Complex Variables for Scientists and Engineers Second Edition PDF

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Complex Variables for Scientists and Engineers SECOND EDITION Complex Variables for Scientists and Engineers SECOND EDITION John D. Paliouras and Douglas S. Meadows Rochester Institute of Technology Dover Publications, Inc. Mineola, New York Copyright Copyright © 1990 by John D. Paliouras and Dougl...

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Complex Variables for Scientists and Engineers SECOND EDITION

Complex Variables for Scientists and Engineers SECOND EDITION

John D. Paliouras and

Douglas S. Meadows Rochester Institute of Technology

Dover Publications, Inc. Mineola, New York

Copyright Copyright © 1990 by John D. Paliouras and Douglas S. Meadows All rights reserved. Bibliographical Note This Dover edition, first published in 2014, is an unabridged republication of the work originally published by Macmillan Publishing Company, New York, in 1990. Library of Congress Cataloging-in-Publication Data Paliouras, John D. Complex variables for scientists and engineers / John D. Paliouras and Douglas S. Meadows.—Second edition. p. cm.—(Dover books on mathematics) Summary: “This outstanding undergraduate text for students of science and engineering requires only a standard course in elementary calculus. Designed to provide a thorough understanding of fundamental concepts and create the basis for higher-level courses, the treatment features numerous examples and extensive exercise sections of varying difficulty, plus answers to selected exercises. 1990 edition”— Provided by publisher. “This Dover edition, first published in 2014, is an unabridged republication of the work originally published by Macmillan Publishing Company, New York, in 1990.” “This Dover edition, first published in 2014, is an unabridged republication of the work originally published by Macmillan Publishing Company, New York, in 1990.” eISBN-13: 978-0-486-78222-5 1. Functions of complex variables. I. Meadows, Douglas S. II. Title. QA331.7.P34 2014 515'.9-dc23 2013029692 Manufactured in the United States by Courier Corporation 49347401 2014 www.doverpublications.com

We gratefully dedicate this edition to our wives Patricia Joyce Paliouras Doris Marguerite Meadows

Preface A first course on complex variables taught to students in the sciences and engineering is invariably faced with the difficult task of meeting two basic objectives: (1) It must create a sound foundation based on the understanding of fundamental concepts and the development of manipulative skills, and (2) it must reach far enough so that the student who completes such a course will be prepared to tackle relatively advanced applications of the subject in subsequent courses that utilize complex variables. This book has been written with those two objectives in mind. Its main goal is to provide a development leading, over a minimal and yet sound path, to the fringes of the promised land of applications of complex variables or to a second course in the theory of analytic functions. The level of the development in Part I is quite elementary and its main theme is the calculus of complex functions. The only prerequisite for its study is a standard course in elementary calculus. The topological aspects of the subject are developed only to the extent necessary to give the reader an intuitive understanding of these matters. Theorems are discussed informally and, whenever possible, are illustrated via examples. Numerous examples illustrate new concepts soon after they are introduced as well as theorems that lend themselves readily to problem solving. Exercises are usually divided into three categories in order to accommodate problems that range from the routine type to the more formidable ones. Among the many changes in this second edition, the most substantive are the inclusion of many more applications of complex function theory. A preliminary discussion of applications of harmonic functions to physical problems is provided in Chapter 3. The concepts introduced in Chapters 1 and 2 provide sufficient background for this material. The more comprehensive applications presented in Chapter 9, some of which are extensions of those in Chapter 3, make full use of the theory of complex variables developed in Chapters 4 through 8. The newly introduced topics for the second edition include the applications in Chapters 3 and 9, the development of the Poisson integral formulas in Section 31, the winding number and its use in the generalization of Cauchy’s integral theorem in Appendix 5(C), the concepts of analytic continuation and the Schwarz reflection principle in Section 38, and a brief discussion of the Riemann

