- 2. COMPLEX VARIABLES AND APPLICATIONS Eighth Edition James Ward Brown Professor of Mathematics The University of Michigan–Dearborn Ruel V. Churchill Late Professor of Mathematics The University of Michigan
- 3. COMPLEX VARIABLES AND APPLICATIONS, EIGHTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright 2009 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions 2004, 1996, 1990, 1984, 1974, 1960, 1948 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 The McGraw-Hill Companies, Inc., 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 0 9 8 ISBN 978–0–07–305194–9 MHID 0–07–305194–2 Editorial Director: Stewart K. Mattson Director of Development: Kristine Tibbetts Senior Sponsoring Editor: Elizabeth Covello Developmental Editor: Michelle Driscoll Editorial Coordinator: Adam Fischer Senior Marketing Manager: Eric Gates Project Manager: April R. Southwood Senior Production Supervisor: Kara Kudronowicz Associate Design Coordinator: Brenda A. Rolwes Cover Designer: Studio Montage, St. Louis, Missouri Project Coordinator: Melissa M. Leick Compositor: Laserwords Private Limited Typeface: 10.25/12 Times Roman Printer: R. R. Donnelly Crawfordsville, IN Library of Congress Cataloging-in-Publication Data Brown, James Ward. Complex variables and applications / James Ward Brown, Ruel V. Churchill.—8th ed. p. cm. Includes bibliographical references and index. ISBN 978–0–07–305194–9—ISBN 0–07–305194–2 (hard copy : acid-free paper) 1. Functions of complex variables. I. Churchill, Ruel Vance, 1899- II. Title. QA331.7.C524 2009 515 .9—dc22 2007043490 www.mhhe.com
- 4. ABOUT THE AUTHORS JAMES WARD BROWN is Professor 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 seventh edition. He has received a research grant from the National Science Foundation as well as a Distinguished Faculty Award from the Michigan Associa- tion 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 70 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. iii
- 5. 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 Myself. JWB
- 6. CONTENTS Preface x 1 Complex Numbers 1 Sums and Products 1 Basic Algebraic Properties 3 Further Properties 5 Vectors and Moduli 9 Complex Conjugates 13 Exponential Form 16 Products and Powers in Exponential Form 18 Arguments of Products and Quotients 20 Roots of Complex Numbers 24 Examples 27 Regions in the Complex Plane 31 2 Analytic Functions 35 Functions of a Complex Variable 35 Mappings 38 Mappings by the Exponential Function 42 Limits 45 Theorems on Limits 48 v
- 7. vi contents Limits Involving the Point at Infinity 50 Continuity 53 Derivatives 56 Differentiation Formulas 60 Cauchy–Riemann Equations 63 Sufficient Conditions for Differentiability 66 Polar Coordinates 68 Analytic Functions 73 Examples 75 Harmonic Functions 78 Uniquely Determined Analytic Functions 83 Reflection Principle 85 3 Elementary Functions 89 The Exponential Function 89 The Logarithmic Function 93 Branches and Derivatives of Logarithms 95 Some Identities Involving Logarithms 98 Complex Exponents 101 Trigonometric Functions 104 Hyperbolic Functions 109 Inverse Trigonometric and Hyperbolic Functions 112 4 Integrals 117 Derivatives of Functions w(t) 117 Definite Integrals of Functions w(t) 119 Contours 122 Contour Integrals 127 Some Examples 129 Examples with Branch Cuts 133 Upper Bounds for Moduli of Contour Integrals 137 Antiderivatives 142 Proof of the Theorem 146 Cauchy–Goursat Theorem 150 Proof of the Theorem 152
- 8. contents vii Simply Connected Domains 156 Multiply Connected Domains 158 Cauchy Integral Formula 164 An Extension of the Cauchy Integral Formula 165 Some Consequences of the Extension 168 Liouville’s Theorem and the Fundamental Theorem of Algebra 172 Maximum Modulus Principle 175 5 Series 181 Convergence of Sequences 181 Convergence of Series 184 Taylor Series 189 Proof of Taylor’s Theorem 190 Examples 192 Laurent Series 197 Proof of Laurent’s Theorem 199 Examples 202 Absolute and Uniform Convergence of Power Series 208 Continuity of Sums of Power Series 211 Integration and Differentiation of Power Series 213 Uniqueness of Series Representations 217 Multiplication and Division of Power Series 222 6 Residues and Poles 229 Isolated Singular Points 229 Residues 231 Cauchy’s Residue Theorem 234 Residue at Infinity 237 The Three Types of Isolated Singular Points 240 Residues at Poles 244 Examples 245 Zeros of Analytic Functions 249 Zeros and Poles 252 Behavior of Functions Near Isolated Singular Points 257
- 9. viii contents 7 Applications of Residues 261 Evaluation of Improper Integrals 261 Example 264 Improper Integrals from Fourier Analysis 269 Jordan’s Lemma 272 Indented Paths 277 An Indentation Around a Branch Point 280 Integration Along a Branch Cut 283 Definite Integrals Involving Sines and Cosines 288 Argument Principle 291 Rouché’s Theorem 294 Inverse Laplace Transforms 298 Examples 301 8 Mapping by Elementary Functions 311 Linear Transformations 311 The Transformation w = 1/z 313 Mappings by 1/z 315 Linear Fractional Transformations 319 An Implicit Form 322 Mappings of the Upper Half Plane 325 The Transformation w = sin z 330 Mappings by z2 and Branches of z1/2 336 Square Roots of Polynomials 341 Riemann Surfaces 347 Surfaces for Related Functions 351 9 Conformal Mapping 355 Preservation of Angles 355 Scale Factors 358 Local Inverses 360 Harmonic Conjugates 363 Transformations of Harmonic Functions 365 Transformations of Boundary Conditions 367
- 10. contents ix 10 Applications of Conformal Mapping 373 Steady Temperatures 373 Steady Temperatures in a Half Plane 375 A Related Problem 377 Temperatures in a Quadrant 379 Electrostatic Potential 385 Potential in a Cylindrical Space 386 Two-Dimensional Fluid Flow 391 The Stream Function 393 Flows Around a Corner and Around a Cylinder 395 11 The Schwarz–Christoffel Transformation 403 Mapping the Real Axis Onto a Polygon 403 Schwarz–Christoffel Transformation 405 Triangles and Rectangles 408 Degenerate Polygons 413 Fluid Flow in a Channel Through a Slit 417 Flow in a Channel With an Offset 420 Electrostatic Potential About an Edge of a Conducting Plate 422 12 Integral Formulas of the Poisson Type 429 Poisson Integral Formula 429 Dirichlet Problem for a Disk 432 Related Boundary Value Problems 437 Schwarz Integral Formula 440 Dirichlet Problem for a Half Plane 441 Neumann Problems 445 Appendixes 449 Bibliography 449 Table of Transformations of Regions 452 Index 461
- 11. PREFACE This book is a revision of the seventh edition, which was published in 2004. That edition has served, just as the earlier ones did, as a textbook for a one-term intro- ductory course in the theory and application of functions of a complex variable. This new edition preserves the basic content and style of the earlier editions, the first two of which were written by the late Ruel V. Churchill alone. The first objective of the book is to develop those parts of the theory that are prominent in applications of the subject. The second objective is to furnish an intro- duction to applications of residues and conformal mapping. With regard to residues, special emphasis is given to their use in evaluating real improper integrals, finding inverse Laplace transforms, and locating zeros of functions. As for conformal map- ping, considerable attention is paid to its use 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 Bound- ary 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 three-hour course given each term at The University of Michigan. The classes have consisted mainly of seniors and graduate students 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. Much of the material in the book need not be covered in the lectures and can be left for self-study or used for reference. If mapping by elementary functions is desired earlier in the course, one can skip to Chap. 8 immediately after Chap. 3 on elementary functions. In order to accommodate as wide a range of readers as possible, there are foot- notes referring to other texts that give proofs and discussions of the more delicate results from calculus and advanced calculus that are occasionally needed. A bibli- ography 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. x
- 12. preface xi The main changes in this edition appear in the first nine chapters. Many of those changes have been suggested by users of the last edition. Some readers have urged that sections which can be skipped or postponed without disruption be more clearly identified. The statements of Taylor’s theorem and Laurent’s theorem, for example, now appear in sections that are separate from the sections containing their proofs. Another significant change involves the extended form of the Cauchy integral formula for derivatives. The treatment of that extension has been completely rewritten, and its immediate consequences are now more focused and appear together in a single section. Other improvements that seemed necessary include more details in arguments involving mathematical induction, a greater emphasis on rules for using complex exponents, some discussion of residues at infinity, and a clearer exposition of real improper integrals and their Cauchy principal values. In addition, some rearrange- ment of material was called for. For instance, the discussion of upper bounds of moduli of integrals is now entirely in one section, and there is a separate section devoted to the definition and illustration of isolated singular points. Exercise sets occur more frequently than in earlier editions and, as a result, concentrate more directly on the material at hand. Finally, there is an Student’s Solutions Manual (ISBN: 978-0-07-333730-2; MHID: 0-07-333730-7) that is available upon request to instructors who adopt the book. It contains solutions of selected exercises in Chapters 1 through 7, covering the material through residues. In the preparation of this edition, continual interest and support has been pro- vided by a variety of people, especially the staff at McGraw-Hill and my wife Jacqueline Read Brown. James Ward Brown
- 14. Brown-chap01-v3 10/29/07 3:32pm 1 C H A P T E R 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, 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) The real numbers x and y are, moreover, known as the real and imaginary parts of z, respectively; and we write x = Re z, y = Im z. (2) Two complex numbers z1 and z2 are equal whenever they have the same real parts and the same imaginary parts. Thus the statement z1 = z2 means that z1 and z2 correspond to the same point in the complex, or z, plane. 1
- 15. Brown-chap01-v3 10/29/07 3:32pm 2 2 Complex Numbers chap. 1 z = (x, y) i = (0, 1) x = (x, 0) x O y FIGURE 1 The sum z1 + z2 and product z1z2 of two complex numbers z1 = (x1, y1) and z2 = (x2, y2) are defined as follows: (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2), (3) (x1, y1)(x2, y2) = (x1x2 − y1y2, y1x2 + x1y2). (4) Note that the operations defined by 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) = (x1x2, 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 + iy. (5) Also, with the convention that z2 = zz, z3 = z2z, etc., we have i2 = (0, 1)(0, 1) = (−1, 0), or i2 = −1. (6) ∗In electrical engineering, the letter j is used instead of i.
- 16. Brown-chap01-v3 10/29/07 3:32pm 3 sec. 2 Basic Algebraic Properties 3 Because (x, y) = x + iy, definitions (3) and (4) become (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2), (7) (x1 + iy1)(x2 + iy2) = (x1x2 − y1y2) + i(y1x2 + x1y2). (8) 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 i2 by −1 when it occurs. Also, observe how equation (8) tells us that any complex number times zero is zero. More precisely, z · 0 = (x + iy)(0 + i0) = 0 + i0 = 0 for any z = x + iy. 2. BASIC ALGEBRAIC PROPERTIES Various properties of addition and multiplication of complex numbers are the same as for real numbers. We list here the more basic of these algebraic properties and verify some of them. Most of the others are verified in the exercises. The commutative laws z1 + z2 = z2 + z1, z1z2 = z2z1 (1) and the associative laws (z1 + z2) + z3 = z1 + (z2 + z3), (z1z2)z3 = z1(z2z3) (2) follow easily from the definitions in Sec. 1 of addition and multiplication of complex numbers and the fact that real numbers obey these laws. For example, if z1 = (x1, y1) and z2 = (x2, y2), then z1 + z2 = (x1 + x2, y1 + y2) = (x2 + x1, y2 + y1) = z2 + z1. Verification of the rest of the above laws, as well as the distributive law z(z1 + z2) = zz1 + zz2, (3) is similar. According to the commutative law for multiplication, iy = yi. Hence one can write z = x + yi instead of z = x + iy. Also, because of the associative laws, a sum z1 + z2 + z3 or a product z1z2z3 is well defined without parentheses, as is the case with real numbers.
