Preface ..... xiii
Symbols and Terms ..... xix

1. PRELIMINARIES ..... 1
   1. Preview ..... 1
      1. It Takes Two Harmonic Functions ..... 3
      2. Heat Flow ..... 6
      3. A Geometric Rule ..... 9
      4. Electrostatics ..... 10
      5. Fluid Flow ..... 13
      6. One Model, Many Applications ..... 14
         
         Exercises ..... 15
   2. Sets, Functions, and Visualization ..... 18
      1. Terminology and Notation for Sets ..... 18
      2. Terminology and Notation for Functions ..... 20
      3. Functions from R to R ..... 25
      4. Functions from the Plane to R ..... 27
      5. Functions from the Plane to the Plane ..... 29
         
         Exercises ..... 30
   3. Structures and Linear Maps on the Plane ..... 34
      1. The Real Line and the Plane ..... 34
      2. Polar Coordinates in the Plane ..... 36
      3. When is a Mapping from the Plane to the Plane
         Linear? ..... 38
      4. Visualizing Nonsingular Linear Mappings ..... 40
      5. The Determinant of a Two-by-Two Matrix ..... 44
      6. Pure Magnifications, Rotations, and Conjugation ..... 45
      7. Conformal Linear Mappings ..... 46
         
         Exercises ..... 48
   4. Open Sets, Open Mappings, Connected Sets ..... 51
      1. Distance, Interior, Boundary, Openness ..... 51
      2. Continuity in Terms of Open Sets ..... 55
      3. Open Mappings ..... 56
      4. Connected Sets ..... 57
         
         Exercises ..... 58
   5. A Review of Some Calculus ..... 61
      1. Integration Theory for Real-Valued Functions ..... 61
      2. Improper Integrals, Principal Values ..... 63
      3. Partial Derivatives ..... 66
      4. Divergence and Curl ..... 68
         
         Exercises ..... 70
   6. Harmonic Functions ..... 71
      1. The Geometry of Laplace's Equation
      2. The Geometry of the Cauchy-Riemann
         Equations ..... 72
      3. The Mean Value Property ..... 73
      4. Changing Variables in a Dirichlet or Neumann
         Problem ..... 76
         
         Exercises ..... 77
2. BASIC TOOLS ..... 83
   1. The Complex Plane ..... 83
      1. The Definition of a Field ..... 83
      2. Complex Multiplication ..... 84
      3. Powers and Roots ..... 87
      4. Conjugation ..... 89
      5. Quotients of Complex Numbers 90
      6. When is a Mapping from C to C Linear? ..... 91
      7. Complex Equations for Lines and Circles ..... 92
      8. The Reciprocal Map, and Reflection in the
         Unit Circle ..... 93
      9. Reflections in Lines and Circles ..... 96
         
         Exercises ..... 97
   2. Visualizing Powers, Exponential, Logarithm, and Since ..... 102
      1. Powers of z ..... 103
      2. Exponential and Logarithms ..... 104
      3. Sin z ..... 106
      4. The Cosine and Sine, and the Hyperbolic
         Cosine and Sine ..... 110
         
         Exercises ..... 97
   3. Differentiability ..... 115
      1. Differentiability at a Point .....115
      2. Differentiability in the Complex Sense:
         Holomorphy ..... 119
      3. Finding Derivatives ..... 122
      4. Picturing the Local Behavior of Holomorphic Mappings ..... 124
         
         Exercises ..... 126
   4. Sequences, Compactness, Convergence ..... 128
      1. Sequences of Complex Numbers ..... 128
      2. The Limit Superior of a Sequence of Reals ..... 131
      3. Implications of Compactness ..... 133
      4. Sequences of Functions ..... 134
         
         Exercises ..... 135
   5. Integrals Over Curves, Paths, and Contours ..... 138
      1. Integrals of Complex-Valued Functions ..... 138
      2. Curves ..... 138
      3. Paths ..... 144
      4. Pathwise Connected Sets ..... 147
      5. Independence of Path and Morera's Theorem ..... 148
      6. Goursat's Lemma ..... 150
      7. The Winding Number ..... 153
      8. Green's Theorem ..... 155
      9. Irrotational and Incompressible Fluid Flow ..... 158
      10. Contours ..... 161
          
