John D. Paliouras - Complex Variables for Scientists and Engineers: Second Edition

Complex Variables for Scientists and Engineers SECOND EDITION Complex Variables for Scientists and Engineers SECOND EDITION John D. PaliourasandDouglas S. MeadowsRochester Institute of Technology Dover Publications, Inc. Mineola, New York Copyright Copyright 1990 by John D. Paliouras and Douglas S. 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. 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. 1990 editionProvided by publisher. 1990 editionProvided 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. II. Title. 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 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 . .

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 . 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.. It has been our experience at Rochester Institute of Technology that 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. D. P.
D. S. M. M.

Complex Variables for Scientists and Engineers SECOND EDITION I Foundations of Complex VariablesC HAPTER 1Complex NumbersS ECTION 1 C OMPLEX N UMBERS AND T HEIR A LGEBRA 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 . 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