- 17. Brown-chap01-v3 10/29/07 3:32pm 4 4 Complex Numbers chap. 1 The additive identity 0 = (0, 0) and the multiplicative identity 1 = (1, 0) for real numbers carry over to the entire complex number system. That is, z + 0 = z and z · 1 = z (4) for every complex number z. Furthermore, 0 and 1 are the only complex numbers with such properties (see Exercise 8). There is associated with each complex number z = (x, y) an additive inverse −z = (−x, −y), (5) satisfying the equation z + (−z) = 0. Moreover, there is only one additive inverse for any given z, since the equation (x, y) + (u, v) = (0, 0) implies that u = −x and v = −y. For any nonzero complex number z = (x, y), there is a number z−1 such that zz−1 = 1. This multiplicative inverse is less obvious than the additive one. To find it, we seek real numbers u and v, expressed in terms of x and y, such that (x, y)(u, v) = (1, 0). According to equation (4), Sec. 1, which defines the product of two complex num- bers, u and v must satisfy the pair xu − yv = 1, yu + xv = 0 of linear simultaneous equations; and simple computation yields the unique solution u = x x2 + y2 , v = −y x2 + y2 . So the multiplicative inverse of z = (x, y) is z−1 = x x2 + y2 , −y x2 + y2 (z = 0). (6) The inverse z−1 is not defined when z = 0. In fact, z = 0 means that x2 + y2 = 0 ; and this is not permitted in expression (6).
- 18. Brown-chap01-v3 10/29/07 3:32pm 6 6 Complex Numbers chap. 1 in Sec. 2. Inasmuch as such properties continue to be anticipated because they also apply to real numbers, the reader can easily pass to Sec. 4 without serious disruption. We begin with the observation that the existence of multiplicative inverses enables us to show that if a product z1z2 is zero, then so is at least one of the factors z1 and z2. For suppose that z1z2 = 0 and z1 = 0. The inverse z−1 1 exists; and any complex number times zero is zero (Sec. 1). Hence z2 = z2 · 1 = z2(z1z−1 1 ) = (z−1 1 z1)z2 = z−1 1 (z1z2) = z−1 1 · 0 = 0. That is, if z1z2 = 0, either z1 = 0 or z2 = 0; or possibly both of the numbers z1 and z2 are zero. Another way to state this result is that if two complex numbers z1 and z2 are nonzero, then so is their product z1z2. Subtraction and division are defined in terms of additive and multiplicative inverses: z1 − z2 = z1 + (−z2), (1) z1 z2 = z1z−1 2 (z2 = 0). (2) Thus, in view of expressions (5) and (6) in Sec. 2, z1 − z2 = (x1, y1) + (−x2, −y2) = (x1 − x2, y1 − y2) (3) and z1 z2 = (x1, y1) x2 x2 2 + y2 2 , −y2 x2 2 + y2 2 = x1x2 + y1y2 x2 2 + y2 2 , y1x2 − x1y2 x2 2 + y2 2 (4) (z2 = 0) when z1 = (x1, y1) and z2 = (x2, y2). Using z1 = x1 + iy1 and z2 = x2 + iy2, one can write expressions (3) and (4) here as z1 − z2 = (x1 − x2) + i(y1 − y2) (5) and z1 z2 = x1x2 + y1y2 x2 2 + y2 2 + i y1x2 − x1y2 x2 2 + y2 2 (z2 = 0). (6) Although expression (6) is not easy to remember, it can be obtained by writing (see Exercise 7) z1 z2 = (x1 + iy1)(x2 − iy2) (x2 + iy2)(x2 − iy2) , (7)
- 19. Brown-chap01-v3 10/29/07 3:32pm 5 sec. 3 Further Properties 5 EXERCISES 1. Verify that (a) ( √ 2 − i) − i(1 − √ 2i) = −2i; (b) (2, −3)(−2, 1) = (−1, 8); (c) (3, 1)(3, −1) 1 5 , 1 10 = (2, 1). 2. Show that (a) Re(iz) = − Im z; (b) Im(iz) = Re z. 3. Show that (1 + z)2 = 1 + 2z + z2 . 4. Verify that each of the two numbers z = 1 ± i satisfies the equation z2 − 2z + 2 = 0. 5. Prove that multiplication of complex numbers is commutative, as stated at the begin- ning of Sec. 2. 6. Verify (a) the associative law for addition of complex numbers, stated at the beginning of Sec. 2; (b) the distributive law (3), Sec. 2. 7. Use the associative law for addition and the distributive law to show that z(z1 + z2 + z3) = zz1 + zz2 + zz3. 8. (a) Write (x, y) + (u, v) = (x, y) and point out how it follows that the complex num- ber 0 = (0, 0) is unique as an additive identity. (b) Likewise, write (x, y)(u, v) = (x, y) and show that the number 1 = (1, 0) is a unique multiplicative identity. 9. Use −1 = (−1, 0) and z = (x, y) to show that (−1)z = −z. 10. Use i = (0, 1) and y = (y, 0) to verify that −(iy) = (−i)y. Thus show that the addi- tive inverse of a complex number z = x + iy can be written −z = −x − iy without ambiguity. 11. Solve the equation z2 + z + 1 = 0 for z = (x, y) by writing (x, y)(x, y) + (x, y) + (1, 0) = (0, 0) and then solving a pair of simultaneous equations in x and y. Suggestion: Use the fact that no real number x satisfies the given equation to show that y = 0. Ans. z = − 1 2 , ± √ 3 2 . 3. FURTHER PROPERTIES In this section, we mention a number of other algebraic properties of addition and multiplication of complex numbers that follow from the ones already described
- 20. Brown-chap01-v3 10/29/07 3:32pm 7 sec. 3 Further Properties 7 multiplying out the products in the numerator and denominator on the right, and then using the property z1 + z2 z3 = (z1 + z2)z−1 3 = z1z−1 3 + z2z−1 3 = z1 z3 + z2 z3 (z3 = 0). (8) The motivation for starting with equation (7) appears in Sec. 5. EXAMPLE. The method is illustrated below: 4 + i 2 − 3i = (4 + i)(2 + 3i) (2 − 3i)(2 + 3i) = 5 + 14i 13 = 5 13 + 14 13 i. There are some expected properties involving quotients that follow from the relation 1 z2 = z−1 2 (z2 = 0), (9) which is equation (2) when z1 = 1. Relation (9) enables us, for instance, to write equation (2) in the form z1 z2 = z1 1 z2 (z2 = 0). (10) Also, by observing that (see Exercise 3) (z1z2)(z−1 1 z−1 2 ) = (z1z−1 1 )(z2z−1 2 ) = 1 (z1 = 0, z2 = 0), and hence that z−1 1 z−1 2 = (z1z2)−1, one can use relation (9) to show that 1 z1 1 z2 = z−1 1 z−1 2 = (z1z2)−1 = 1 z1z2 (z1 = 0, z2 = 0). (11) Another useful property, to be derived in the exercises, is z1 z3 z2 z4 = z1z2 z3z4 (z3 = 0, z4 = 0). (12) Finally, we note that the binomial formula involving real numbers remains valid with complex numbers. That is, if z1 and z2 are any two nonzero complex numbers, then (z1 + z2)n = n k=0 n k zk 1zn−k 2 (n = 1, 2, . . .) (13) where n k = n! k!(n − k)! (k = 0, 1, 2, . . . , n) and where it is agreed that 0! = 1. The proof is left as an exercise.
- 21. Brown-chap01-v3 10/29/07 3:32pm 8 8 Complex Numbers chap. 1 EXERCISES 1. Reduce each of these quantities to a real number: (a) 1 + 2i 3 − 4i + 2 − i 5i ; (b) 5i (1 − i)(2 − i)(3 − i) ; (c) (1 − i)4 . Ans. (a) −2/5; (b) −1/2; (c) −4. 2. Show that 1 1/z = z (z = 0). 3. Use the associative and commutative laws for multiplication to show that (z1z2)(z3z4) = (z1z3)(z2z4). 4. Prove that if z1z2z3 = 0, then at least one of the three factors is zero. Suggestion: Write (z1z2)z3 = 0 and use a similar result (Sec. 3) involving two factors. 5. Derive expression (6), Sec. 3, for the quotient z1/z2 by the method described just after it. 6. With the aid of relations (10) and (11) in Sec. 3, derive the identity z1 z3 z2 z4 = z1z2 z3z4 (z3 = 0, z4 = 0). 7. Use the identity obtained in Exercise 6 to derive the cancellation law z1z z2z = z1 z2 (z2 = 0, z = 0). 8. Use mathematical induction to verify the binomial formula (13) in Sec. 3. More pre- cisely, note that the formula is true when n = 1. Then, assuming that it is valid when n = m where m denotes any positive integer, show that it must hold when n = m + 1. Suggestion: When n = m + 1, write (z1 + z2)m+1 = (z1 + z2)(z1 + z2)m = (z2 + z1) m k=0 m k zk 1zm−k 2 = m k=0 m k zk 1zm+1−k 2 + m k=0 m k zk+1 1 zm−k 2 and replace k by k − 1 in the last sum here to obtain (z1 + z2)m+1 = zm+1 2 + m k=1 m k + m k − 1 zk 1zm+1−k 2 + zm+1 1 .
- 22. Brown-chap01-v3 10/29/07 3:32pm 9 sec. 4 Vectors and Moduli 9 Finally, show how the right-hand side here becomes zm+1 2 + m k=1 m + 1 k zk 1zm+1−k 2 + zm+1 1 = m+1 k=0 m + 1 k zk 1zm+1−k 2 . 4. VECTORS AND MODULI It is natural to associate any nonzero complex number z = x + iy with the directed line segment, or vector, from the origin to the point (x, y) that represents z in the complex plane. In fact, we often refer to z as the point z or the vector z. In Fig. 2 the numbers z = x + iy and −2 + i are displayed graphically as both points and radius vectors. z = (x, y) z = x + iy –2 + i x O –2 (–2, 1) 1 y FIGURE 2 When z1 = x1 + iy1 and z2 = x2 + iy2, the sum z1 + z2 = (x1 + x2) + i(y1 + y2) corresponds to the point (x1 + x2, y1 + y2). It also corresponds to a vector with those coordinates as its components. Hence z1 + z2 may be obtained vectorially as shown in Fig. 3. x O y z1 z1 + z2 z2 z2 FIGURE 3 Although the product of two complex numbers z1 and z2 is itself a complex number represented by a vector, that vector lies in the same plane as the vectors for z1 and z2. Evidently, then, this product is neither the scalar nor the vector product used in ordinary vector analysis.
- 23. Brown-chap01-v3 10/29/07 3:32pm 10 10 Complex Numbers chap. 1 The vector interpretation of complex numbers is especially helpful in extending the concept of absolute values of real numbers to the complex plane. The modulus, or absolute value, of a complex number z = x + iy is defined as the nonnegative real number x2 + y2 and is denoted by |z|; that is, |z| = x2 + y2. (1) Geometrically, the number |z| is the distance between the point (x, y) and the origin, or the length of the radius vector representing z. It reduces to the usual absolute value in the real number system when y = 0. Note that while the inequality z1 z2 is meaningless unless both z1 and z2 are real, the statement |z1| |z2| means that the point z1 is closer to the origin than the point z2 is. EXAMPLE 1. Since |− 3 + 2i| = √ 13 and |1 + 4i| = √ 17, we know that the point −3 + 2i is closer to the origin than 1 + 4i is. The distance between two points (x1, y1) and (x2, y2) is |z1 − z2|. This is clear from Fig. 4, since |z1 − z2| is the length of the vector representing the number z1 − z2 = z1 + (−z2); and, by translating the radius vector z1 − z2, one can interpret z1 − z2 as the directed line segment from the point (x2, y2) to the point (x1, y1). Alternatively, it follows from the expression z1 − z2 = (x1 − x2) + i(y1 − y2) and definition (1) that |z1 − z2| = (x1 − x2)2 + (y1 − y2)2. x O y z1 |z1 – z2| z1 – z2 z2 –z2 (x2, y2) (x1, y1) FIGURE 4 The complex numbers z corresponding to the points lying on the circle with center z0 and radius R thus satisfy the equation |z − z0| = R, and conversely. We refer to this set of points simply as the circle |z − z0| = R. EXAMPLE 2. The equation |z − 1 + 3i| = 2 represents the circle whose center is z0 = (1, −3) and whose radius is R = 2.