          Exercises ..... 162
   6. Power Series ..... 166
      1. Infinite Series ..... 166
      2. The Geometric Series ..... 167
      3. An Improved Root Test ..... 171
      4. Power Series and the Cauchy-Hadamard
         Theorem ..... 172
      5. Uniqueness of the Power Series Representation ..... 174
      6. Integrals That Give Rise to Power Series ..... 178
         
         Exercises ..... 180
3. THE CAUCHY THEORY ..... 187
   1. Fundamental Properties of Holomorphic Functions ..... 188
      1. Integral and Series Representations ..... 188
      2. Eight Ways to Say "Holomorphic" ..... 193
      3. Determinism ..... 193
      4. Liouville's Theorem ..... 196
      5. The Fundamental Theorem of Algebra ..... 196
      6. Subuniform Convergence Preserves
         Holomorphy ..... 197
         
         Exercises ..... 198
   2. Cauchy's Theorem ..... 204
      1. Cerny's 1976 Proof ..... 205
      2. Simply Connected Sets ..... 208
      3. Subuniform Boundedness, Subuniform Convergence ..... 209
   3. Isolated Singularities ..... 212
      1. The Laurent Series Representation on an
         Annulus ..... 212
      2. Benavior Near an Isolated Singularity in the
         Plane ..... 216
      3. Examples: Classifying Singularities, Finding
         Residues ..... 219
      4. Behavior Near a Singularity at Infinity ..... 225
      5. A Digression: Picard's Great Theorem ..... 229
         
         Exercises ..... 229
   4. The Residue Theorem and the Argument Principle ..... 236
      1. Meromorphic Functions and the Extended Plane .....236
      2. The Residue Theorem ..... 239
      3. Multiplicity and Valence ..... 242
      4. Valence for a Rational Function ..... 243
      5. The Argument Principle: Integrals That Count ..... 243
         
         Exercises ..... 249
   5. Mapping Properties ..... 251
      
      Exercises ..... 259
   6. The Riemann Sphere ..... 260
      
      Exercises ..... 264
4. THE RESIDUE CALCULUS ..... 167
   1. Integrals of Trigonometric Functions ..... 268
      
      Exercises ..... 270
   2. Estimating Complex Integrals ..... 273
      
      Exercises ..... 276
   3. Integrals of Rational Functions Over the Line ..... 277
      
      Exercises ..... 280
   4. Integrals Involving the Exponential ..... 282
      1. Integrals Giving Fourier Transforms ..... 286
         
         Exercises ..... 290
   5. Integrals Involving a Logarithm ..... 293
      
      Exercises ..... 301
   6. Integration on a Riemann Surface ..... 302
      1. Mellin Transforms ..... 306
         
         Exercises ..... 307
   7. The Inverse Laplace Transform ..... 309
      
      Exercises ..... 315
5. BOUNDARY VALUE PROBLEMS ..... 317
   1. Examples ..... 318
      1. Easy Problems ..... 318
      2. The Conformal Mapping Method ..... 323
         
         Exercises ..... 326
   2. The Mobius Maps ..... 317
      
      Exercises ..... 338
   3. Electric Fields ..... 341
      1. A Point Charge in 3-Space ..... 341
      2. Uniform Charge on One or More Long Wires ..... 342
      3. Examples with Bounded Potentials ..... 347
         
         Exercises ..... 350
   4. Steady Flow of a Perfect Fluid ..... 350
      
      Exercises ..... 354
   5. Using the Poisson Integral to Obtain Solutions ..... 355
      1. The Poisson Integral on a Disk ..... 355
      2. Solutions on the Disk by the Poisson Integral ..... 358
      3. Geometry of the Poisson Integral ..... 358
      4. Harmonic Functions and the Mean Value
         Property ..... 363
      5. The Neumann Problem on a Disk ..... 364
      6. The Poisson Integral on a Half-Plane, and on
         Other Domains ..... 365
         
         Exercises ..... 366
   6. When is the Solution Unique? ..... 368
      
      Exercises ..... 370
   7. The Schwarz Reflection Principle ..... 370
   8. Schwarz-Christoffel Formulas ..... 374
      1. Triangles ..... 375
      2. Rectangles and Other Polygons ..... 385
      3. Generalized Polygons ..... 389
         
         Exercises ..... 390
6. LAGNIAPPE ..... 393
   1. Dixon's 1971 Proof of Cauchy's Theorem ..... 394
   2. Runge's Theorem ..... 398
      
      Exercises ..... 403
   3. The Riemann Mapping Theorem ..... 404
      
      Exercises
   4. The Osgood-Taylor-Caratheodory Theorem ..... 406

References ..... 413
Index ..... 419

To Students, About Prerequisites

This text is primarily for use in a one-quarter or one-semester undergraduate course. Such a course should cover a reasonable amount of material chosen from the first five chapters to fit the backgrounds and interests of the students. The prerequisites consist of the standard calculus sequence, including the differentiation and integration of functions of two or more real variables.

