A first course in complex analysis with applications

517 Pages·2003·2.987 MB·English

by  Dennis Zill

A First Course in
Complex
Analysis
with Applications
Dennis G. Zill
Loyola Marymount University
Patrick D. Shanahan
Loyola Marymount University
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Library of Congress Cataloging-in-Publication Data
Zill, Dennis G., 1940-
Afirst 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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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 zn 81
2.4.2 The Power Function z1/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 and Residues 301
6.1 Sequences and Series 302
6.2 Taylor Series 313
6.3 Laurent Series 324
6.4 Zeros and Poles 335
6.5 Residues and Residue Theorem 342
6.6 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
6.7 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 Proof of Theorem 2.1 APP-2
II Proof of the Cauchy-Goursat Theorem APP-4
III Table of Conformal Mappings APP-9
Answers for Selected Odd-Numbered Problems ANS-1
Index IND-1

Preface

7.2 Preface
Philosophy Thistextgrewoutofchapters17-20inAdvancedEngineer-
ing Mathematics, Second Edition (Jones and Bartlett Publishers), by Dennis
G. Zill and the late Michael R. Cullen. This present work represents an ex-
pansionandrevisionofthatoriginalmaterialandisintendedforuseineither
aone-semesteroraone-quartercourse. Itsaimistointroducethebasicprin-
ciples and applications of complex analysis to undergraduates who have no
prior knowledge of this subject.
The motivation to adapt the material from Advanced Engineering Math-
ematics into a stand-alone text sprang from our dissatisfaction with the suc-
cession of textbooks that we have used over the years in our departmental
undergraduatecourseofferingincomplexanalysis. Ithasbeenourexperience
thatbooksclaimingtobeaccessibletoundergraduateswereoftenwrittenata
level that was too advanced for our audience. The “audience” for our junior-
level course consists of some majors in mathematics, some majors in physics,
but mostly majors from electrical engineering and computer science. At our
institution, atypicalstudentmajoringinscienceorengineeringdoesnottake
theory-oriented mathematics courses in methods of proof, linear algebra, ab-
stract 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, con-
centrating 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
calculusoffunctionsofacomplexvariable. Donotmisinterpretthepreceding
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 ma-
jors 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
toseven. Thiswasdonefortwo“reasons”: toprovideanappropriatequantity
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 logarith-
mic functions are covered. The subtle notions of multiple-valued func-
tions and branches are also discussed.
• Chapter 5 The chapter begins with a review of real integrals (in-
cluding line integrals). The definitions of real line integrals are used to
motivate the definition of the complex integral. The famous Cauchy-
Goursat theorem and the Cauchy integral formulas are introduced in
this chapter. Although we use Green’s theorem to prove Cauchy’s the-
orem, a sketch of the proof of Goursat’s version of this same theorem is
given in an appendix.
• Chapter6 Thischapterintroducestheconceptsofcomplexsequences
andinfiniteseries. ThefocusofthechapterisonLaurentseries,residues,
andtheresiduetheorem. Evaluationofcomplexaswellasrealintegrals,
summation of infinite series, and calculation of inverse Laplace and in-
verse 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 Eachchapterbeginswithitsownopeningpagethatincludesa
tableofcontentsandabriefintroductiondescribingthematerialtobecovered
in the chapter. Moreover, each section in a chapter starts with introduc-
tory 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