- 24. Brown-chap01-v3 10/29/07 3:32pm 11 sec. 4 Vectors and Moduli 11 It also follows from definition (1) that the real numbers |z|, Re z = x, and Im z = y are related by the equation |z|2 = (Re z)2 + (Im z)2 . (2) Thus Re z ≤ |Re z| ≤ |z| and Im z ≤ |Im z| ≤ |z|. (3) We turn now to the triangle inequality, which provides an upper bound for the modulus of the sum of two complex numbers z1 and z2: |z1 + z2| ≤ |z1| + |z2|. (4) This important inequality is geometrically evident in Fig. 3, since it is merely a statement that the length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides. We can also see from Fig. 3 that inequality (4) is actually an equality when 0, z1, and z2 are collinear. Another, strictly algebraic, derivation is given in Exercise 15, Sec. 5. An immediate consequence of the triangle inequality is the fact that |z1 + z2| ≥ ||z1| − |z2||. (5) To derive inequality (5), we write |z1| = |(z1 + z2) + (−z2)| ≤ |z1 + z2| + |− z2|, which means that |z1 + z2| ≥ |z1| − |z2|. (6) This is inequality (5) when |z1| ≥ |z2|. If |z1| |z2|, we need only interchange z1 and z2 in inequality (6) to arrive at |z1 + z2| ≥ −(|z1| − |z2|), which is the desired result. Inequality (5) tells us, of course, that the length of one side of a triangle is greater than or equal to the difference of the lengths of the other two sides. Because |− z2| = |z2|, one can replace z2 by −z2 in inequalities (4) and (5) to summarize these results in a particularly useful form: |z1 ± z2| ≤ |z1| + |z2|, (7) |z1 ± z2| ≥ ||z1| − |z2||. (8) When combined, inequalities (7) and (8) become ||z1| − |z2|| ≤ |z1 ± z2| ≤ |z1| + |z2|. (9)
- 25. Brown-chap01-v3 10/29/07 3:32pm 12 12 Complex Numbers chap. 1 EXAMPLE 3. If a point z lies on the unit circle |z| = 1 about the origin, it follows from inequalities (7) and (8) that |z − 2| ≤ |z| + 2 = 3 and |z − 2| ≥ ||z| − 2| = 1. The triangle inequality (4) can be generalized by means of mathematical induc- tion to sums involving any finite number of terms: |z1 + z2 + · · · + zn| ≤ |z1| + |z2| + · · · + |zn| (n = 2, 3, . . .). (10) To give details of the induction proof here, we note that when n = 2, inequality (10) is just inequality (4). Furthermore, if inequality (10) is assumed to be valid when n = m, it must also hold when n = m + 1 since, by inequality (4), |(z1 + z2 + · · · + zm) + zm+1| ≤ |z1 + z2 + · · · + zm| + |zm+1| ≤ (|z1| + |z2| + · · · + |zm|) + |zm+1|. EXERCISES 1. Locate the numbers z1 + z2 and z1 − z2 vectorially when (a) z1 = 2i, z2 = 2 3 − i; (b) z1 = (− √ 3, 1), z2 = ( √ 3, 0); (c) z1 = (−3, 1), z2 = (1, 4); (d) z1 = x1 + iy1, z2 = x1 − iy1. 2. Verify inequalities (3), Sec. 4, involving Re z, Im z, and |z|. 3. Use established properties of moduli to show that when |z3| = |z4|, Re(z1 + z2) |z3 + z4| ≤ |z1| + |z2| ||z3| − |z4|| . 4. Verify that √ 2 |z| ≥ |Re z| + |Im z|. Suggestion: Reduce this inequality to (|x| − |y|)2 ≥ 0. 5. In each case, sketch the set of points determined by the given condition: (a) |z − 1 + i| = 1; (b) |z + i| ≤ 3 ; (c) |z − 4i| ≥ 4. 6. Using the fact that |z1 − z2| is the distance between two points z1 and z2, give a geometric argument that (a) |z − 4i| + |z + 4i| = 10 represents an ellipse whose foci are (0, ±4) ; (b) |z − 1| = |z + i| represents the line through the origin whose slope is −1.
- 26. Brown-chap01-v3 10/29/07 3:32pm 13 sec. 5 Complex Conjugates 13 5. COMPLEX CONJUGATES The complex conjugate, or simply the conjugate, of a complex number z = x + iy is defined as the complex number x − iy and is denoted by z ; that is, z = x − iy. (1) The number z is represented by the point (x, −y), which is the reflection in the real axis of the point (x, y) representing z (Fig. 5). Note that z = z and |z| = |z| for all z. x O y z – z (x, y) (x, –y) FIGURE 5 If z1 = x1 + iy1 and z2 = x2 + iy2, then z1 + z2 = (x1 + x2) − i(y1 + y2) = (x1 − iy1) + (x2 − iy2). So the conjugate of the sum is the sum of the conjugates: z1 + z2 = z1 + z2. (2) In like manner, it is easy to show that z1 − z2 = z1 − z2, (3) z1z2 = z1 z2, (4) and z1 z2 = z1 z2 (z2 = 0). (5) The sum z + z of a complex number z = x + iy and its conjugate z = x − iy is the real number 2x, and the difference z − z is the pure imaginary number 2iy. Hence Re z = z + z 2 and Im z = z − z 2i . (6)
- 27. Brown-chap01-v3 10/29/07 3:32pm 14 14 Complex Numbers chap. 1 An important identity relating the conjugate of a complex number z = x + iy to its modulus is z z = |z|2 , (7) where each side is equal to x2 + y2. It suggests the method for determining a quotient z1/z2 that begins with expression (7), Sec. 3. That method is, of course, based on multiplying both the numerator and the denominator of z1/z2 by z2, so that the denominator becomes the real number |z2|2 . EXAMPLE 1. As an illustration, −1 + 3i 2 − i = (−1 + 3i)(2 + i) (2 − i)(2 + i) = −5 + 5i |2 − i|2 = −5 + 5i 5 = −1 + i. See also the example in Sec. 3. Identity (7) is especially useful in obtaining properties of moduli from properties of conjugates noted above. We mention that |z1z2| = |z1||z2| (8) and z1 z2 = |z1| |z2| (z2 = 0). (9) Property (8) can be established by writing |z1z2|2 = (z1z2)(z1z2) = (z1z2)(z1 z2) = (z1z1)(z2z2) = |z1|2 |z2|2 = (|z1||z2|)2 and recalling that a modulus is never negative. Property (9) can be verified in a similar way. EXAMPLE 2. Property (8) tells us that |z2| = |z|2 and |z3| = |z|3. Hence if z is a point inside the circle centered at the origin with radius 2, so that |z| 2, it follows from the generalized triangle inequality (10) in Sec. 4 that |z3 + 3z2 − 2z + 1| ≤ |z|3 + 3|z|2 + 2|z| + 1 25. EXERCISES 1. Use properties of conjugates and moduli established in Sec. 5 to show that (a) z + 3i = z − 3i; (b) iz = −iz; (c) (2 + i)2 = 3 − 4i; (d) |(2z + 5)( √ 2 − i)| = √ 3 |2z + 5|. 2. Sketch the set of points determined by the condition (a) Re(z − i) = 2; (b) |2z + i| = 4.
- 28. Brown-chap01-v3 10/29/07 3:32pm 15 sec. 5 Exercises 15 3. Verify properties (3) and (4) of conjugates in Sec. 5. 4. Use property (4) of conjugates in Sec. 5 to show that (a) z1z2z3 = z1 z2 z3 ; (b) z4 = z4 . 5. Verify property (9) of moduli in Sec. 5. 6. Use results in Sec. 5 to show that when z2 and z3 are nonzero, (a) z1 z2z3 = z1 z2 z3 ; (b) z1 z2z3 = |z1| |z2||z3| . 7. Show that |Re(2 + z + z3 )| ≤ 4 when |z| ≤ 1. 8. It is shown in Sec. 3 that if z1z2 = 0, then at least one of the numbers z1 and z2 must be zero. Give an alternative proof based on the corresponding result for real numbers and using identity (8), Sec. 5. 9. By factoring z4 − 4z2 + 3 into two quadratic factors and using inequality (8), Sec. 4, show that if z lies on the circle |z| = 2, then 1 z4 − 4z2 + 3 ≤ 1 3 . 10. Prove that (a) z is real if and only if z = z; (b) z is either real or pure imaginary if and only if z2 = z2. 11. Use mathematical induction to show that when n = 2, 3, . . . , (a) z1 + z2 + · · · + zn = z1 + z2 + · · · + zn; (b) z1z2 · · · zn = z1 z2 · · · zn. 12. Let a0, a1, a2, . . . , an (n ≥ 1) denote real numbers, and let z be any complex number. With the aid of the results in Exercise 11, show that a0 + a1z + a2z2 + · · · + anzn = a0 + a1z + a2z2 + · · · + anz n . 13. Show that the equation |z − z0| = R of a circle, centered at z0 with radius R, can be written |z|2 − 2 Re(zz0) + |z0|2 = R2 . 14. Using expressions (6), Sec. 5, for Re z and Im z, show that the hyperbola x2 − y2 = 1 can be written z2 + z2 = 2. 15. Follow the steps below to give an algebraic derivation of the triangle inequality (Sec. 4) |z1 + z2| ≤ |z1| + |z2|. (a) Show that |z1 + z2|2 = (z1 + z2)(z1 + z2) = z1z1 + (z1z2 + z1z2) + z2z2.
- 29. Brown-chap01-v3 10/29/07 3:32pm 16 16 Complex Numbers chap. 1 (b) Point out why z1z2 + z1z2 = 2 Re(z1z2) ≤ 2|z1||z2|. (c) Use the results in parts (a) and (b) to obtain the inequality |z1 + z2|2 ≤ (|z1| + |z2|)2 , and note how the triangle inequality follows. 6. EXPONENTIAL FORM Let r and θ be polar coordinates of the point (x, y) that corresponds to a nonzero complex number z = x + iy. Since x = r cos θ and y = r sin θ, the number z can be written in polar form as z = r(cos θ + i sin θ). (1) If z = 0, the coordinate θ is undefined; and so it is understood that z = 0 whenever polar coordinates are used. In complex analysis, the real number r is not allowed to be negative and is the length of the radius vector for z ; that is, r = |z|. The real number θ represents the angle, measured in radians, that z makes with the positive real axis when z is inter- preted as a radius vector (Fig. 6). As in calculus, θ has an infinite number of possible values, including negative ones, that differ by integral multiples of 2π. Those values can be determined from the equation tan θ = y/x, where the quadrant containing the point corresponding to z must be specified. Each value of θ is called an argument of z, and the set of all such values is denoted by arg z. The principal value of arg z, denoted by Arg z, is that unique value such that −π ≤ π. Evidently, then, arg z = Arg z + 2nπ (n = 0, ±1, ±2, . . .). (2) Also, when z is a negative real number, Arg z has value π, not −π. x y z = x + iy r FIGURE 6 EXAMPLE 1. The complex number −1 − i, which lies in the third quadrant, has principal argument −3π/4. That is, Arg(−1 − i) = − 3π 4 .
- 30. Brown-chap01-v3 10/29/07 3:32pm 17 sec. 6 Exponential Form 17 It must be emphasized that because of the restriction −π ≤ π of the principal argument , it is not true that Arg(−1 − i) = 5π/4. According to equation (2), arg(−1 − i) = − 3π 4 + 2nπ (n = 0, ±1, ±2, . . .). Note that the term Arg z on the right-hand side of equation (2) can be replaced by any particular value of arg z and that one can write, for instance, arg(−1 − i) = 5π 4 + 2nπ (n = 0, ±1, ±2, . . .). The symbol eiθ , or exp(iθ), is defined by means of Euler’s formula as eiθ = cos θ + i sin θ, (3) where θ is to be measured in radians. It enables one to write the polar form (1) more compactly in exponential form as z = reiθ . (4) The choice of the symbol eiθ will be fully motivated later on in Sec. 29. Its use in Sec. 7 will, however, suggest that it is a natural choice. EXAMPLE 2. The number −1 − i in Example 1 has exponential form −1 − i = √ 2 exp i − 3π 4 . (5) With the agreement that e−iθ = ei(−θ) , this can also be written −1 − i = √ 2 e−i3π/4 . Expression (5) is, of course, only one of an infinite number of possibilities for the exponential form of −1 − i: −1 − i = √ 2 exp i − 3π 4 + 2nπ (n = 0, ±1, ±2, . . .). (6) Note how expression (4) with r = 1 tells us that the numbers eiθ lie on the circle centered at the origin with radius unity, as shown in Fig. 7. Values of eiθ are, then, immediate from that figure, without reference to Euler’s formula. It is, for instance, geometrically obvious that eiπ = −1, e−iπ/2 = −i, and e−i4π = 1.
- 31. Brown-chap01-v3 10/29/07 3:32pm 18 18 Complex Numbers chap. 1 x O 1 y FIGURE 7 Note, too, that the equation z = Reiθ (0 ≤ θ ≤ 2π) (7) is a parametric representation of the circle |z| = R, centered at the origin with radius R. As the parameter θ increases from θ = 0 to θ = 2π, the point z starts from the positive real axis and traverses the circle once in the counterclockwise direction. More generally, the circle |z − z0| = R, whose center is z0 and whose radius is R, has the parametric representation z = z0 + Reiθ (0 ≤ θ ≤ 2π). (8) This can be seen vectorially (Fig. 8) by noting that a point z traversing the circle |z − z0| = R once in the counterclockwise direction corresponds to the sum of the fixed vector z0 and a vector of length R whose angle of inclination θ varies from θ = 0 to θ = 2π. x O y z z0 FIGURE 8 7. PRODUCTS AND POWERS IN EXPONENTIAL FORM Simple trigonometry tells us that eiθ has the familiar additive property of the expo- nential function in calculus: eiθ1 eiθ2 = (cos θ1 + i sin θ1)(cos θ2 + i sin θ2) = (cos θ1 cos θ2 − sin θ1 sin θ2) + i(sin θ1 cos θ2 + cos θ1 sin θ2) = cos(θ1 + θ2) + i sin(θ1 + θ2) = ei(θ1+θ2) .
