numbers of the form (0, y)

correspond to points on the y axis and are called pure imaginary numbers. The y axis is, then, referred to as the imaginary axis. It is customary to denote a complex number (x, y) by z, so that

The real numbers x and y are, moreover, known as the real and imaginary parts of z, respectively; and we write

l h o complex numbers zl = ( x l , yl) and z2 = (x2, y2) are equal whenever they have the same real parts and the same imaginary parts. Thus the statement zl = means

that zl and z2 correspond to the same point in the complex, or z, plane.

2 COMPLEXNUMBERS CHAP. I

The sum zl + zz and the product zlz2 of two complex numbers zl = (xl, yl) and 22 = (x2, y2) are defined as follows:

Note that the operations defined by equations (3) and (4) become the usual operations of addition and multiplication when restricted to the real numbers:

The complex number system is, therefore, a natural extension of the real number system. Any complex number z = (x, y) can be written z = (x, 0) + (0, y), and it is easy to see that (0, l)(y, 0) = (0, y). Hence

and, if we think of a real number as either x or (x, 0) and let i denote the imaginary number (0, 1 ) (see Fig. I), it is clear that*

Also, with the convention z2 = zz, z3 = zz2, etc,, we find that

i (^2) = (0, l)(O, 1) = (-1, O),

i = (0, 1)

I FIGURE 1

In view of expression (5), definitions (3) and (4) become

*In electrical engineering, the letter j is used instead of i

4 COMPLEXNUMBERS CHAP. I

(Exercise 8) - (i y ) = (-i) y = i (- y ) .Additive inverses are used to define subtraction:

SO if z l = (xl, yl) and z2 = ( ~ 2 ,~ 2 then) ~

For any nonzero complex number z = (x, y), there is a number z-I such that

zz-' = 1. This multiplicative inverse is less obvious than the additive one. To find it,

we seek real numbers u and v, expressed in terms of x and y ,such that

According to equation (4), Sec. 1, which defines the product of two complex numbers,

u and v must satisfy the pair

of linear simultaneous equations; and simple computation yields the unique solution

So the multiplicative inverse of z = ( x , y) is

The inverse z-' is not defined when z = 0. In fact, z = 0 means that x2 + y2 = 0; and this is not permitted in expression (8).

EXERCf SES

1. Verify that (a) i - 1 - i - 2 (b) (2, -3)(-2, 1)=(-1,8); 2. Show that (a)Re(iz)=-Irnz; (b)Im(iz)=ReZ. 3. Show that (1 + z ) ~ = 1 + 2z + z2. 4. Verify that each of the two numbers z = 1 & i satisfies the equation z2 - 22 + 2 = 0. S. Prove that multiplicationis commutative, as stated in the second of equations (I), Sec. 2.

6. Verify (a) the associative law for addition, stated in the first of equations (2), Sec. 2; (b) the distributive law (3), Sec. 2.

SEC. 3 FURTHERPROPERTIES 5

7. Use the associative law for addition and the distributive law to show that
8. By writing i = (0, 1) and y = (y, 0), show that -(iy) = (-i)y = i(-y). 9. (a) Write (x, y ) + ( u , v) = (x ,y) and point out how it follows that the complex number 0 = (0, 0) is unique as an additive identity. (b) Likewise, write (x, y ) ( u , v) = (x, y) and show that the number 1 = (1,O) is a unique multiplicative identity. 10. Solve the equation z2+ z + 1 = 0 for z = (x, y) by writing

and then solving a pair of simultaneous equations in x and y. Suggestion: Use the fact that no real number x satisfies the given equation to show

that y # 0.

3. FURTHER PROPERTIES

In this section, we mention a number of other algebraic properties of addition and multiplication of complex numbers that follow from the ones already described in

Sec. 2. Inasmuch as such properties continue to be anticipated because they also apply

to real numbers, the reader can easily pass to Sec. 4 without serious disruption. We begin with the observation that the existence of multiplicative inverses enables us to show that ifa product zlz2 is zero, then s o is at least one of the factors zl and

22. For suppose that t l z z = 0 and zl# 0. The inverse 'z; exists; and, according to the definition of multiplication, any complex number times zero is zero. Hence

That is, if zlz2 = 0, either z1 = 0 or zz = 0; or possibly both zl and z2 equal zero. Another way to state this result is that iftwo complex numbers zl and z2 are nonzero, then so is their product z lz2. Division by a nonzero complex number is defined as follows:

If z 1 = (xl, y l ) and 22 = (xZ, y2), equation (1) here and expression (8) in Sec. 2 tell us that

SEC. 3 EXERCISES 7

EXAMPLE. Computations such as the following are now justified:

Finally, we note that the binomial formula involving real numbers remains valid

with complex numbers. That is, if sl and z2 are any two complex numbers,

where

n! (;) = k! ( n - k )!

( k = O , 1 , 2 ,... , n )

and where it is agreed that O! = 1. The proof, by mathematical induction, is left as an exercise.