mapping theorem and the Bieberbach conjecture in Appendix 9. Other changes in the second edition include the following: • The study of mapping properties of analytic functions has been greatly expanded and placed later in the book, in Chapter 8. • The practice we followed in the first edition of placing the proofs of all theorems in appendixes to the chapters has been modified, in that in this edition, we include in the bodies of the chapters those proofs which we believe provide a constructive understanding of the theorems, while proofs that are largely technical are again placed in the appendixes. The symbol □ □ is used to indicate the end of a proof. • We have substantially expanded the material on conformal mapping, Riemann surfaces, branch points and branch cuts, the behavior of functions at infinity, and the Schwarz-Christoffel integral. The overall structure of the book has been revised into two parts. Part I, which consists of Chapters 1 through 7, provides the core of a first course in complex variables with applications. It includes the development of the primary concept of analytic function, the Cauchy integral theory, the series development of analytic functions through evaluation of integrals by residues, and some elementary applications of harmonic functions. The relatively elementary level of the first edition of this book has been retained in Part I, with the exception of Chapter 3. For although Chapters 1 and 2 provide sufficient background for it, this chapter does place stronger demands on the student. Since Chapters 4 through 8 do not depend on the material in Chapter 3, it can be omitted if desired. Part II of this edition, consisting of Chapters 8 through 10, presents an introduction to some of the deeper aspects of complex function theory. It includes a discussion of mapping properties of analytic functions, applications to various vector field problems with boundary conditions, and a collection of further theoretical results. The level of the material in Part II is somewhat more sophisticated and more demanding than that in Part I. It has been our experience at Rochester Institute of Technology that Chapters 1, 2, and 4 through 7 constitute a briskly paced one-quarter introductory course in complex variables. The students who take the course are primarily engineering majors, with a sprinkling of science and mathematics majors. With Chapter 3 and parts or all of Part II, the book provides ample material for a course covering one semester, two quarters, or a full year. Also, for courses of longer duration than one quarter, the structure of the book permits a high degree of flexibility in the choice of material. Inclusion of Chapters 3 and 9 provides a

curricular path with an emphasis on applications. However, omission of part or all of Chapters 3 and 9 will not affect the continuity if a less applied course is desired. Additional flexibility is provided by the fact that many of the proofs of theorems are placed in appendixes to the chapters. By the inclusion of material from the appendixes, a mathematically rigorous and complete course may be developed. On the other hand, for a course that must cover a great deal of material in a brief time at the expense of complete mathematical rigor, the book provides such a pathway, without loss of continuity, if one omits most of the material in the chapter appendixes. The authors of this edition would like to express their thanks and appreciation to several members of the Mathematics Department at Rochester Institute of Technology. In particular, our thanks go to Charles Haines, Edwin Hoefer, Pasquale Saeva, Richard Orr, and Patricia Clark, who very kindly read portions of the manuscript and provided many valuable suggestions and comments. J. D. P. D. S. M.

Contents Preface PART I



FOUNDATIONS OF COMPLEX VARIABLES

Chapter Section Appendix

1 1 2 1

Complex Numbers Complex Numbers and Their Algebra Geometry of Complex Numbers Part A: A Formal Look at Complex Numbers Part Part B: Stereographic Projection

Chapter Section

2 3 4 5 6 7 8

Appendix

9 2

Complex Functions Preliminaries Definition and Elementary Geometry of a Complex Function Limits, Continuity Differentiation The Cauchy-Riemann Equations Elementary Complex Functions: Definitions and Basic Properties Analytic Functions; Domains of Analyticity Proofs of Theorems

Chapter Section Appendix

3 10 11 12 3

Chapter

4

Harmonic Functions with Applications Harmonic Functions Applications to Fluid Flow Applications to Electrostatics Part A: The Equations of Fluid Flow Part B: Basic Laws of Electrostatics Complex Integration

Section Appendix

13 14 15 4

Paths; Connectedness Line Integrals The Complex Integral Proofs of Theorems

Chapter Section Appendix

5 16 17 18 5

Cauchy Theory of Integration Integrals of Analytic Functions; Cauchy’s Theorem The Annulus Theorem and Its Extension The Cauchy Integral Formulas; Morera’s Theorem Part A: Proofs of Theorems Part B: Proof of the Cauchy Integral Theorem Part C: The Winding Number and the Generalized Cauchy Theorems

Chapter Section Appendix

6 19 20 21 22 6

Complex Power Series Sequences and Series of Complex Numbers Power Series Power Series as Analytic Functions Analytic Functions as Power Series Part A: Proofs of Theorems Part B: More on Sequences and Series; The CauchyHadamard Theorem