- 32. Brown-chap01-v3 10/29/07 3:32pm 19 sec. 7 Products and Powers in Exponential Form 19 Thus, if z1 = r1eiθ1 and z2 = r2eiθ2 , the product z1z2 has exponential form z1z2 = r1eiθ1 r2eiθ2 = r1r2eiθ1 eiθ2 = (r1r2)ei(θ1+θ2) . (1) Furthermore, z1 z2 = r1eiθ1 r2eiθ2 = r1 r2 · eiθ1 e−iθ2 eiθ2 e−iθ2 = r1 r2 · ei(θ1−θ2) ei0 = r1 r2 ei(θ1−θ2) . (2) Note how it follows from expression (2) that the inverse of any nonzero complex number z = reiθ is z−1 = 1 z = 1ei0 reiθ = 1 r ei(0−θ) = 1 r e−iθ . (3) Expressions (1), (2), and (3) are, of course, easily remembered by applying the usual algebraic rules for real numbers and ex . Another important result that can be obtained formally by applying rules for real numbers to z = reiθ is zn = rn einθ (n = 0, ±1, ±2, . . .). (4) It is easily verified for positive values of n by mathematical induction. To be specific, we first note that it becomes z = reiθ when n = 1. Next, we assume that it is valid when n = m, where m is any positive integer. In view of expression (1) for the product of two nonzero complex numbers in exponential form, it is then valid for n = m + 1: zm+1 = zm z = rm eimθ reiθ = (rm r)ei(mθ+θ) = rm+1 ei(m+1)θ . Expression (4) is thus verified when n is a positive integer. It also holds when n = 0, with the convention that z0 = 1. If n = −1, −2, . . . , on the other hand, we define zn in terms of the multiplicative inverse of z by writing zn = (z−1 )m where m = −n = 1, 2, . . . . Then, since equation (4) is valid for positive integers, it follows from the exponential form (3) of z−1 that zn = 1 r ei(−θ) m = 1 r m eim(−θ) = 1 r −n ei(−n)(−θ) = rn einθ (n = −1, −2, . . .). Expression (4) is now established for all integral powers. Expression (4) can be useful in finding powers of complex numbers even when they are given in rectangular form and the result is desired in that form.
- 33. Brown-chap01-v3 10/29/07 3:32pm 20 20 Complex Numbers chap. 1 EXAMPLE 1. In order to put ( √ 3 + 1)7 in rectangular form, one need only write ( √ 3 + i)7 = (2eiπ/6 )7 = 27 ei7π/6 = (26 eiπ )(2eiπ/6 ) = −64( √ 3 + i). Finally, we observe that if r = 1, equation (4) becomes (eiθ )n = einθ (n = 0, ±1, ±2, . . .). (5) When written in the form (cos θ + i sin θ)n = cos nθ + i sin nθ (n = 0, ±1, ±2, . . .), (6) this is known as de Moivre’s formula. The following example uses a special case of it. EXAMPLE 2. Formula (6) with n = 2 tells us that (cos θ + i sin θ)2 = cos 2θ + i sin 2θ, or cos2 θ − sin2 θ + i2 sin θ cos θ = cos 2θ + i sin 2θ. By equating real parts and then imaginary parts here, we have the familiar trigono- metric identities cos 2θ = cos2 θ − sin2 θ, sin 2θ = 2 sin θ cos θ. (See also Exercises 10 and 11, Sec. 8.) 8. ARGUMENTS OF PRODUCTS AND QUOTIENTS If z1 = r1eiθ1 and z2 = r2eiθ2 , the expression z1z2 = (r1r2)ei(θ1+θ2) (1) in Sec. 7 can be used to obtain an important identity involving arguments: arg(z1z2) = arg z1 + arg z2. (2) This result is to be interpreted as saying that if values of two of the three (multiple- valued) arguments are specified, then there is a value of the third such that the equation holds. We start the verification of statement (2) by letting θ1 and θ2 denote any values of arg z1 and arg z2, respectively. Expression (1) then tells us that θ1 + θ2 is a value of arg(z1z2). (See Fig. 9.) If, on the other hand, values of arg(z1z2) and
- 34. Brown-chap01-v3 10/29/07 3:32pm 21 sec. 8 Arguments of Products and Quotients 21 x O y z1z2 z1 z2 FIGURE 9 arg z1 are specified, those values correspond to particular choices of n and n1 in the expressions arg(z1z2) = (θ1 + θ2) + 2nπ (n = 0, ±1, ±2, . . .) and arg z1 = θ1 + 2n1π (n1 = 0, ±1, ±2, . . .). Since (θ1 + θ2) + 2nπ = (θ1 + 2n1π) + [θ2 + 2(n − n1)π], equation (2) is evidently satisfied when the value arg z2 = θ2 + 2(n − n1)π is chosen. Verification when values of arg(z1z2) and arg z2 are specified follows by symmetry. Statement (2) is sometimes valid when arg is replaced everywhere by Arg (see Exercise 6). But, as the following example illustrates, that is not always the case. EXAMPLE 1. When z1 = −1 and z2 = i, Arg(z1z2) = Arg(−i) = − π 2 but Arg z1 + Arg z2 = π + π 2 = 3π 2 . If, however, we take the values of arg z1 and arg z2 just used and select the value Arg(z1z2) + 2π = − π 2 + 2π = 3π 2 of arg(z1z2), we find that equation (2) is satisfied. Statement (2) tells us that arg z1 z2 = arg(z1z−1 2 ) = arg z1 + arg(z−1 2 );
- 35. Brown-chap01-v3 10/29/07 3:32pm 22 22 Complex Numbers chap. 1 and, since (Sec. 7) z−1 2 = 1 r2 e−iθ2 , one can see that arg(z−1 2 ) = −arg z2. (3) Hence arg z1 z2 = arg z1 − arg z2. (4) Statement (3) is, of course, to be interpreted as saying that the set of all values on the left-hand side is the same as the set of all values on the right-hand side. Statement (4) is, then, to be interpreted in the same way that statement (2) is. EXAMPLE 2. In order to find the principal argument Arg z when z = −2 1 + √ 3i , observe that arg z = arg(−2) − arg(1 + √ 3i). Since Arg(−2) = π and Arg(1 + √ 3i) = π 3 , one value of arg z is 2π/3; and, because 2π/3 is between −π and π, we find that Arg z = 2π/3. EXERCISES 1. Find the principal argument Arg z when (a) z = i −2 − 2i ; (b) z = ( √ 3 − i)6. Ans. (a) −3π/4; (b) π. 2. Show that (a) |eiθ | = 1; (b) eiθ = e−iθ . 3. Use mathematical induction to show that eiθ1 eiθ2 · · · eiθn = ei(θ1+θ2+···+θn) (n = 2, 3, . . .). 4. Using the fact that the modulus |eiθ − 1| is the distance between the points eiθ and 1 (see Sec. 4), give a geometric argument to find a value of θ in the interval 0 ≤ θ 2π that satisfies the equation |eiθ − 1| = 2. Ans. π.
- 36. Brown-chap01-v3 10/29/07 3:32pm 23 sec. 8 Exercises 23 5. By writing the individual factors on the left in exponential form, performing the needed operations, and finally changing back to rectangular coordinates, show that (a) i(1 − √ 3i)( √ 3 + i) = 2(1 + √ 3i); (b) 5i/(2 + i) = 1 + 2i; (c) (−1 + i)7 = −8(1 + i); (d) (1 + √ 3i)−10 = 2−11(−1 + √ 3i). 6. Show that if Re z1 0 and Re z2 0, then Arg(z1z2) = Arg z1 + Arg z2, where principal arguments are used. 7. Let z be a nonzero complex number and n a negative integer (n = −1, −2, . . .). Also, write z = reiθ and m = −n = 1, 2, . . . . Using the expressions zm = rm eimθ and z−1 = 1 r ei(−θ) , verify that (zm)−1 = (z−1)m and hence that the definition zn = (z−1)m in Sec. 7 could have been written alternatively as zn = (zm )−1 . 8. Prove that two nonzero complex numbers z1 and z2 have the same moduli if and only if there are complex numbers c1 and c2 such that z1 = c1c2 and z2 = c1c2. Suggestion: Note that exp i θ1 + θ2 2 exp i θ1 − θ2 2 = exp(iθ1) and [see Exercise 2(b)] exp i θ1 + θ2 2 exp i θ1 − θ2 2 = exp(iθ2). 9. Establish the identity 1 + z + z2 + · · · + zn = 1 − zn+1 1 − z (z = 1) and then use it to derive Lagrange’s trigonometric identity: 1 + cos θ + cos 2θ + · · · + cos nθ = 1 2 + sin[(2n + 1)θ/2] 2 sin(θ/2) (0 θ 2π). Suggestion: As for the first identity, write S = 1 + z + z2 + · · · + zn and consider the difference S − zS. To derive the second identity, write z = eiθ in the first one. 10. Use de Moivre’s formula (Sec. 7) to derive the following trigonometric identities: (a) cos 3θ = cos3 θ − 3 cos θ sin2 θ; (b) sin 3θ = 3 cos2 θ sin θ − sin3 θ.
- 37. Brown-chap01-v3 10/29/07 3:32pm 24 24 Complex Numbers chap. 1 11. (a) Use the binomial formula (Sec. 3) and de Moivre’s formula (Sec. 7) to write cos nθ + i sin nθ = n k=0 n k cosn−k θ (i sin θ)k (n = 0, 1, 2, . . .). Then define the integer m by means of the equations m = n/2 if n is even, (n − 1)/2 if nis odd and use the above summation to show that [compare with Exercise 10(a)] cos nθ = m k=0 n 2k (−1)k cosn−2k θ sin2k θ (n = 0, 1, 2, . . .). (b) Write x = cos θ in the final summation in part (a) to show that it becomes a polynomial Tn(x) = m k=0 n 2k (−1)k xn−2k (1 − x2 )k of degree n (n = 0, 1, 2, . . .) in the variable x.∗ 9. ROOTS OF COMPLEX NUMBERS Consider now a point z = reiθ , lying on a circle centered at the origin with radius r (Fig. 10). As θ is increased, z moves around the circle in the counterclockwise direction. In particular, when θ is increased by 2π, we arrive at the original point; and the same is true when θ is decreased by 2π. It is, therefore, evident from Fig. 10 that two nonzero complex numbers z1 = r1eiθ1 and z2 = r2eiθ2 x O r y FIGURE 10 ∗These are called Chebyshev polynomials and are prominent in approximation theory.
