Complex Variables for Scientists and Engineers: Second Edition

512 Pages·2014·14.51 MB·English

by  John D. Paliouras

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 1 Complex Numbers
Section 1 Complex Numbers and Their Algebra
2 Geometry of Complex Numbers
Appendix 1 Part A: A Formal Look at Complex Numbers Part
Part B: Stereographic Projection
Chapter 2 Complex Functions
Section 3 Preliminaries
4 Definition and Elementary Geometry of a Complex Function
5 Limits, Continuity
6 Differentiation
7 The Cauchy-Riemann Equations
8 Elementary Complex Functions: Definitions and Basic
Properties
9 Analytic Functions; Domains of Analyticity
Appendix 2 Proofs of Theorems
Chapter 3 Harmonic Functions with Applications
Section 10 Harmonic Functions
11 Applications to Fluid Flow
12 Applications to Electrostatics
Appendix 3 Part A: The Equations of Fluid Flow
Part B: Basic Laws of Electrostatics
Chapter 4 Complex Integration
Section 13 Paths; Connectedness
14 Line Integrals
15 The Complex Integral
Appendix 4 Proofs of Theorems
Chapter 5 Cauchy Theory of Integration
Section 16 Integrals of Analytic Functions; Cauchy’s Theorem
17 The Annulus Theorem and Its Extension
18 The Cauchy Integral Formulas; Morera’s Theorem
Appendix 5 Part A: Proofs of Theorems
Part B: Proof of the Cauchy Integral Theorem
Part C: The Winding Number and the Generalized Cauchy
Theorems
Chapter 6 Complex Power Series
Section 19 Sequences and Series of Complex Numbers
20 Power Series
21 Power Series as Analytic Functions
22 Analytic Functions as Power Series
Appendix 6 Part A: Proofs of Theorems
Part B: More on Sequences and Series; The Cauchy-
Hadamard Theorem
Chapter 7 Laurent Series; Residues
Section 23 Laurent Series
24 Singularities and Zeros of an Analytic Function
25 Theory of Residues
26 Evaluation of Certain Real Integrals by Use of Residues
Appendix 7 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