# COMPLEX VARIABLES AND APPLICATIONS

Midterm II (At-Home, Summer I 2021) MATH 147 Sec A 4PM July 27 – 4PM July 28 Student ID: Name: | {z by writing my name i swear by the honor code } Read all of the following information before starting the exam: • This is a 24-hour “at-home” exam. Books and notes are allowed. No online searching, no calculator! Please complete the assignments by yourself. Any form of cheating, no matter how small (e.g., even copy the solution of others [including some online solution resources]) will result in an automatic F in the course, and will be pursuant to further disciplinary sanctions. • Must show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct). • Justify your answers algebraically whenever possible to ensure full credit. Circle or otherwise indicate your final answers. • Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements. • This exam has 5 problems and is worth 50 points. It is your responsibility to make sure that you have all of the pages! • Good luck! 1. (10 points) Find Z |z|=1 sin z z− π 2 4 dz . 2. (10 points) Find Z |z|=2 cos z z+ π 2 2 dz . 3. (10 points) Find ez 2 + dz . z+3 |z|=4 z − i Z 4. (10 points) Let a > 1, find Z 0 +∞ x sin(ax) dx . x 2 + a2 5. (10 points) Find Z +∞ −∞ cos(2x) dx . − 4x + 8 x2 Scrap Page (please do not remove this page from the test packet) COMPLEX VARIABLES AND APPLICATIONS Brown and Churchill Series Complex Variables and Applications, 9th Edition Fourier Series and Boundary Value Problems, 8th Edition The Walter Rudin Student Series in Advanced Mathematics Bóna, Miklós: Introduction to Enumerative Combinatorics Chartrand, Gary and Ping Zhang: Introduction to Graph Theory Davis, Sheldon: Topology Rudin, Walter: Principles of Mathematical Analysis Rudin, Walter: Real and Complex Analysis Other McGraw-Hill Titles in Higher Mathematics Ahlfors, Lars: Complex Analysis Burton, David M.: Elementary Number Theory Burton, David M.: The History of Mathematics: An Introduction Hvidsten, Michael: Geometry with Geometry Explorer COMPLEX VARIABLES AND APPLICATIONS Ninth Edition James Ward Brown Professor Emeritus of Mathematics The University of Michigan-Dearborn Ruel V. Churchill Late Professor of Mathematics The University of Michigan-Dearborn COMPLEX VARIABLES AND APPLICATIONS, NINTH EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2014 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2009, 2004, & 1996. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOC/DOC 1 0 9 8 7 6 5 4 3 ISBN 978-0-07-338317-0 MHID 0-07-338317-1 Senior Vice President, Products & Markets: Kurt L. Strand Vice President, General Manager: Marty Lange Vice President, Content Production & Technology Services: Kimberly Meriwether David Managing Director: Thomas Timp Executive Brand Manager: Bill Stenquist Executive Marketing Manager: Curt Reynolds Development Editors: Lorraine Buczek and Samantha Donisi-Hamm Director, Content Production: Terri Schiesl Senior Project Manager: Melissa M. Leick Cover Designer: Studio Montage, St. Louis, MO Buyer: Jennifer Pickel Media Project Manager: Sridevi Palani Compositor: MPS Limited Typeface: 10.25 × 12 Times Roman Printer: R. R. Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Churchill, Ruel V. (Ruel Vance), 1899–1987. Complex variables and applications / James Ward Brown, professor of mathematics, The University of Michigan/Dearborn, Ruel V. Churchill, late professor of mathematics, The University of Michigan. – Ninth edition. pages cm Churchill’s name appears first on the earlier editions. Includes bibliographical references and index. ISBN 978-0-07-338317-0 (alk. paper) 1. Functions of complex variables. I. Brown, James Ward. II. Title. QA331.7.C524 2014 515 .9—dc23 2013018657 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. www.mhhe.com ABOUT THE AUTHORS JAMES WARD BROWN is Professor Emeritus of Mathematics at The University of Michigan–Dearborn. He earned his A.B. in physics from Harvard University and his A.M. and Ph.D. in mathematics from The University of Michigan in Ann Arbor, where he was an Institute of Science and Technology Predoctoral Fellow. He is coauthor with Dr. Churchill of Fourier Series and Boundary Value Problems, now in its eighth edition. He has received a research grant from the National Science Foundation as well as a Distinguished Faculty Award from the Michigan Association of Governing Boards of Colleges and Universities. Dr. Brown is listed in Who’s Who in the World. RUEL V. CHURCHILL was, at the time of his death in 1987, Professor Emeritus of Mathematics