- 38. Brown-chap01-v3 10/29/07 3:32pm 25 sec. 9 Roots of Complex Numbers 25 are equal if and only if r1 = r2 and θ1 = θ2 + 2kπ, where k is some integer (k = 0, ±1, ±2, . . .). This observation, together with the expression zn = rn einθ in Sec. 7 for integral powers of complex numbers z = reiθ , is useful in finding the nth roots of any nonzero complex number z0 = r0eiθ0 , where n has one of the values n = 2, 3, . . . . The method starts with the fact that an nth root of z0 is a nonzero number z = reiθ such that zn = z0, or rn einθ = r0eiθ0 . According to the statement in italics just above, then, rn = r0 and nθ = θ0 + 2kπ, where k is any integer (k = 0, ±1, ±2, . . .). So r = n √ r0, where this radical denotes the unique positive nth root of the positive real number r0, and θ = θ0 + 2kπ n = θ0 n + 2kπ n (k = 0, ±1, ±2, . . .). Consequently, the complex numbers z = n √ r0 exp i θ0 n + 2kπ n (k = 0, ±1, ±2, . . .) are the nth roots of z0. We are able to see immediately from this exponential form of the roots that they all lie on the circle |z| = n √ r0 about the origin and are equally spaced every 2π/n radians, starting with argument θ0/n. Evidently, then, all of the distinct roots are obtained when k = 0, 1, 2, . . . , n − 1, and no further roots arise with other values of k. We let ck (k = 0, 1, 2, . . . , n − 1) denote these distinct roots and write ck = n √ r0 exp i θ0 n + 2kπ n (k = 0, 1, 2, . . . , n − 1). (1) (See Fig. 11.) x O y n ck–1 ck √ r0 n FIGURE 11
- 39. Brown-chap01-v3 10/29/07 3:32pm 26 26 Complex Numbers chap. 1 The number n √ r0 is the length of each of the radius vectors representing the n roots. The first root c0 has argument θ0/n; and the two roots when n = 2 lie at the opposite ends of a diameter of the circle |z| = n √ r0, the second root being −c0. When n ≥ 3, the roots lie at the vertices of a regular polygon of n sides inscribed in that circle. We shall let z 1/n 0 denote the set of nth roots of z0. If, in particular, z0 is a positive real number r0, the symbol r 1/n 0 denotes the entire set of roots; and the symbol n √ r0 in expression (1) is reserved for the one positive root. When the value of θ0 that is used in expression (1) is the principal value of arg z0 (−π θ0 ≤ π), the number c0 is referred to as the principal root. Thus when z0 is a positive real number r0, its principal root is n √ r0. Observe that if we write expression (1) for the roots of z0 as ck = n √ r0 exp i θ0 n exp i 2kπ n (k = 0, 1, 2, . . . , n − 1), and also write ωn = exp i 2π n , (2) it follows from property (5), Sec. 7. of eiθ that ωk n = exp i 2kπ n (k = 0, 1, 2, . . . , n − 1) (3) and hence that ck = c0ωk n (k = 0, 1, 2, . . . , n − 1). (4) The number c0 here can, of course, be replaced by any particular nth root of z0, since ωn represents a counterclockwise rotation through 2π/n radians. Finally, a convenient way to remember expression (1) is to write z0 in its most general exponential form (compare with Example 2 in Sec. 6) z0 = r0 ei(θ0+2kπ) (k = 0, ±1, ±2, . . .) (5) and to formally apply laws of fractional exponents involving real numbers, keeping in mind that there are precisely n roots: z 1/n 0 = r0 ei(θ0+2kπ) 1/n = n √ r0 exp i(θ0 + 2kπ) n = n √ r0 exp i θ0 n + 2kπ n (k = 0, 1, 2, . . . , n − 1).
- 40. Brown-chap01-v3 10/29/07 3:32pm 27 sec. 10 Examples 27 The examples in the next section serve to illustrate this method for finding roots of complex numbers. 10. EXAMPLES In each of the examples here, we start with expression (5), Sec. 9, and proceed in the manner described just after it. EXAMPLE 1. Let us find all values of (−8i)1/3, or the three cube roots of the number −8i. One need only write −8i = 8 exp i − π 2 + 2kπ (k = 0, ±1, ±2, . . .) to see that the desired roots are ck = 2 exp i − π 6 + 2kπ 3 (k = 0, 1, 2). (1) They lie at the vertices of an equilateral triangle, inscribed in the circle |z| = 2, and are equally spaced around that circle every 2π/3 radians, starting with the principal root (Fig. 12) c0 = 2 exp i − π 6 = 2 cos π 6 − i sin π 6 = √ 3 − i. Without any further calculations, it is then evident that c1 = 2i; and, since c2 is symmetric to c0 with respect to the imaginary axis, we know that c2 = − √ 3 − i. Note how it follows from expressions (2) and (4) in Sec. 9 that these roots can be written c0, c0ω3, c0ω2 3 where ω3 = exp i 2π 3 . x y c1 c0 c2 2 FIGURE 12
- 41. Brown-chap01-v3 10/29/07 3:32pm 28 28 Complex Numbers chap. 1 EXAMPLE 2. In order to determine the nth roots of unity, we start with 1 = 1 exp[i(0 + 2kπ)] (k = 0, ±1, ±2 . . .) and find that 11/n = n √ 1 exp i 0 n + 2kπ n = exp i 2kπ n (k = 0, 1, 2, . . . , n − 1). (2) When n = 2, these roots are, of course, ±1. When n ≥ 3, the regular polygon at whose vertices the roots lie is inscribed in the unit circle |z| = 1, with one vertex corresponding to the principal root z = 1 (k = 0). In view of expression (3), Sec. 9, these roots are simply 1, ωn, ω2 n, . . . , ωn−1 n where ωn = exp i 2π n . See Fig. 13, where the cases n = 3, 4, and 6 are illustrated. Note that ωn n = 1. x y 1 x y 1 x y 1 FIGURE 13 EXAMPLE 3. The two values ck (k = 0, 1) of ( √ 3 + i)1/2 , which are the square roots of √ 3 + i, are found by writing √ 3 + i = 2 exp i π 6 + 2kπ (k = 0, ±1, ±2, . . .) and (see Fig. 14) ck = √ 2 exp i π 12 + kπ (k = 0, 1). (3) Euler’s formula tells us that c0 = √ 2 exp i π 12 = √ 2 cos π 12 + i sin π 12 , and the trigonometric identities
- 42. Brown-chap01-v3 10/29/07 3:32pm 29 sec. 10 Exercises 29 x y c0 c1 = –c0 2 √ FIGURE 14 cos2 α 2 = 1 + cos α 2 , sin2 α 2 = 1 − cos α 2 (4) enable us to write cos2 π 12 = 1 2 1 + cos π 6 = 1 2 1 + √ 3 2 = 2 + √ 3 4 , sin2 π 12 = 1 2 1 − cos π 6 = 1 2 1 − √ 3 2 = 2 − √ 3 4 . Consequently, c0 = √ 2 2 + √ 3 4 + i 2 − √ 3 4 = 1 √ 2 2 + √ 3 + i 2 − √ 3 . Since c1 = −c0, the two square roots of √ 3 + i are, then, ± 1 √ 2 2 + √ 3 + i 2 − √ 3 . (5) EXERCISES 1. Find the square roots of (a) 2i; (b) 1 − √ 3i and express them in rectangular coordi- nates. Ans. (a) ± (1 + i); (b) ± √ 3 − i √ 2 . 2. In each case, find all the roots in rectangular coordinates, exhibit them as vertices of certain squares, and point out which is the principal root: (a) (−16)1/4; (b) (−8 − 8 √ 3i)1/4. Ans. (a) ± √ 2(1 + i), ± √ 2(1 − i); (b) ±( √ 3 − i), ±(1 + √ 3i).
- 43. Brown-chap01-v3 10/29/07 3:32pm 30 30 Complex Numbers chap. 1 3. In each case, find all the roots in rectangular coordinates, exhibit them as vertices of certain regular polygons, and identify the principal root: (a) (−1)1/3; (b) 81/6. Ans. (b) ± √ 2, ± 1 + √ 3i √ 2 , ± 1 − √ 3i √ 2 . 4. According to Sec. 9, the three cube roots of a nonzero complex number z0 can be written c0, c0ω3, c0ω2 3 where c0 is the principal cube root of z0 and ω3 = exp i 2π 3 = −1 + √ 3i 2 . Show that if z0 = −4 √ 2 + 4 √ 2i, then c0 = √ 2(1 + i) and the other two cube roots are, in rectangular form, the numbers c0ω3 = −( √ 3 + 1) + ( √ 3 − 1)i √ 2 , c0ω2 3 = ( √ 3 − 1) − ( √ 3 + 1)i √ 2 . 5. (a) Let a denote any fixed real number and show that the two square roots of a + i are ± √ A exp i α 2 where A = √ a2 + 1 and α = Arg(a + i). (b) With the aid of the trigonometric identities (4) in Example 3 of Sec. 10, show that the square roots obtained in part (a) can be written ± 1 √ 2 √ A + a + i √ A − a . (Note that this becomes the final result in Example 3, Sec. 10, when a = √ 3.) 6. Find the four zeros of the polynomial z4 + 4, one of them being z0 = √ 2 eiπ/4 = 1 + i. Then use those zeros to factor z2 + 4 into quadratic factors with real coefficients. Ans. (z2 + 2z + 2)(z2 − 2z + 2). 7. Show that if c is any nth root of unity other than unity itself, then 1 + c + c2 + · · · + cn−1 = 0. Suggestion: Use the first identity in Exercise 9, Sec. 8. 8. (a) Prove that the usual formula solves the quadratic equation az2 + bz + c = 0 (a = 0) when the coefficients a, b, and c are complex numbers. Specifically, by completing the square on the left-hand side, derive the quadratic formula z = −b + (b2 − 4ac)1/2 2a , where both square roots are to be considered when b2 − 4ac = 0,
- 44. Brown-chap01-v3 10/29/07 3:32pm 31 sec. 11 Regions in the Complex Plane 31 (b) Use the result in part (a) to find the roots of the equation z2 + 2z + (1 − i) = 0. Ans. (b) −1 + 1 √ 2 + i √ 2 , −1 − 1 √ 2 − i √ 2 . 9. Let z = reiθ be a nonzero complex number and n a negative integer (n = −1, −2, . . .). Then define z1/n by means of the equation z1/n = (z−1 )1/m where m = −n. By showing that the m values of (z1/m)−1 and (z−1)1/m are the same, verify that z1/n = (z1/m)−1. (Compare with Exercise 7, Sec. 8.) 11. REGIONS IN THE COMPLEX PLANE In this section, we are concerned with sets of complex numbers, or points in the z plane, and their closeness to one another. Our basic tool is the concept of an ε neighborhood |z − z0| ε (1) of a given point z0. It consists of all points z lying inside but not on a circle centered at z0 and with a specified positive radius ε (Fig. 15). When the value of ε is understood or is immaterial in the discussion, the set (1) is often referred to as just a neighborhood. Occasionally, it is convenient to speak of a deleted neighborhood, or punctured disk, 0 |z − z0| ε (2) consisting of all points z in an ε neighborhood of z0 except for the point z0 itself. x z |z – z0| z0 O y ε FIGURE 15 A point z0 is said to be an interior point of a set S whenever there is some neighborhood of z0 that contains only points of S; it is called an exterior point of S when there exists a neighborhood of it containing no points of S. If z0 is neither of these, it is a boundary point of S. A boundary point is, therefore, a point all of whose neighborhoods contain at least one point in S and at least one point not in S. The totality of all boundary points is called the boundary of S. The circle |z| = 1, for instance, is the boundary of each of the sets |z| 1 and |z| ≤ 1. (3)
- 45. Brown-chap01-v3 10/29/07 3:32pm 32 32 Complex Numbers chap. 1 A set is open if it contains none of its boundary points. It is left as an exercise to show that a set is open if and only if each of its points is an interior point. A set is closed if it contains all of its boundary points, and the closure of a set S is the closed set consisting of all points in S together with the boundary of S. Note that the first of the sets (3) is open and that the second is its closure. Some sets are, of course, neither open nor closed. For a set to be not open, there must be a boundary point that is contained in the set; and if a set is not closed, there exists a boundary point not contained in the set. Observe that the punctured disk 0 |z| ≤ 1 is neither open nor closed. The set of all complex numbers is, on the other hand, both open and closed since it has no boundary points. An open set S is connected if each pair of points z1 and z2 in it can be joined by a polygonal line, consisting of a finite number of line segments joined end to end, that lies entirely in S. The open set |z| 1 is connected. The annulus 1 |z| 2 is, of course, open and it is also connected (see Fig. 16). A nonempty open set that is connected is called a domain. Note that any neighborhood is a domain. A domain together with some, none, or all of its boundary points is referred to as a region. x z1 z2 O 1 2 y FIGURE 16 A set S is bounded if every point of S lies inside some circle |z| = R; otherwise, it is unbounded. Both of the sets (3) are bounded regions, and the half plane Re z ≥ 0 is unbounded. A point z0 is said to be an accumulation point of a set S if each deleted neighborhood of z0 contains at least one point of S. It follows that if a set S is closed, then it contains each of its accumulation points. For if an accumulation point z0 were not in S, it would be a boundary point of S; but this contradicts the fact that a closed set contains all of its boundary points. It is left as an exercise to show that the converse is, in fact, true. Thus a set is closed if and only if it contains all of its accumulation points. Evidently, a point z0 is not an accumulation point of a set S whenever there exists some deleted neighborhood of z0 that does not contain at least one point of S. Note that the origin is the only accumulation point of the set zn = i/n (n = 1, 2, . . .).