In teaching Mathematics 4036 here at Louisiana State University, I've found that I need to review and restate for my students some of the necessary ideas from the calculus while setting the stage for the problems and methods of complex analysis. Chapter 1 is devoted to such preparation. I offer the following remarks about specific background material that you need.

• From the very beginning, Chapter 1 assumes a certain familiarity with partial derivatives for real-valued functions of two real variables. Section 1.5 states and reviews the most advanced results concerning partial derivatives that you need to understand. In Section 1.6 the multidimensional Chain Rule is used twice in the text and in several of the problems.
• In Section 1.2, I present notation and terminology for sets and functions. The examples presented make use of certain real-valued functions of a real variable, which ought to be familiar from your previous studies: polynomials, exponential, logarithm, sine, cosine, hyperbolic cosine, and hyperbolic sine.
• In Section 1.3, I develop from scratch the modest amount of linear algebra that you need. It will be helpful if you have previously dealt with a system of two linear equations in two unknowns, and understand how a two-by-two matrix determines a linear transformation of the plane.
• I assume that you are familiar with the Riemann integral as taught in first-year calculus. Section 1.5 summarizes the integration theory that you need. It will be helpful if you have had some experience with integrals over curves and with integrals over two- and three-dimensional domains.
• Sections 2.4 and 2.6 present everything you need about complex-valued sequences and series. You may find this material easier if you take some time to review your previous studies of sequences, infinite series, and Taylor series.

From time to time, you may need to review or look up some topic you're not familiar with, even beyond the fairly complete list above. I have tried to be careful, when using prerequisite material, to make clear what I am assuming and how I am applying it.

This book uses some concepts which occur also in upper-division advanced calculus or real-variables courses, but I do not assume you have had those courses. Such topics as open sets, compact sets, and uniform convergence are important for making complex analysis simple and elegant. I develop those topics as needed.

All veterans of the calculus sequence have no doubt acquired a quite adequate mental concept of the real number system R. It is good to be aware of the completeness property of R, and to recognize it when it is used. I will state and explain this version of it in Section 2.4: Every nonempty set of real numbers which is bounded above has a least upper bound.

I have in mind an audience of students like the ones to whom I've taught the course here at LSU. Most have completed the calculus sequence at some time in the past. Most have taken other mathematics courses as well, such as differential equations. About half of them are majoring in some field of engineering or natural science; they take complex analysis as an elective, or perhaps as part of a minor or double major. They bring to the course an awareness of certain physical and computational problems, and they want this course to provide them some new and useful ideas and techniques. Some have a well-formed curiosity about, for example, the integration methods that come from the Residue Theorem, and the conformal mapping method for solving Dirichlet problems.

At the very beginning of the book, I try to respond to these students' curiosity and motivation by describing some of the results we'll obtain, and some of the physical problems we'll be able to solve. I present the physics briefly, just enough to explain how the mathematical models may be interpreted in physical terms.

The audience also includes mathematics majors whose priority is to master the theory and prepare for more advanced mathematics. But the physics is for their benefit too. The applications serve to illuminate the theory, provide sources of intuition, create historical awareness, and make a connection with other knowledge.
 

To All Readers, About the Book

I've thought about this project, the writing of this book, from time to time since the late 1960s when I first taught complex analysis at the University of California at Berkeley. Having taught the course often, I've encountered a number of good books, old and new, which are serviceable as undergraduate texts. Nevertheless, I have felt moved to write a new one. Perhaps teachers selecting a text will be glad to have one more choice. In recent decades, there have been developments in what we know, and in how we understand the subject, which need to be incorporated in a text. If I've done so successfully, then perhaps this book will have a good influence on the teaching of the subject. Beyond what I've already said, I offer the following remarks about the book I have aimed for.