at The University of Michigan, where he began teaching in 1922. He received his B.S. in physics from the University of Chicago and his M.S. in physics and Ph.D. in mathematics from The University of Michigan. He was coauthor with Dr. Brown of Fourier Series and Boundary Value Problems, a classic text that he first wrote almost 75 years ago. He was also the author of Operational Mathematics. Dr. Churchill held various offices in the Mathematical Association of America and in other mathematical societies and councils. This page intentionally left blank TO THE MEMORY OF MY FATHER, GEORGE H. BROWN, AND OF MY LONG-TIME FRIEND AND COAUTHOR, RUEL V. CHURCHILL. THESE DISTINGUISHED MEN OF SCIENCE FOR YEARS INFLUENCED THE CAREERS OF MANY PEOPLE, INCLUDING MINE. J.W.B. This page intentionally left blank CONTENTS Preface 1 xv Complex Numbers Sums and Products 1 1 Basic Algebraic Properties 3 Further Algebraic Properties Vectors and Moduli 8 Triangle Inequality 11 Complex Conjugates Exponential Form 5 14 17 Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Analytic Functions Functions and Mappings The Mapping w = z 2 Limits 21 25 28 Regions in the Complex Plane 2 20 32 37 37 40 44 Theorems on Limits 47 Limits Involving the Point at Infinity 50 ix x CONTENTS Continuity 52 Derivatives 55 Rules for Differentiation 59 Cauchy–Riemann Equations Examples 62 64 Sufficient Conditions for Differentiability Polar Coordinates 68 Analytic Functions Further Examples 65 72 74 Harmonic Functions 77 Uniquely Determined Analytic Functions Reflection Principle 3 82 Elementary Functions The Exponential Function 87 The Logarithmic Function 90 Examples 87 92 Branches and Derivatives of Logarithms Some Identities Involving Logarithms The Power Function Examples 80 93 97 100 101 The Trigonometric Functions sin z and cos z 103 Zeros and Singularities of Trigonometric Functions Hyperbolic Functions 109 Inverse Trigonometric and Hyperbolic Functions 4 Integrals 115 Derivatives of Functions w(t) 115 Definite Integrals of Functions w(t) Contours 120 Contour Integrals 125 105 117 112 CONTENTS Some Examples 127 Examples Involving Branch Cuts 131 Upper Bounds for Moduli of Contour Integrals Antiderivatives 135 140 Proof of the Theorem 144 Cauchy–Goursat Theorem Proof of the Theorem 148 150 Simply Connected Domains 154 Multiply Connected Domains Cauchy Integral Formula 156 162 An Extension of the Cauchy Integral Formula Verification of the Extension 164 166 Some Consequences of the Extension 168 Liouville’s Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series 179 Convergence of Sequences Convergence of Series Taylor Series 186 187 189 Negative Powers of (z − z 0 ) Laurent Series 193 197 Proof of Laurent’s Theorem Examples 179 182 Proof of Taylor’s Theorem Examples 173 199 202 Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series 211 Integration and Differentiation of Power Series Uniqueness of Series Representations 213 216 Multiplication and Division of Power Series 221 208 172 xi xii 6 CONTENTS Residues and Poles Isolated Singular Points Residues 227 227 229 Cauchy’s Residue Theorem Residue at Infinity 233 235 The Three Types of Isolated Singular Points Examples 240 Residues at Poles Examples 238 242 244 Zeros of Analytic Functions Zeros and Poles 248 251 Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example 259 259 262 Improper Integrals from Fourier Analysis Jordan’s Lemma An Indented Path 267 269 274 An Indentation Around a Branch Point Integration Along a Branch Cut 277 280 Definite Integrals Involving Sines and Cosines Argument Principle Rouché’s Theorem 284 287 290 Inverse Laplace Transforms 8 255 294 Mapping by Elementary Functions Linear Transformations 299 The Transformation w = 1/z Mappings by 1/z 303 301 299 CONTENTS Linear Fractional Transformations An Implicit Form 310 Mappings of the Upper Half Plane Examples 307 313 315 Mappings by the Exponential Function 318 Mapping Vertical Line Segments by w = sin z Mapping Horizontal Line Segments by w = sin z Some Related Mappings Mappings by z 2 326 Square Roots of Polynomials Riemann Surfaces 328 332 338 Surfaces for Related Functions Conformal Mapping 341 345 Preservation of Angles and Scale Factors Further Examples Local Inverses 322 324 Mappings by Branches of z 1/2 9 320 345 348 350 Harmonic Conjugates 354 Transformations of Harmonic Functions 357 Transformations of Boundary Conditions 360 10 Applications of Conformal Mapping Steady Temperatures 365 Steady Temperatures in a Half Plane A Related Problem 369 Temperatures in a Quadrant