- 46. Brown-chap01-v3 10/29/07 3:32pm 33 sec. 11 Exercises 33 EXERCISES 1. Sketch the following sets and determine which are domains: (a) |z − 2 + i| ≤ 1; (b) |2z + 3| 4; (c) Im z 1; (d) Im z = 1; (e) 0 ≤ arg z ≤ π/4 (z = 0); (f) |z − 4| ≥ |z|. Ans. (b), (c) are domains. 2. Which sets in Exercise 1 are neither open nor closed? Ans. (e). 3. Which sets in Exercise 1 are bounded? Ans. (a). 4. In each case, sketch the closure of the set: (a) −π arg z π (z = 0); (b) |Re z| |z|; (c) Re 1 z ≤ 1 2 ; (d) Re(z2) 0. 5. Let S be the open set consisting of all points z such that |z| 1 or |z − 2| 1. State why S is not connected. 6. Show that a set S is open if and only if each point in S is an interior point. 7. Determine the accumulation points of each of the following sets: (a) zn = in (n = 1, 2, . . .); (b) zn = in /n (n = 1, 2, . . .); (c) 0 ≤ arg z π/2 (z = 0); (d) zn = (−1)n(1 + i) n − 1 n (n = 1, 2, . . .). Ans. (a) None; (b) 0; (d) ±(1 + i). 8. Prove that if a set contains each of its accumulation points, then it must be a closed set. 9. Show that any point z0 of a domain is an accumulation point of that domain. 10. Prove that a finite set of points z1, z2, . . . , zn cannot have any accumulation points.
- 48. C H A P T E R 2 ANALYTIC FUNCTIONS We now consider functions of a complex variable and develop a theory of differ- entiation for them. The main goal of the chapter is to introduce analytic functions, which play a central role in complex analysis. 12. FUNCTIONS OF A COMPLEX VARIABLE Let S be a set of complex numbers. A function f defined on S is a rule that assigns to each z in S a complex number w. The number w is called the value of f at z and is denoted by f (z); that is, w = f (z). The set S is called the domain of definition of f .∗ It must be emphasized that both a domain of definition and a rule are needed in order for a function to be well defined. When the domain of definition is not mentioned, we agree that the largest possible set is to be taken. Also, it is not always convenient to use notation that distinguishes between a given function and its values. EXAMPLE 1. If f is defined on the set z = 0 by means of the equation w = 1/z, it may be referred to only as the function w = 1/z, or simply the func- tion 1/z. Suppose that w = u + iv is the value of a function f at z = x + iy, so that u + iv = f (x + iy). ∗Although the domain of definition is often a domain as defined in Sec. 11, it need not be. 35
- 49. 36 Analytic Functions chap. 2 Each of the real numbers u and v depends on the real variables x and y, and it follows that f (z) can be expressed in terms of a pair of real-valued functions of the real variables x and y: f (z) = u(x, y) + iv(x, y). (1) If the polar coordinates r and θ, instead of x and y, are used, then u + iv = f (reiθ ) where w = u + iv and z = reiθ . In that case, we may write f (z) = u(r, θ) + iv(r, θ). (2) EXAMPLE 2. If f (z) = z2 , then f (x + iy) = (x + iy)2 = x2 − y2 + i2xy. Hence u(x, y) = x2 − y2 and v(x, y) = 2xy. When polar coordinates are used, f (reiθ ) = (reiθ )2 = r2 ei2θ = r2 cos 2θ + ir2 sin 2θ. Consequently, u(r, θ) = r2 cos 2θ and v(r, θ) = r2 sin 2θ. If, in either of equations (1) and (2), the function v always has value zero, then the value of f is always real. That is, f is a real-valued function of a complex variable. EXAMPLE 3. A real-valued function that is used to illustrate some important concepts later in this chapter is f (z) = |z|2 = x2 + y2 + i0. If n is zero or a positive integer and if a0, a1, a2, . . . , an are complex constants, where an = 0, the function P (z) = a0 + a1z + a2z2 + · · · + anzn is a polynomial of degree n. Note that the sum here has a finite number of terms and that the domain of definition is the entire z plane. Quotients P (z)/Q(z) of
- 50. sec. 12 Exercises 37 polynomials are called rational functions and are defined at each point z where Q(z) = 0. Polynomials and rational functions constitute elementary, but important, classes of functions of a complex variable. A generalization of the concept of function is a rule that assigns more than one value to a point z in the domain of definition. These multiple-valued func- tions occur in the theory of functions of a complex variable, just as they do in the case of a real variable. When multiple-valued functions are studied, usually just one of the possible values assigned to each point is taken, in a systematic manner, and a (single-valued) function is constructed from the multiple-valued function. EXAMPLE 4. Let z denote any nonzero complex number. We know from Sec. 9 that z1/2 has the two values z1/2 = ± √ r exp i 2 , where r = |z| and (−π ≤ π) is the principal value of arg z. But, if we choose only the positive value of ± √ r and write f (z) = √ r exp i 2 (r 0, −π ≤ π), (3) the (single-valued) function (3) is well defined on the set of nonzero numbers in the z plane. Since zero is the only square root of zero, we also write f (0) = 0. The function f is then well defined on the entire plane. EXERCISES 1. For each of the functions below, describe the domain of definition that is understood: (a) f (z) = 1 z2 + 1 ; (b) f (z) = Arg 1 z ; (c) f (z) = z z + z ; (d) f (z) = 1 1 − |z|2 . Ans. (a) z = ±i; (c) Re z = 0. 2. Write the function f (z) = z3 + z + 1 in the form f (z) = u(x, y) + iv(x, y). Ans. f (z) = (x3 − 3xy2 + x + 1) + i(3x2y − y3 + y). 3. Suppose that f (z) = x2 − y2 − 2y + i(2x − 2xy), where z = x + iy. Use the expres- sions (see Sec. 5) x = z + z 2 and y = z − z 2i to write f (z) in terms of z, and simplify the result. Ans. f (z) = z2 + 2iz.
- 51. 38 Analytic Functions chap. 2 4. Write the function f (z) = z + 1 z (z = 0) in the form f (z) = u(r, θ) + iv(r, θ). Ans. f (z) = r + 1 r cos θ + i r − 1 r sin θ. 13. MAPPINGS Properties of a real-valued function of a real variable are often exhibited by the graph of the function. But when w = f (z), where z and w are complex, no such convenient graphical representation of the function f is available because each of the numbers z and w is located in a plane rather than on a line. One can, however, display some information about the function by indicating pairs of corresponding points z = (x, y) and w = (u, v). To do this, it is generally simpler to draw the z and w planes separately. When a function f is thought of in this way, it is often referred to as a mapping, or transformation. The image of a point z in the domain of definition S is the point w = f (z), and the set of images of all points in a set T that is contained in S is called the image of T . The image of the entire domain of definition S is called the range of f . The inverse image of a point w is the set of all points z in the domain of definition of f that have w as their image. The inverse image of a point may contain just one point, many points, or none at all. The last case occurs, of course, when w is not in the range of f . Terms such as translation, rotation, and reflection are used to convey domi- nant geometric characteristics of certain mappings. In such cases, it is sometimes convenient to consider the z and w planes to be the same. For example, the mapping w = z + 1 = (x + 1) + iy, where z = x + iy, can be thought of as a translation of each point z one unit to the right. Since i = eiπ/2 , the mapping w = iz = r exp i θ + π 2 , where z = reiθ , rotates the radius vector for each nonzero point z through a right angle about the origin in the counterclockwise direction; and the mapping w = z = x − iy transforms each point z = x + iy into its reflection in the real axis. More information is usually exhibited by sketching images of curves and regions than by simply indicating images of individual points. In the following three examples, we illustrate this with the transformation w = z2 . We begin by finding the images of some curves in the z plane.
- 52. sec. 13 Mappings 39 EXAMPLE 1. According to Example 2 in Sec. 12, the mapping w = z2 can be thought of as the transformation u = x2 − y2 , v = 2xy (1) from the xy plane into the uv plane. This form of the mapping is especially useful in finding the images of certain hyperbolas. It is easy to show, for instance, that each branch of a hyperbola x2 − y2 = c1 (c1 0) (2) is mapped in a one to one manner onto the vertical line u = c1. We start by noting from the first of equations (1) that u = c1 when (x, y) is a point lying on either branch. When, in particular, it lies on the right-hand branch, the second of equations (1) tells us that v = 2y y2 + c1. Thus the image of the right-hand branch can be expressed parametrically as u = c1, v = 2y y2 + c1 (−∞ y ∞); and it is evident that the image of a point (x, y) on that branch moves upward along the entire line as (x, y) traces out the branch in the upward direction (Fig. 17). Likewise, since the pair of equations u = c1, v = −2y y2 + c1 (−∞ y ∞) furnishes a parametric representation for the image of the left-hand branch of the hyperbola, the image of a point going downward along the entire left-hand branch is seen to move up the entire line u = c1. x O y u u = c1 0 v = c2 0 O v FIGURE 17 w = z2. On the other hand, each branch of a hyperbola 2xy = c2 (c2 0) (3) is transformed into the line v = c2, as indicated in Fig. 17. To verify this, we note from the second of equations (1) that v = c2 when (x, y) is a point on either
- 53. 40 Analytic Functions chap. 2 branch. Suppose that (x, y) is on the branch lying in the first quadrant. Then, since y = c2/(2x), the first of equations (1) reveals that the branch’s image has parametric representation u = x2 − c2 2 4x2 , v = c2 (0 x ∞). Observe that lim x→0 x0 u = −∞ and lim x→∞ u = ∞. Since u depends continuously on x, then, it is clear that as (x, y) travels down the entire upper branch of hyperbola (3), its image moves to the right along the entire horizontal line v = c2. Inasmuch as the image of the lower branch has parametric representation u = c2 2 4y2 − y2 , v = c2 (−∞ y 0) and since lim y→−∞ u = −∞ and lim y→0 y0 u = ∞, it follows that the image of a point moving upward along the entire lower branch also travels to the right along the entire line v = c2 (see Fig. 17). We shall now use Example 1 to find the image of a certain region. EXAMPLE 2. The domain x 0, y 0, xy 1 consists of all points lying on the upper branches of hyperbolas from the family 2xy = c, where 0 c 2 (Fig. 18). We know from Example 1 that as a point travels downward along the entirety of such a branch, its image under the transformation w = z2 moves to the right along the entire line v = c. Since, for all values of c between 0 and 2, these upper branches fill out the domain x 0, y 0, xy 1, that domain is mapped onto the horizontal strip 0 v 2. x B C E A D D′ A′ B′ C′ E′ 2i y u v FIGURE 18 w = z2.
- 54. sec. 13 Mappings 41 In view of equations (1), the image of a point (0, y) in the z plane is (−y2 , 0). Hence as (0, y) travels downward to the origin along the y axis, its image moves to the right along the negative u axis and reaches the origin in the w plane. Then, since the image of a point (x, 0) is (x2, 0), that image moves to the right from the origin along the u axis as (x, 0) moves to the right from the origin along the x axis. The image of the upper branch of the hyperbola xy = 1 is, of course, the horizontal line v = 2. Evidently, then, the closed region x ≥ 0, y ≥ 0, xy ≤ 1 is mapped onto the closed strip 0 ≤ v ≤ 2, as indicated in Fig. 18. Our last example here illustrates how polar coordinates can be useful in ana- lyzing certain mappings. EXAMPLE 3. The mapping w = z2 becomes w = r2 ei2θ (4) when z = reiθ . Evidently, then, the image w = ρeiφ of any nonzero point z is found by squaring the modulus r = |z| and doubling the value θ of arg z that is used: ρ = r2 and φ = 2θ. (5) Observe that points z = r0eiθ on a circle r = r0 are transformed into points w = r2 0 ei2θ on the circle ρ = r2 0 . As a point on the first circle moves counterclock- wise from the positive real axis to the positive imaginary axis, its image on the second circle moves counterclockwise from the positive real axis to the negative real axis (see Fig. 19). So, as all possible positive values of r0 are chosen, the corresponding arcs in the z and w planes fill out the first quadrant and the upper half plane, respectively. The transformation w = z2 is, then, a one to one map- ping of the first quadrant r ≥ 0, 0 ≤ θ ≤ π/2 in the z plane onto the upper half ρ ≥ 0, 0 ≤ φ ≤ π of the w plane, as indicated in Fig. 19. The point z = 0 is, of course, mapped onto the point w = 0. x O y r0 u O v r2 0 FIGURE 19 w = z2. The transformation w = z2 also maps the upper half plane r ≥ 0, 0 ≤ θ ≤ π onto the entire w plane. However, in this case, the transformation is not one to one since both the positive and negative real axes in the z plane are mapped onto the positive real axis in the w plane.