• When teaching the one-semester course here at LSU, I have covered most of Chapters 1--3, presented just a sample of the integration techniques in Chapter 4, and devoted a few weeks to a careful treatment of selected topics from Chapter 5. When the students have especially good backgrounds, Chapter 1 can be covered quickly and Chapter 2 in less time than otherwise; and then more of Chapters 3--5 can be treated. For more suggestions for planning the course and using the book, visit the web site at http://www.math.lsu.edu/~mcgehee.
• The book is also usable for a first course in complex analysis for graduate students in mathematics, especially if one objective is to acquaint them with applications, such as they may wish to teach some day. Such a course might cover Chapter 3 thoroughly and also reach the advanced topics of Chapter 6.
• The book engages the students in a review, and in the active use, of their calculus background. Chapter 1 states typical boundary value problems, and previews the methods and solutions, in terms of the real-variable calculus. In Chapter 2, the complex-valued exponential, sine, and cosine are defined and developed in terms of the familiar real-valued versions of those functions. I frequently give specific page references to material in two currently popular calculus texts. I realize that those editions will be superseded in a few years, but I wish to make the point that this book is designed to fit and to follow what the students have studied before.
• Chapter 2 introduces complex numbers, and it soon becomes clear that a useful extra structure is being added to the already-understood real setting. By thus waiting to add the complex structure until the real preliminaries have been discussed, the book avoids the complex-from-the-first-day approach that often (I think) leaves a student feeling that real and complex methods are like oil and water, and wondering how they mix.
• I define conformality separately for linear maps, and emphasize the nature of differentiability as approximability by linear maps. I think this is the best way to remove the mystery about complex versus real differentiability for functions from the plane to the plane, a mystery which (I think) sometimes haunts a student throughout the course and for years afterward.
• The book follows an efficient path through the development of the theory. I've included what I consider important for understanding, and for use. I have tried to give short proofs, of the kind whose motivating idea is clear, and in which the techniques and the reasoning will be helpful later on. I think I've largely succeeded in doing so up to Section 5.7.
• It seems to me that the modern homology (or winding-number) version of Cauchy's Theorem deserves the spotlight in an introductory course, especially a course for engineering students. It is easy to state, easy to understand, and easy to prove. It is also stunningly powerful and convenient to use. This book does not speak of homotopies or deformations.
• I emphasize visualization. I often leave a drawing for the reader to do, saying "Make a sketch!" But there are many figures, chosen to provide helpful examples and to represent important ideas to the reader's mind. Some of them will require, and I hope reward, a bit of careful attention. I drew all the figures using the graphics commands of Mathematica 3.0. All show precisely rendered actual objects, with two exceptions: Figure 5.7-3, page 374, is frankly schematic; and in Figure 5.8-1, the shaded region is an "artist's conception."

Acknowledgments

Many people have influenced this work. Many have given me help, encouragement, and inspiration. I should like to mention just a few.

There are influences from long ago. In 1961--1962, I was taught an excellent one-year graduate course at Yale University by Gustav Hedlund, using the text by Einar Hille.

Although I never met Maurice Heins, I've benefited from having a set of notes taken from the lectures in an undergraduate course that he taught in the Fall of 1963.

On a personal level, I wish to thank most warmly three persons who have been good friends to me and to this work: Kevin Scott Brown, currently a graduate student in physics at Cornell University; the mathematician Robert B. Burckel, Professor at Kansas State University; and the mathematician David W. Kammler, Professor at Southern Illinois University.

I wish also to thank the several other people, some of whom I do not know by name, who reviewed early drafts of this book and gave encouragement and helpful advice.

I am grateful for the many students in my classes whom I've had the pleasure of watching as they learned and enjoyed complex analysis.

I thank my colleagues in the Mathematics Department at LSU for their consideration and support for this undertaking. Particular thanks to Professor Neal Stoltzfus for his advice and help with the production of this book on our computer system; and also to Loc Thi Stewart, the Department's Computer Analyst, for her reliable technical assistance.

I'm of course thankful for the modern tools that make the life of a writer so much easier these days, LaTeX and Mathematica.

Finally, I thank the editors at John Wiley & Sons, who have been superb.

O. Carruth McGehee

Baton Rouge, Louisiana

July 11, 2000