Complex Analysis for Mathematics and Engineering, 3rd Edition

528 Pages·2009·25.12 MB·English

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Complex Analysis for Mathematics and Engineering
Third Edition
John H. Mathews
California State UniversityFullerton
Russell W. Howell
Westmont College
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Copyright ©1997 by Jones and Bartlett Publishers. © 1996 Times Mirror Higher
Education Group. Inc.
All rights reserved. No part of this publication may be reproduced or transmitted in any
form or by any means, electronic or mechanical, including photocopy, recording, or
any information storage or retrieval system, without permission in writing from the
publisher.
Library of Congress Catalog Number: 95-76589
ISBN 0-7637-0270-6
Printed in the United States of America
00 99 10 9 8 7 6 5 4 3
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Contents
Preface ix
Chapter 1 1
Complex Numbers
1
1.1 The Origin of Complex Numbers
5
1.2 The Algebra of Complex Numbers
12
1.3 The Geometry of Complex Numbers
18
1.4 The Geometry of Complex Numbers, Continued
24
1.5 The Algebra of Complex Numbers, Revisited
30
1.6 The Topology of Complex Numbers
Chapter 2 38
Complex Functions
38
2.1 Functions of a Complex Variable
41
2.2 Transformations and Linear Mappings
47
2.3 The Mappings w = zn and w = z1/n
53
2.4 Limits and Continuity
60
2.5 Branches of Functions
64
2.6 The Reciprocal Transformation w = 1/z (Prerequisite for Section 9.2)
Chapter 3 71
Analytic and Harmonic Functions
71
3.1 Differentiable Functions
76
3.2 The Cauchy-Riemann Equations
84
3.3 Analytic Functions and Harmonic Functions
Chapter 4 95
Sequences, Series, and Julia and Mandelbrot Sets
95
4.1 Definitions and Basic Theorems for Sequences and Series
109
4.2 Power Series Functions
116
4.3 Julia and Mandelbrot Sets
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Chapter 5 125
Elementary Functions
125
5.1 The Complex Exponential Function
132
5.2 Branches of the Complex Logarithm Function
138
5.3 Complex Exponents
143
5.4 Trigonometric and Hyperbolic Functions
152
5.5 Inverse Trigonometric and Hyperbolic Functions
Chapter 6 157
Complex Integration
157
6.1 Complex Integrals
160
6.2 Contours and Contour Integrals
175
6.3 The Cauchy-Goursat Theorem
189
6.4 The Fundamental Theorems of Integration
195
6.5 Integral Representations for Analytic Functions
6.6 The Theorems of Morera and Liouville and Some
201
Applications
Chapter 7 208
Taylor and Laurent Series
208
7.1 Uniform Convergence
214
7.2 Taylor Series Representations
223
7.3 Laurent Series Representations
232
7.4 Singularities, Zeros, and Poles
239
7.5 Applications of Taylor and Laurent Series
Chapter 8 244
Residue Theory
244
8.1 The Residue Theorem
246
8.2 Calculation of Residues
252
8.3 Trigonometric Integrals
256
8.4 Improper Integrals of Rational Functions
260
8.5 Improper Integrals Involving Trigonometric Functions
264
8.6 Indented Contour Integrals
270
8.7 Integrands with Branch Points
274
8.8 The Argument Principle and Rouché's Theorem
Chapter 9 281
Conformal Mapping
281
9.1 Basic Properties of Conformal Mappings
287
9.2 Bilinear Transformations
294
9.3 Mappings Involving Elementary Functions
303
9.4 Mapping by Trigonometric Functions
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Chapter 10 310
Applications of Harmonic Functions
310
10.1 Preliminaries
312
10.2 Invariance of Laplace's Equation and the Dirichlet Problem
323
10.3 Poisson's Integral Formula for the Upper Half Plane
327
10.4 Two-Dimensional Mathematical Models
329
10.5 Steady State Temperatures
342
10.6 Two-Dimensional Electrostatics
349
10.7 Two-Dimensional Fluid Flow
360
10.8 The Joukowski Airfoil
369
10.9 The Schwarz-Christoffel Transformation
380
10.10 Image of a Fluid Flow
384
10.11 Sources and Sinks
Chapter 11 397
Fourier Series and the Laplace Transform
397
11.1 Fourier Series
406
11.2 The Dirichlet Problem for the Unit Disk
412
11.3 Vibrations in Mechanical Systems
418
11.4 The Fourier Transform
422
11.5 The Laplace Transform
430
11.6 Laplace Transforms of Derivatives and Integrals
433
11.7 Shifting Theorems and the Step Function
438
11.8 Multiplication and Division by t
441
11.9 Inverting the Laplace Transform
448
11.10 Convolution
Appendix A 456
Undergraduate Student Research Projects
Bibliography 458
Answers to Selected Problems 466
Index 477
Preface
Approach This text is intended for undergraduate students in mathematics,
physics, and engineering. We have attempted to strike a balance between the pure
and applied aspects of complex analysis and to present concepts in a clear writing
style that is understandable to students at the junior or senior undergraduate level.
A wealth of exercises that vary in both difficulty and substance gives the text flex
ibility. Sufficient applications are included to illustrate how complex analysis is
used in science and engineering. The use of computer graphics gives insight for
understanding that complex analysis is a computational tool of practical value. The
exercise sets offer a wide variety of choices for computational skills, theoretical
understanding, and applications that have been class tested for two editions of the
text. Student research projects are suggested throughout the text and citations are
made to the bibliography of books and journal articles.
The purpose of the first six chapters is to lay the foundations for the study of
complex analysis and develop the topics of analytic and harmonic functions, the
elementary functions, and contour integration. If the goal is to study series and the
residue calculus and applications, then Chapters 7 and 8 can be covered. If con-
formal mapping and applications of harmonic functions are desired, then Chapters
9 and 10 can be studied after Chapter 6. A new Chapter 11 on Fourier and Laplace
transforms has been added for courses that emphasize more applications.
Proofs are kept at an elementary level and are presented in a self-contained
manner that is understandable for students having a sophomore calculus back
ground. For example, Green's theorem is included and it is used to prove the
Cauchy-Goursat theorem. The proof by Goursat is included. The development of
series is aimed at practical applications.
Features Conformal mapping is presented in a visual and geometric manner so
that compositions and images of curves and regions can be understood. Boundary
value problems for harmonic functions are first solved in the upper half-plane so
that conformal mapping by elementary functions can be used to find solutions in
other domains. The Schwarz-Christoffel formula is carefully developed and appli
cations are given. Two-dimensional mathematical models are used for applications
in the area of ideal fluid flow, steady state temperatures and electrostatics. Computer
drawn figures accurately portray streamlines, isothermals, and equipotential curves.
New for this third edition is a historical introduction of the origin of complex
numbers in Chapter 1. An early introduction to sequences and series appears in
Chapter 4 and facilitates the definition of the exponential function via series. A new
section on the Julia and Mandelbrot sets shows how complex analysis is connected
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