- 55. 42 Analytic Functions chap. 2 When n is a positive integer greater than 2, various mapping properties of the transformation w = zn , or w = rn einθ , are similar to those of w = z2 . Such a transformation maps the entire z plane onto the entire w plane, where each nonzero point in the w plane is the image of n distinct points in the z plane. The circle r = r0 is mapped onto the circle ρ = rn 0 ; and the sector r ≤ r0, 0 ≤ θ ≤ 2π/n is mapped onto the disk ρ ≤ rn 0 , but not in a one to one manner. Other, but somewhat more involved, mappings by w = z2 appear in Example 1, Sec. 97, and Exercises 1 through 4 of that section. 14. MAPPINGS BY THE EXPONENTIAL FUNCTION In Chap. 3 we shall introduce and develop properties of a number of elemen- tary functions which do not involve polynomials. That chapter will start with the exponential function ez = ex eiy (z = x + iy), (1) the two factors ex and eiy being well defined at this time (see Sec. 6). Note that definition (1), which can also be written ex+iy = ex eiy , is suggested by the familiar additive property ex1+x2 = ex1 ex2 of the exponential function in calculus. The object of this section is to use the function ez to provide the reader with additional examples of mappings that continue to be reasonably simple. We begin by examining the images of vertical and horizontal lines. EXAMPLE 1. The transformation w = ez (2) can be written w = ex eiy , where z = x + iy, according to equation (1). Thus, if w = ρeiφ , transformation (2) can be expressed in the form ρ = ex , φ = y. (3) The image of a typical point z = (c1, y) on a vertical line x = c1 has polar coordinates ρ = exp c1 and φ = y in the w plane. That image moves counterclock- wise around the circle shown in Fig. 20 as z moves up the line. The image of the line is evidently the entire circle; and each point on the circle is the image of an infinite number of points, spaced 2π units apart, along the line.
- 56. sec. 14 Mappings by the Exponential Function 43 x x = c1 y = c2 O y u exp c1 c2 O v FIGURE 20 w = exp z. A horizontal line y = c2 is mapped in a one to one manner onto the ray φ = c2. To see that this is so, we note that the image of a point z = (x, c2) has polar coordinates ρ = ex and φ = c2. Consequently, as that point z moves along the entire line from left to right, its image moves outward along the entire ray φ = c2, as indicated in Fig. 20. Vertical and horizontal line segments are mapped onto portions of circles and rays, respectively, and images of various regions are readily obtained from obser- vations made in Example 1. This is illustrated in the following example. EXAMPLE 2. Let us show that the transformation w = ez maps the rect- angular region a ≤ x ≤ b, c ≤ y ≤ d onto the region ea ≤ ρ ≤ eb , c ≤ φ ≤ d. The two regions and corresponding parts of their boundaries are indicated in Fig. 21. The vertical line segment AD is mapped onto the arc ρ = ea , c ≤ φ ≤ d, which is labeled AD. The images of vertical line segments to the right of AD and join- ing the horizontal parts of the boundary are larger arcs; eventually, the image of the line segment BC is the arc ρ = eb , c ≤ φ ≤ d, labeled B C . The mapping is one to one if d − c 2π. In particular, if c = 0 and d = π, then 0 ≤ φ ≤ π; and the rectangular region is mapped onto half of a circular ring, as shown in Fig. 8, Appendix 2. x b a B B′ A A′ C C′ D D′ O y d c u O v FIGURE 21 w = exp z.
- 57. 44 Analytic Functions chap. 2 Our final example here uses the images of horizontal lines to find the image of a horizontal strip. EXAMPLE 3. When w = ez , the image of the infinite strip 0 ≤ y ≤ π is the upper half v ≥ 0 of the w plane (Fig. 22). This is seen by recalling from Example 1 how a horizontal line y = c is transformed into a ray φ = c from the origin. As the real number c increases from c = 0 to c = π, the y intercepts of the lines increase from 0 to π and the angles of inclination of the rays increase from φ = 0 to φ = π. This mapping is also shown in Fig. 6 of Appendix 2, where corresponding points on the boundaries of the two regions are indicated. x ci i O y u O v FIGURE 22 w = exp z. EXERCISES 1. By referring to Example 1 in Sec. 13, find a domain in the z plane whose image under the transformation w = z2 is the square domain in the w plane bounded by the lines u = 1, u = 2, v = 1, and v = 2. (See Fig. 2, Appendix 2.) 2. Find and sketch, showing corresponding orientations, the images of the hyperbolas x2 − y2 = c1 (c1 0) and 2xy = c2 (c2 0) under the transformation w = z2 . 3. Sketch the region onto which the sector r ≤ 1, 0 ≤ θ ≤ π/4 is mapped by the trans- formation (a) w = z2; (b) w = z3; (c) w = z4. 4. Show that the lines ay = x (a = 0) are mapped onto the spirals ρ = exp(aφ) under the transformation w = exp z, where w = ρ exp(iφ). 5. By considering the images of horizontal line segments, verify that the image of the rectangular region a ≤ x ≤ b, c ≤ y ≤ d under the transformation w = exp z is the region ea ≤ ρ ≤ eb, c ≤ φ ≤ d, as shown in Fig. 21 (Sec. 14). 6. Verify the mapping of the region and boundary shown in Fig. 7 of Appendix 2, where the transformation is w = exp z. 7. Find the image of the semi-infinite strip x ≥ 0, 0 ≤ y ≤ π under the transformation w = exp z, and label corresponding portions of the boundaries.
- 58. sec. 15 Limits 45 8. One interpretation of a function w = f (z) = u(x, y) + iv(x, y) is that of a vector field in the domain of definition of f . The function assigns a vector w, with components u(x, y) and v(x, y), to each point z at which it is defined. Indicate graphically the vector fields represented by (a) w = iz; (b) w = z/|z|. 15. LIMITS Let a function f be defined at all points z in some deleted neighborhood (Sec. l1) of z0. The statement that the limit of f (z) as z approaches z0 is a number w0, or that lim z→z0 f (z) = w0, (1) means that the point w = f (z) can be made arbitrarily close to w0 if we choose the point z close enough to z0 but distinct from it. We now express the definition of limit in a precise and usable form. Statement (1) means that for each positive number ε, there is a positive number δ such that |f (z) − w0| ε whenever 0 |z − z0| δ. (2) Geometrically, this definition says that for each ε neighborhood |w − w0| ε of w0, there is a deleted δ neighborhood 0 |z − z0| δ of z0 such that every point z in it has an image w lying in the ε neighborhood (Fig. 23). Note that even though all points in the deleted neighborhood 0 |z − z0| δ are to be considered, their images need not fill up the entire neighborhood |w − w0| ε. If f has the constant value w0, for instance, the image of z is always the center of that neighborhood. Note, too, that once a δ has been found, it can be replaced by any smaller positive number, such as δ/2. x O y u O v z0 w0 z w ε FIGURE 23 It is easy to show that when a limit of a function f (z) exists at a point z0, it is unique. To do this, we suppose that lim z→z0 f (z) = w0 and lim z→z0 f (z) = w1. Then, for each positive number ε, there are positive numbers δ0 and δ1 such that |f (z) − w0| ε whenever 0 |z − z0| δ0
- 59. 46 Analytic Functions chap. 2 and |f (z) − w1| ε whenever 0 |z − z0| δ1. So if 0 |z − z0| δ, where δ is any positive number that is smaller than δ0 and δ1, we find that |w1 − w0| = |[f (z) − w0] − [f (z) − w1]| ≤ |f (z) − w0| + |f (z) − w1| ε + ε = 2ε. But |w1 − w0| is a nonnegative constant, and ε can be chosen arbitrarily small. Hence w1 − w0 = 0, or w1 = w0. Definition (2) requires that f be defined at all points in some deleted neigh- borhood of z0. Such a deleted neighborhood, of course, always exists when z0 is an interior point of a region on which f is defined. We can extend the definition of limit to the case in which z0 is a boundary point of the region by agreeing that the first of inequalities (2) need be satisfied by only those points z that lie in both the region and the deleted neighborhood. EXAMPLE 1. Let us show that if f (z) = iz/2 in the open disk |z| 1, then lim z→1 f (z) = i 2 , (3) the point 1 being on the boundary of the domain of definition of f . Observe that when z is in the disk |z| 1, f (z) − i 2 = iz 2 − i 2 = |z − 1| 2 . Hence, for any such z and each positive number ε (see Fig. 24), f (z) − i 2 ε whenever 0 |z − 1| 2ε. Thus condition (2) is satisfied by points in the region |z| 1 when δ is equal to 2ε or any smaller positive number. x O 1 y u O v – 2 i z = 2 δ ε ε f(z) FIGURE 24
- 60. sec. 15 Limits 47 If limit (1) exists, the symbol z → z0 implies that z is allowed to approach z0 in an arbitrary manner, not just from some particular direction. The next example emphasizes this. EXAMPLE 2. If f (z) = z z , (4) the limit lim z→0 f (z) (5) does not exist. For, if it did exist, it could be found by letting the point z = (x, y) approach the origin in any manner. But when z = (x, 0) is a nonzero point on the real axis (Fig. 25), f (z) = x + i0 x − i0 = 1; and when z = (0, y) is a nonzero point on the imaginary axis, f (z) = 0 + iy 0 − iy = −1. Thus, by letting z approach the origin along the real axis, we would find that the desired limit is 1. An approach along the imaginary axis would, on the other hand, yield the limit −1. Since a limit is unique, we must conclude that limit (5) does not exist. x z = (x, 0) z = (0, y) (0, 0) y FIGURE 25 While definition (2) provides a means of testing whether a given point w0 is a limit, it does not directly provide a method for determining that limit. Theorems on limits, presented in the next section, will enable us to actually find many limits.
- 61. 48 Analytic Functions chap. 2 16. THEOREMS ON LIMITS We can expedite our treatment of limits by establishing a connection between limits of functions of a complex variable and limits of real-valued functions of two real variables. Since limits of the latter type are studied in calculus, we use their definition and properties freely. Theorem 1. Suppose that f (z) = u(x, y) + iv(x, y) (z = x + iy) and z0 = x0 + iy0, w0 = u0 + iv0. Then lim z→z0 f (z) = w0 (1) if and only if lim (x,y)→(x0,y0) u(x, y) = u0 and lim (x,y)→(x0,y0) v(x, y) = v0. (2) To prove the theorem, we first assume that limits (2) hold and obtain limit (1). Limits (2) tell us that for each positive number ε, there exist positive numbers δ1 and δ2 such that |u − u0| ε 2 whenever 0 (x − x0)2 + (y − y0)2 δ1 (3) and |v − v0| ε 2 whenever 0 (x − x0)2 + (y − y0)2 δ2. (4) Let δ be any positive number smaller than δ1 and δ2. Since |(u + iv) − (u0 + iv0)| = |(u − u0) + i(v − v0)| ≤ |u − u0| + |v − v0| and (x − x0)2 + (y − y0)2 = |(x − x0) + i(y − y0)| = |(x + iy) − (x0 + iy0)|, it follows from statements (3) and (4) that |(u + iv) − (u0 + iv0)| ε 2 + ε 2 = ε whenever 0 |(x + iy) − (x0 + iy0)| δ. That is, limit (1) holds.
- 62. sec. 16 Theorems on Limits 49 Let us now start with the assumption that limit (1) holds. With that assump- tion, we know that for each positive number ε, there is a positive number δ such that |(u + iv) − (u0 + iv0)| ε (5) whenever 0 |(x + iy) − (x0 + iy0)| δ. (6) But |u − u0| ≤ |(u − u0) + i(v − v0)| = |(u + iv) − (u0 + iv0)|, |v − v0| ≤ |(u − u0) + i(v − v0)| = |(u + iv) − (u0 + iv0)|, and |(x + iy) − (x0 + iy0)| = |(x − x0) + i(y − y0)| = (x − x0)2 + (y − y0)2. Hence it follows from inequalities (5) and (6) that |u − u0| ε and |v − v0| ε whenever 0 (x − x0)2 + (y − y0)2 δ. This establishes limits (2), and the proof of the theorem is complete. Theorem 2. Suppose that lim z→z0 f (z) = w0 and lim z→z0 F(z) = W0. (7) Then lim z→z0 [f (z) + F(z)] = w0 + W0, (8) lim z→z0 [f (z)F(z)] = w0W0 ; (9) and, if W0 = 0 , lim z→z0 f (z) F(z) = w0 W0 . (10) This important theorem can be proved directly by using the definition of the limit of a function of a complex variable. But, with the aid of Theorem 1, it follows almost immediately from theorems on limits of real-valued functions of two real variables.