EXERCISES

1. Reduce each of these quantities to a real number:

Ans. (a) - 2 / 5 ; ( b - 1 ; (c) -4.

2. Show that 1

(a) (-l)z = -z; (b) - = z (z # 0).

l/z

3. Use the associative and commutative laws for multiplication to show that
4. Prove that if ~ 1 ~ = 2 0, ~then at least one of the three factors 3 is zero. Suggestion: Write (z1z2)z3= 0 and use a similar result (Sec. 3) involving two factors.

5. Derive expression (2), Sec. 3, for the quotient z1/z2 by the method described just after

it.

6. With the aid of relations (6) and (7) in Sec. 3, derive identity (8) there.
7. Use identity (8) in Sec. 3 to derive the cancellation law:

8 COMPLEXNUMBERS CHAP. I

8. Use mathematical induction to verify the binomial formula (9) in Sec. 3. More precisely, note first that the formula is true when n = 1. Then, assuming that it is valid when n = rn where m denotes any positive integer, show that it must hold when n = m + 1. 4. MODULI

It is natural to associate any nonzero complex number z = x + iy with the directed line segment, or vector, from the origin to the point (x, y) that represents z (Sec. 1) in the complex plane. In fact, we often refer to z as the point z or the vector z. In Fig. 2 the numbers z = x + iy and -2 + i are displayed graphically as both points and radius vectors.

I -2 X FIGURE 2

According to the definition of the sum of two complex numbers z l = x l + iyl and 22 = x2 + iy2, the number zl + z2 corresponds to the point ( x l + x2, y1 + y2). It also corresponds to a vector with those coordinates as its components. Hence z l + z

may be obtained vectorially as shown in Fig. 3. The difference z l - z 2 = z l + (-z2)

corresponds to the sum of the vectors for z l and -22 (Fig. 4).

Although the product of two complex numbers z l and 22 is itself a complex number represented by a vector, that vector lies in the same plane as the vectors for z 1 and 22. Evidently, then, this product is neither the scalar nor the vector product used in ordinary vector analysis. The vector interpretation of complex numbers is especially helpful in extending the concept of absolute values of real numbers to the complex plane. The modulus, or absolute value, of a complex number z = x + iy is defined as the nonnegative real

CHAP. I

Thus

(3) R e z s 1 R e z l ~ l z l and^ I m z s 1 I m z l S l z l +

We turn now to the triangle inequality, which provides an upper bound for the modulus of the sum of two complex numbers z l and 22:

This important inequality is geometrically evident in Fig. 3, since it is merely a statement that the length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides. We can also see from Fig. 3 that inequality (4)

is actually an equality when 0, zl, and z2 are collinear, Another, strictly algebraic,

derivation is given in Exercise 16, Sec. 5. An immediate consequence of the triangle inequality is the fact that

To derive inequality (5), we write

which means that

This is inequality (5) when lz 1 2 1 z21. If 1 z 11 < 1 z2 1, we need only interchange z 1 and 22 in inequality ( 6 ) to get

which is the desired result. Inequality (5) tells us, of course, that the length of one side of a triangle is greater than or equal to the difference of the lengths of the other two sides. Because I - 221 = 1z21, one can replace z2 by -z2 in inequalities ( 4 ) and (5) to summarize these results in a particularly useful form:

EXAMPLE 3. If a point z lies on the unit circle 12) = 1 about the origin, then

and

The triangle inequality (4) can be generalized by means of mathematical induc- tion to sums involving any finite number of terms:

To give details of the induction proof here, we note that when n = 2, inequality (9) is just inequality (4). Furthermore, if inequality (9) is assumed to be valid when n = m, it must also hold when n = m + 1 since, by inequality (4),

EXERCISES

1. Locate the numbers zl+ z2 and zl - z2 vectorially when

(c) z1 = (-3, l), zz = (1,4); ( d ) zl = X I + iyl,^22 = X I^ -^ i ~ 1.

2. Verify inequalities (3), Sec. 4, involving Re z , Im z, and lzl. 3. verify that &lzl 2 IRezl + IImzl. Suggestion: Reduce this inequality to (Ix I - 1 ~ 1 2 ) ~ 0.

4. In each case, sketch the set of points determined by the given condition:

5. Using the fact that lz - z2 I is the distance between two points z I and ZZ, give a geometric

argument that (a) 1 z - 4 i I + lz + 4i I = 10 represents an ellipse whose foci are (0, f4); (b) [z - 11 = [ Z + i 1 represents the line through the origin whose slope is - 1.

5. COMPLEX CONJUGATES The complex conjugate, or simply the conjugate, of a complex number z = x + iy is

defined as the complex number x - iy and is denoted by 7; that is,

The number 'Z is represented by the point (x, -y), which is the reflection in the real

axis of the point ( x , y) representing z (Fig. 5). Note that

Z = Z and l? t = l z l

for all z. If z l = xl + iyl and 22 = x2 + i ~ 2 , then

zl + z2 = ( x l + x2) - i(yl + y2) = ( X I - i ~ i ) + ( ~ 2 - i~2).