Electrostatic Potential Examples 371 376 377 Two-Dimensional Fluid Flow 382 367 365 xiii xiv CONTENTS The Stream Function 384 Flows Around a Corner and Around a Cylinder 386 11 The Schwarz–Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz–Christoffel Transformation Triangles and Rectangles Degenerate Polygons 393 395 399 402 Fluid Flow in a Channel through a Slit Flow in a Channel with an Offset 407 409 Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula 417 Dirichlet Problem for a Disk Examples 420 422 Related Boundary Value Problems Schwarz Integral Formula Appendixes Bibliography 451 430 433 437 437 Table of Transformations of Regions Index 426 428 Dirichlet Problem for a Half Plane Neumann Problems 393 441 412 417 PREFACE This book is a thorough revision of its earlier eighth edition, which was published in 2009. That edition has served, just as the earlier ones did, as a textbook for a one-term introductory course in the theory and application of functions of a complex variable. This new edition preserves the basic content and style of the earlier ones, the first two of which were written by the late Ruel V. Churchill alone. The book has always had two main objectives. (a) The first is to develop those parts of the theory that are prominent in applications of the subject. (b) The second objective is to furnish an introduction to applications of residues and conformal mapping. The applications of residues include their use in evaluating real improper integrals, finding inverse Laplace transforms, and locating zeros of functions. Considerable attention is paid to the use of conformal mapping in solving boundary value problems that arise in studies of heat conduction and fluid flow. Hence the book may be considered as a companion volume to the authors’ text Fourier Series and Boundary Value Problems, where another classical method for solving boundary value problems in partial differential equations is developed. The first nine chapters of this book have for many years formed the basis of a threehour course given each term at The University of Michigan. The final three chapters have fewer changes and are mostly intended for self-study and reference. The classes using the book have consisted mainly of seniors concentrating in mathematics, engineering, or one of the physical sciences. Before taking the course, the students have completed at least a three-term calculus sequence and a first course in ordinary differential equations. If mapping by elementary functions is desired earlier in the course, one can skip to Chap. 8 immediately after Chap. 3 on elementary functions and then return to Chap. 4 on integrals. We mention here a sample of the changes in this edition, some of which were suggested by students and people teaching from the book. A number of topics have been moved from where they were. For example, although harmonic functions are still xv xvi PREFACE introduced in Chap. 2, harmonic conjugates have been moved to Chap. 9, where they are actually needed. Another example is the moving of the derivation of an important inequality needed in proving the fundamental theorem of algebra (Chap. 4) to Chap. 1, where related inequalities are introduced. This has the advantage of enabling the reader to concentrate on such inequalities when they are grouped together and also of making the proof of the fundamental theorem of algebra reasonably brief and efficient without taking the reader on a distracting side-trip. The introduction to the concept of mapping in Chap. 2 is shortened somewhat in this edition, and only the mapping w = z 2 is emphasized in that chapter. This was suggested by some users of the last edition, who felt that a detailed consideration of w = z 2 is sufficient in Chap. 2 in order to illustrate concepts needed there. Finally, since most of the series, both Taylor and Laurent, that are found and discussed in Chap. 5 rely on the reader’s familiarity with just six Maclaurin series, those series are now grouped together so that the reader is not forced to hunt around for them whenever they are needed in finding other series expansions. Also, Chap. 5 now contains a separate section, following Taylor’s theorem, devoted entirely to series representations involving negative powers of z − z 0 . Experience has shown that this is especially valuable in making the transformation from Taylor to Laurent series a natural one. This edition contains many new examples, sometimes taken from the exercises in the last edition. Quite often the examples follow in a separate