- 63. 50 Analytic Functions chap. 2 To verify property (9), for example, we write f (z) = u(x, y) + iv(x, y), F(z) = U(x, y) + iV (x, y), z0 = x0 + iy0, w0 = u0 + iv0, W0 = U0 + iV0. Then, according to hypotheses (7) and Theorem 1, the limits as (x, y) approaches (x0, y0) of the functions u, v, U, and V exist and have the values u0, v0, U0, and V0, respectively. So the real and imaginary components of the product f (z)F(z) = (uU − vV ) + i(vU + uV ) have the limits u0U0 − v0V0 and v0U0 + u0V0, respectively, as (x, y) approaches (x0, y0). Hence, by Theorem 1 again, f (z)F(z) has the limit (u0U0 − v0V0) + i(v0U0 + u0V0) as z approaches z0 ; and this is equal to w0W0. Property (9) is thus established. Corresponding verifications of properties (8) and (10) can be given. It is easy to see from definition (2), Sec. 15, of limit that lim z→z0 c = c and lim z→z0 z = z0, where z0 and c are any complex numbers; and, by property (9) and mathematical induction, it follows that lim z→z0 zn = zn 0 (n = 1, 2, . . .). So, in view of properties (8) and (9), the limit of a polynomial P (z) = a0 + a1z + a2z2 + · · · + anzn as z approaches a point z0 is the value of the polynomial at that point: lim z→z0 P (z) = P (z0). (11) 17. LIMITS INVOLVING THE POINT AT INFINITY It is sometimes convenient to include with the complex plane the point at infinity, denoted by ∞, and to use limits involving it. The complex plane together with this point is called the extended complex plane. To visualize the point at infinity, one can think of the complex plane as passing through the equator of a unit sphere centered at the origin (Fig. 26). To each point z in the plane there corresponds exactly one point P on the surface of the sphere. The point P is the point where the line through z and the north pole N intersects the sphere. In like manner, to each point P on the surface of the sphere, other than the north pole N, there corresponds exactly one
- 64. sec. 17 Limits Involving the Point at Infinity 51 P O N z FIGURE 26 point z in the plane. By letting the point N of the sphere correspond to the point at infinity, we obtain a one to one correspondence between the points of the sphere and the points of the extended complex plane. The sphere is known as the Riemann sphere, and the correspondence is called a stereographic projection. Observe that the exterior of the unit circle centered at the origin in the complex plane corresponds to the upper hemisphere with the equator and the point N deleted. Moreover, for each small positive number ε, those points in the complex plane exterior to the circle |z| = 1/ε correspond to points on the sphere close to N. We thus call the set |z| 1/ε an ε neighborhood, or neighborhood, of ∞. Let us agree that in referring to a point z, we mean a point in the finite plane. Hereafter, when the point at infinity is to be considered, it will be specifically mentioned. A meaning is now readily given to the statement lim z→z0 f (z) = w0 when either z0 or w0, or possibly each of these numbers, is replaced by the point at infinity. In the definition of limit in Sec. 15, we simply replace the appropriate neighborhoods of z0 and w0 by neighborhoods of ∞. The proof of the following theorem illustrates how this is done. Theorem. If z0 and w0 are points in the z and w planes, respectively, then lim z→z0 f (z) = ∞ if and only if lim z→z0 1 f (z) = 0 (1) and lim z→∞ f (z) = w0 if and only if lim z→0 f 1 z = w0. (2) Moreover, lim z→∞ f (z) = ∞ if and only if lim z→0 1 f (1/z) = 0. (3)
- 65. 52 Analytic Functions chap. 2 We start the proof by noting that the first of limits (1) means that for each positive number ε, there is a positive number δ such that |f (z)| 1 ε whenever 0 |z − z0| δ. (4) That is, the point w = f (z) lies in the ε neighborhood |w| 1/ε of ∞ whenever z lies in the deleted neighborhood 0 |z − z0| δ of z0. Since statement (4) can be written 1 f (z) − 0 ε whenever 0 |z − z0| δ, the second of limits (1) follows. The first of limits (2) means that for each positive number ε, a positive number δ exists such that |f (z) − w0| ε whenever |z| 1 δ . (5) Replacing z by 1/z in statement (5) and then writing the result as f 1 z − w0 ε whenever 0 |z − 0| δ, we arrive at the second of limits (2). Finally, the first of limits (3) is to be interpreted as saying that for each positive number ε, there is a positive number δ such that |f (z)| 1 ε whenever |z| 1 δ . (6) When z is replaced by 1/z, this statement can be put in the form 1 f (1/z) − 0 ε whenever 0 |z − 0| δ; and this gives us the second of limits (3). EXAMPLES. Observe that lim z→−1 iz + 3 z + 1 = ∞ since lim z→−1 z + 1 iz + 3 = 0 and lim z→∞ 2z + i z + 1 = 2 since lim z→0 (2/z) + i (1/z) + 1 = lim z→0 2 + iz 1 + z = 2. Furthermore, lim z→∞ 2z3 − 1 z2 + 1 = ∞ since lim z→0 (1/z2 ) + 1 (2/z3) − 1 = lim z→0 z + z3 2 − z3 = 0.
- 66. sec. 18 Continuity 53 18. CONTINUITY A function f is continuous at a point z0 if all three of the following conditions are satisfied: lim z→z0 f (z) exists, (1) f (z0) exists, (2) lim z→z0 f (z) = f (z0). (3) Observe that statement (3) actually contains statements (1) and (2), since the exis- tence of the quantity on each side of the equation there is needed. Statement (3) says, of course, that for each positive number ε, there is a positive number δ such that |f (z) − f (z0)| ε whenever |z − z0| δ. (4) A function of a complex variable is said to be continuous in a region R if it is continuous at each point in R. If two functions are continuous at a point, their sum and product are also contin- uous at that point; their quotient is continuous at any such point if the denominator is not zero there. These observations are direct consequences of Theorem 2, Sec. 16. Note, too, that a polynomial is continuous in the entire plane because of limit (11) in Sec. 16. We turn now to two expected properties of continuous functions whose veri- fications are not so immediate. Our proofs depend on definition (4) of continuity, and we present the results as theorems. Theorem 1. A composition of continuous functions is itself continuous. A precise statement of this theorem is contained in the proof to follow. We let w = f (z) be a function that is defined for all z in a neighborhood |z − z0| δ of a point z0 , and we let W = g(w) be a function whose domain of definition contains the image (Sec. 13) of that neighborhood under f . The composition W = g[f (z)] is, then, defined for all z in the neighborhood |z − z0| δ. Suppose now that f is continuous at z0 and that g is continuous at the point f (z0) in the w plane. In view of the continuity of g at f (z0), there is, for each positive number ε, a positive number γ such that |g[f (z)] − g[f (z0)]| ε whenever |f (z) − f (z0)| γ. (See Fig. 27.) But the continuity of f at z0 ensures that the neighborhood |z − z0| δ can be made small enough that the second of these inequalities holds. The continuity of the composition g[f (z)] is, therefore, established.
- 67. 54 Analytic Functions chap. 2 u x O y z0 v z O f(z0) f(z) U V O g[f(z0)] g[f(z)] ε FIGURE 27 Theorem 2. If a function f (z) is continuous and nonzero at a point z0 , then f (z) = 0 throughout some neighborhood of that point. Assuming that f (z) is, in fact, continuous and nonzero at z0, we can prove Theorem 2 by assigning the positive value |f (z0)|/2 to the number ε in statement (4). This tells us that there is a positive number δ such that |f (z) − f (z0)| |f (z0)| 2 whenever |z − z0| δ. So if there is a point z in the neighborhood |z − z0| δ at which f (z) = 0, we have the contradiction |f (z0)| |f (z0)| 2 ; and the theorem is proved. The continuity of a function f (z) = u(x, y) + iv(x, y) (5) is closely related to the continuity of its component functions u(x, y) and v(x, y). We note, for instance, how it follows from Theorem 1 in Sec. 16 that the function (5) is continuous at a point z0 = (x0, y0) if and only if its component functions are continuous there. Our proof of the next theorem illustrates the use of this state- ment. The theorem is extremely important and will be used often in later chapters, especially in applications. Before stating the theorem, we recall from Sec. 11 that a region R is closed if it contains all of its boundary points and that it is bounded if it lies inside some circle centered at the origin. Theorem 3. If a function f is continuous throughout a region R that is both closed and bounded, there exists a nonnegative real number M such that |f (z)| ≤ M for all points z inR, (6) where equality holds for at least one such z.
- 68. sec. 18 Exercises 55 To prove this, we assume that the function f in equation (5) is continuous and note how it follows that the function [u(x, y)]2 + [v(x, y)]2 is continuous throughout R and thus reaches a maximum value M somewhere in R.∗ Inequality (6) thus holds, and we say that f is bounded on R. EXERCISES 1. Use definition (2), Sec. 15, of limit to prove that (a) lim z→z0 Re z = Re z0 ; (b) lim z→z0 z = z0 ; (c) lim z→0 z2 z = 0. 2. Let a, b, and c denote complex constants. Then use definition (2), Sec. 15, of limit to show that (a) lim z→z0 (az + b) = az0 + b; (b) lim z→z0 (z2 + c) = z2 0 + c; (c) lim z→1−i [x + i(2x + y)] = 1 + i (z = x + iy). 3. Let n be a positive integer and let P (z) and Q(z) be polynomials, where Q(z0) = 0. Use Theorem 2 in Sec. 16, as well as limits appearing in that section, to find (a) lim z→z0 1 zn (z0 = 0); (b) lim z→i iz3 − 1 z + i ; (c) lim z→z0 P (z) Q(z) . Ans. (a) 1/zn 0; (b) 0; (c) P (z0)/Q(z0). 4. Use mathematical induction and property (9), Sec. 16, of limits to show that lim z→z0 zn = zn 0 when n is a positive integer (n = 1, 2, . . .). 5. Show that the limit of the function f (z) = z z 2 as z tends to 0 does not exist. Do this by letting nonzero points z = (x, 0) and z = (x, x) approach the origin. [Note that it is not sufficient to simply consider points z = (x, 0) and z = (0, y), as it was in Example 2, Sec. 15.] 6. Prove statement (8) in Theorem 2 of Sec. 16 using (a) Theorem 1 in Sec. 16 and properties of limits of real-valued functions of two real variables; (b) definition (2), Sec. 15, of limit. ∗See, for instance, A. E. Taylor and W. R. Mann, “Advanced Calculus,” 3d ed., pp. 125–126 and p. 529, 1983.
- 69. 56 Analytic Functions chap. 2 7. Use definition (2), Sec. 15, of limit to prove that if lim z→z0 f (z) = w0, then lim z→z0 |f (z)| = |w0|. Suggestion: Observe how the first of inequalities (9), Sec. 4, enables one to write ||f (z)| − |w0|| ≤ |f (z) − w0|. 8. Write z = z − z0 and show that lim z→z0 f (z) = w0 if and only if lim z→0 f (z0 + z) = w0. 9. Show that lim z→z0 f (z)g(z) = 0 if lim z→z0 f (z) = 0 and if there exists a positive number M such that |g(z)| ≤ M for all z in some neighborhood of z0. 10. Use the theorem in Sec. 17 to show that (a) lim z→∞ 4z2 (z − 1)2 = 4; (b) lim z→1 1 (z − 1)3 = ∞; (c) lim z→∞ z2 + 1 z − 1 = ∞. 11. With the aid of the theorem in Sec. 17, show that when T (z) = az + b cz + d (ad − bc = 0), (a) lim z→∞ T (z) = ∞ if c = 0; (b) lim z→∞ T (z) = a c and lim z→−d/c T (z) = ∞ if c = 0. 12. State why limits involving the point at infinity are unique. 13. Show that a set S is unbounded (Sec. 11) if and only if every neighborhood of the point at infinity contains at least one point in S. 19. DERIVATIVES Let f be a function whose domain of definition contains a neighborhood |z − z0| ε of a point z0. The derivative of f at z0 is the limit f (z0) = lim z→z0 f (z) − f (z0) z − z0 , (1) and the function f is said to be differentiable at z0 when f (z0) exists. By expressing the variable z in definition (1) in terms of the new complex variable z = z − z0 (z = z0),
- 70. sec. 19 Derivatives 57 one can write that definition as f (z0) = lim z→0 f (z0 + z) − f (z0) z . (2) Because f is defined throughout a neighborhood of z0, the number f (z0 + z) is always defined for | z| sufficiently small (Fig. 28). x O y z0 ε FIGURE 28 When taking form (2) of the definition of derivative, we often drop the subscript on z0 and introduce the number w = f (z + z) − f (z), which denotes the change in the value w = f (z) of f corresponding to a change z in the point at which f is evaluated. Then, if we write dw/dz for f (z), equation (2) becomes dw dz = lim z→0 w z . (3) EXAMPLE 1. Suppose that f (z) = z2 . At any point z, lim z→0 w z = lim z→0 (z + z)2 − z2 z = lim z→0 (2z + z) = 2z since 2z + z is a polynomial in z. Hence dw/dz = 2z, or f (z) = 2z. EXAMPLE 2. If f (z) = z, then w z = z + z − z z = z + z − z z = z z . (4)