Preface Preface
After six editions of publishing with Jones and Bartlett Learning we are delighted that we can now make this text available as an open-source document, and have designed it for students in mathematics, physics, and engineering at the undergraduate level. Our goal is to illustrate the theoretical concepts and proofs with practical applications, and to present them in a style that is enjoyable for students to read. We believe both mathematicians and scientists should be exposed to a careful presentation of mathematics. Our use of the term “careful” here means paying attention to such things as ensuring required assumptions are met before using a theorem, checking that algebraic operations are valid, and confirming that formulas have not been blindly applied. We do not mean to equate care with rigor, as we present our proofs in a self-contained manner that is understandable by students having studied multivariable calculus. For example, we include Green’s theorem and use it to prove the Cauchy-Goursat theorem, although we also include the proof by Goursat. Depending on the level of rigor desired, students may look at one or the other—or both.
We aim to give sufficient applications to motivate and illustrate how complex analysis is used in applied fields. Computer graphics help show that complex analysis is a computational tool of practical value. The exercise sets offer a wide variety of choices for computational skills, theoretical understanding, and applications that were class tested for six editions of the text when it was available for purchase (i.e., prior to this free open-source edition). We provide answers to most odd-numbered problems. For those problems that require proofs, we attempt to model what a good proof should look like, often guiding students up to a point and then asking them to fill in the details.
The purpose of the first six chapters is to lay the foundation for the study of complex analysis and develop the topics of analytic and harmonic functions, the elementary functions, and contour integration. Chapters 7 and Chapter 8, dealing with residue calculus and applications, may be skipped if there is more interest in conformal mapping and applications of harmonic functions, which are the topics of Chapters 9 and Chapter 10 respectively. For courses requiring even more applications, Chapter 11 investigates Fourier and Laplace transforms.
Features: The answers to most odd-numbered exercises should help instructors as they deliberate on problem assignments, and should help students as they review material. We present conformal mapping in a visual and geometric manner so that compositions and images of curves and regions can be more easily understood. We first solve boundary value problems for harmonic functions in the upper half-plane so that we can use conformal mapping by elementary functions to obtain solutions in other domains. We carefully develop the Schwarz-Christoffel transformation and present applications. We use two-dimensional mathematical models for applications in the areas of ideal fluid flow, steady-state temperatures, and electrostatics. We accurately portray streamlines, isothermals, and equipotential curves with computer-drawn figures.
An early introduction to sequences and series appears in Chapter 4, which facilitates the definition of the exponential function via series. The section on Julia and Mandelbrot sets illustrates how complex analysis connects with contemporary topics in mathematics. Included are computer-generated illustrations such as Riemann surfaces, contour and surface graphics for harmonic functions, the Dirichlet problem, streamlines involving harmonic and analytic functions, and conformal mapping. We also have a section on the Joukowski airfoil.
Acknowledgments: A textbook does not make it to the sixth edition without the support of a long list of colleagues from various institutions. Their help has been invaluable, and we owe them much more than the brief acknowledgment we are able to provide here. Alphabetically by institution they are: Edward G. Thurber (Biola University); Robert A. Calabretta (Boeing Corporation); Vencil Skarda (Brigham Young University; Stuart Goldenberg (California Polytechnic State University, San Luis Obispo); Vuryl Klassen, Gerald Marley, and Harris Shultz (California State University, Fullerton); Michael Stob (Calvin College); Al Hibbard(Central College); Paul Martin (Colorado School of Mines); R.E. Williamson (Dartmouth College); William Trench (Drexel University); Arlo Davis (Indiana University of Pennsylvania); Elgin H. Johnston (Iowa State University); Richard A. Alo (Lamar University); Martin Bazant (Massachusetts Institute of Technology); Carroll O. Wilde (Naval Postgraduate School); Holland Filgo (Northeastern University); E. Melvin J. Jacobsen (Rensselaer Polytechnic Institute); Christine Black (Seattle University); Geoffrey Prince and John Trienz (United States Naval Academy); William Yslas Velez (University of Arizona); Charles P. Luehr (University of Florida); Robert D. Brown and T.E. Duncan (University of Kansas); Donald Hadwin (University of New Hampshire); Calvin Wilcox (University of Utah); Robert Heal (Utah State University); and C. Ray Rosentrater (Westmont College).
We also wish to thank the students of California Baptist University, Cal State Fullerton, the United States Air Force Academy, University of Maryland, and Westmont College for their helpful suggestions and words of encouragement. In production matters we are indebted to David Farmer of the American Institute of Mathematics. He generously gave an enormous amount of time in helping the PreTeXt version of this text come to fruition. Finally, we thank in advance those of you who will make suggestions for improvements to the text as it now stands, whether that be the PreTeXt or PDF version. We welcome correspondence via surface or e-mail.