Chapter Section Appendix

7 23 24 25 26 7

Laurent Series; Residues Laurent Series Singularities and Zeros of an Analytic Function Theory of Residues Evaluation of Certain Real Integrals by Use of Residues Proof of Laurent’s Theorem; Uniqueness of Taylor and Laurent Expansions

PART II



FURTHER THEORY AND APPLICATIONS OF COMPLEX VARIABLES

Chapter

8

Mapping Properties of Analytic Functions

Section Appendix

27 28 29 8

Algebraic Functions Transcendental Functions Behavior of Functions at Infinity Part A: Riemann Surfaces of Multivalued Functions Part B: Integration Involving Branch Points

Chapter Section Appendix

9 30 31 32 33 34 9

Conformai Mapping with Applications Conformality and Analytic Functions Laplace’s Equation Applications to Boundary Value Problems Applications to Aerodynamics The Schwarz-Christoffel Integral Univalent Functions

Chapter Section

10 35 36 37 38

Further Theoretical Results The Maximum Modulus Principle Liouville’s Theorem ; The Fundamental Theorem of Algebra Behavior of Functions Near Isolated Singularities Analytic Continuation and the Schwarz Reflection Principle

Bibliography Answers to Selected Exercises Index

Complex Variables for Scientists and Engineers SECOND EDITION

I Foundations of Complex Variables

CHAPTER 1 Complex Numbers SECTION 1

SECTION 2

APPENDIX 1

Definition of a complex number. Some special complex numbers. Equality, sum, difference, product, and quotient of complex numbers. Conjugation. Basic algebraic laws. The complex plane; real and imaginary axes. Modulus and argument of a complex number. Distance between two complex numbers. Principal value of the argument. Properties of the modulus. Complex form of twodimensional curves. Polar form of a complex number. Equality in polar form. Roots of complex numbers; roots of unity. Geometry of rational operations on complex numbers. Part A: A formal look at complex numbers. Part B: Stereographic projection.

SECTION 1 COMPLEX NUMBERS AND THEIR ALGEBRA It is assumed that the reader is familiar with the system of real numbers and their elementary algebraic properties. Our work in this book will take us to a larger system of numbers that have been given the unfortunate name “imaginary” or “ complex numbers.” A historical account of the discovery of such numbers and of their development into prominence in the world of mathematics is outside the scope of this book. Suffice it to say that the need for such numbers arose from the need to find square roots of negative numbers. The system of complex numbers can be formally introduced by use of the concept of an “ordered pair” (a, b) of real numbers. The set of all such pairs with

appropriate operations defined on them can be defined to constitute the system of complex numbers. The reader who is interested in this formal approach is referred to Appendix 1(A). Here, with due apologies to the formalists, we shall proceed to define the complex numbers in the more conventional, if somewhat incomplete manner. We will see that the system of complex numbers is a “natural extension” of the real numbers in the sense that a real number is a special case of a complex number. The set of complex numbers is defined to be the totality of all quantities of the form

where a and b are real numbers and i2 = − 1. To the reader who may wonder what is so incomplete about this approach of defining the complex numbers, we point out that nothing is said as to the meaning of the implied multiplication in the terms ib and bi. If z = a + ib is any complex number, a is called the real part or real component of z and b is called the imaginary part or imaginary component of z; we sometimes denote them

respectively, and reemphasize the fact that both Re (z) and Im (z) are real numbers. If Re (z) = 0 and Im (z) ≠ 0, then z is called pure imaginary; for example, z = 3i is such a number. In particular, if Re (z) = 0 and Im (z) = 1, we write z = i and we call this number the imaginary unit. If Im (z) = 0, z reduces to the real number Re (z); in that sense, one can think of any real number x as being a complex number of the form z = x + 0i. This illustrates the fact that was noted earlier, namely, that the system of complex numbers is an extension of the system of real numbers; equivalently, we say that the latter is a special case of the former. We now proceed to define some of the basic operations on complex numbers. For the remainder of this section,

are three arbitrary complex numbers. Equality of complex numbers is defined quite naturally. Thus two complex numbers are equal provided that their real parts and their imaginary parts are,

respectively, equal; that is,

The sum of two complex numbers is obtained by adding the real parts and the imaginary parts, respectively; that is,