section immediately following a section that develops the theory to be illustrated. The clarity of the presentation has been enhanced in other ways. Boldface letters have been used to make definitions more easily identified. The book has fifteen new figures, as well as a number of existing ones that have been improved. Finally, when the proofs of theorems are unusually long, those proofs are clearly divided into parts. This happens, for instance, in the proof (Sec. 49) of the three-part theorem regarding the existence and use of antiderivatives. The same is true of the proof (Sec. 51) of the Cauchy-Goursat theorem. Finally, there is a Student’s Solutions Manual (ISBN: 978-0-07-352899-1; MHID: 0-07-352899-4) that is available. It contains solutions of selected exercises in Chapters 1 through 7, covering the material through residues. In order to accommodate as wide a range of readers as possible, there are footnotes referring to other texts that give proofs and discussions of the more delicate results from calculus and advanced calculus that are occasionally needed. A bibliography of other books on complex variables, many of which are more advanced, is provided in Appendix 1. A table of conformal transformations that are useful in applications appears in Appendix 2. As already indicated, some of the changes in this edition have been suggested by users of the earlier edition. Moreover, in the preparation of this new edition, continual interest and support has been provided by a variety of other people, especially the staff at McGraw-Hill and my wife Jacqueline Read Brown. James Ward Brown CHAPTER 1 COMPLEX NUMBERS In this chapter, we survey the algebraic and geometric structure of the complex number system. We assume various corresponding properties of real numbers to be known. 1. SUMS AND PRODUCTS Complex numbers can be defined as ordered pairs (x, y) of real numbers that are to be interpreted as points in the complex plane, with rectangular coordinates x and y, just as real numbers x are thought of as points on the real line. When real numbers x are displayed as points (x, 0) on the real axis, we write x = (x, 0); and it is clear that the set of complex numbers includes the real numbers as a subset. Complex numbers of the form (0, y) correspond to points on the y axis and are called pure imaginary numbers when y = 0. The y axis is then referred to as the imaginary axis. It is customary to denote a complex number (x, y) by z, so that (see Fig. 1) z = (x, y). (1) y z = (x, y) i = (0, 1) O x = (x, 0) x FIGURE 1 1 2 COMPLEX NUMBERS CHAP. 1 The real numbers x and y are, moreover, known as the real and imaginary parts of z, respectively, and we write x = Re z, (2) y = Im z. Two complex numbers z 1 and z 2 are equal whenever they have the same real parts and the same imaginary parts. Thus the statement z 1 = z 2 means that z 1 and z 2 correspond to the same point in the complex, or z, plane. The sum z 1 + z 2 and product z 1 z 2 of two complex numbers z 1 = (x1 , y1 ) and z 2 = (x1 , y1 ) are defined as follows: (3) (4) (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y2 ), (x1 , y1 )(x2 , y2 ) = (x1 x2 − y1 y2 , y1 x2 + x1 y2 ). Note that the operations defined by means of equations (3) and (4) become the usual operations of addition and multiplication when restricted to the real numbers: (x1 , 0) + (x2 , 0) = (x1 + x2 , 0), (x1 , 0)(x2 , 0) = (x1 x2 , 0). The complex number system is, therefore, a natural extension of the real number system. Any complex number z = (x, y) can be written z = (x, 0) + (0, y), and it is easy to see that (0, 1)(y, 0) = (0, y). Hence z = (x, 0) + (0, 1)(y, 0); and if we think of a real number as either x or (x, 0) and let i denote the pure imaginary number (0,1), as shown in Fig. 1, it is clear that∗ z = x + i y. (5) Also, with the convention that z = zz, z 3 = z 2 z, etc., we have 2 i 2 = (0, 1)(0, 1) = (−1, 0), or (6) i 2 = −1. Because (x, y) = x + i y, definitions (3) and (4) become (7) (8) ∗ (x1 + i y1 ) + (x2 + i y2 ) = (x1 + x2 ) + i(y1 + y2 ), (x1 + i y1 )(x2 + i y2 ) = (x1 x2 − y1 y2 ) + i(y1 x2 + x1 y2 ). In electrical engineering, the letter j is used instead of i. SEC. BASIC ALGEBRAIC PROPERTIES 2 3 Observe that the right-hand sides of these equations can be obtained by formally manipulating the terms on the left as if they involved only real numbers and by replacing i 2 by −1 when it occurs. Also, observe how equation (8) tells us that any complex number times zero is zero. More precise…

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