The product of z1 and z2 is found by multiplying the two numbers as if they were two binomials, using the reduction formula i2 = − 1 and collecting “like terms”; thus

Then, using the preceding formula, one defines the nonnegative integral powers of a complex number z as in the case of real numbers. Thus

The zero (additive identity) of the system of complex numbers is the number

which we simply write 0, and the unity (multiplicative identity) is the number

which we write simply as 1. Using the definitions of addition and multiplication given above, we find that it is very easy to show that for any complex number z = x + iy,

thus verifying that these two numbers are, indeed, the additive and multiplicative identities of the system. If z ≠ 0, the zero power of z is 1:

Again, if z is any complex number, there is one and only one complex number,

which we will denote by − z, such that

− z is called the negative of z and it is easy to verify that

For any nonzero complex number z = x + iy there is one and only one complex number, which we will denote by z−1, such that

z−1 is called the reciprocal (multiplicative inverse) of z and a direct calculation from the preceding equation yields

See the Note prior to Example 1. To facilitate further algebraic manipulations, we now define the difference of two numbers by

which, through an easy calculation, yields

Finally, we define the quotient of two numbers by

In particular, 1 / z = z−1. A straightforward, if somewhat involved, calculation in which we utilize the formula for the reciprocal, above, yields the formula

See the Note prior to Example 1. In addition to the operations defined above, we have a “new” operation, called conjugation, defined on complex numbers as follows: If z = x + iy, then the conjugate of z, denoted , is defined by

Unlike the four “binary” operations defined earlier, conjugation is a “unary” operation; that is, it acts on one number at a time and has the effect of negating the imaginary part of the number.

ALGEBRAIC PROPERTIES OF COMPLEX NUMBERS The operations defined above obey the following laws. 1. Commutative laws:

2. Associative laws:

3. Distributive law of multiplication over addition:

4. Distributive laws of conjugation:

5. 6. z = [Re (z)]2 + [Im (z)]2. Some of these properties are proved in the examples that follow; the remaining ones are left for the exercises. NOTE:

With the concept of the conjugate at our disposal, calculation of the reciprocal of a complex number and of the quotient of two numbers becomes much easier than by use of the method suggested earlier. The reason for this revolves around the fact that the product of a complex number and its conjugate, which appears in the denominator of the following formulas, is a real number that makes the calculation of the quotient easier to effect. Specifically, we have the following two formulas :

In other words, in order to find the quotient of two complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator. As an exercise, the reader should verify that the results obtained by use of these two formulas agree with those obtained earlier. EXAMPLE 1 If z = 5 − 5i and w = − 3 + 4i, find z + w, z − w, z − w, , and z/w. Using the definitions of the respective operations, we find that

EXAMPLE 2

Prove the commutative law for addition: z1 + z2 = z2 + z1. We carry out this proof by using the corresponding law for real numbers, which states that for any two real numbers a and b, a + b = b + a. Thus we have

EXAMPLE 3 Prove that conjugation distributes over multiplication: Let z = a + bi and w = c + di. On the one hand, we have

.

On the other hand,

Clearly, the two sides are equal and the proof is complete. EXAMPLE 4 Prove property 6: If z = x + iy, then z = x2 + y2.

This property says that given any complex number, the product of the number and its conjugate is always a nonnegative real number, since it is the sum of squares of two real numbers.

By now it should be apparent to the reader that most of the familiar algebraic properties of the real numbers are shared by the complex numbers. There is, however, a particular property of the real numbers, namely, the property of order, which does not carry over to the complex case. By this we mean that given two arbitrary complex numbers z and w such that z ≠ w, no reasonable meaning can be attached to the expression

discussion and proof of this fact are left as an exercise for the reader. See Review Exercise 19 at the end of the chapter. EXERCISE 1 A In Exercises 1.1 – 1.10, perform the operations indicated, reducing the answer to the form A + Bi. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15

(5 − 2i)+ (2+ 3i). (2 − i) − (6 − 3i). (2 + 3i)(−2 − 3i) −i(5 + i). i · ī. (a + bi)(a − bi). 6i/(6 − 5i). (a + bi)...