Complex Variables

PREFACE Take care of the sense, and the sounds will take care of themselves. Alice in Wonderland After having lectured for several decades on complex variables to prospective engineers and physicists, I have definite and, I hope, not unrealistic ideas about their requirements and preferences. Students are students. Since they are required to take several courses, they may study some subjects just for the examination, with the intention of forgetting what they have learned after the examination. Yet they may (and the more intelligent and purposeful students do) ask pertinent questions about the subject: Is it interesting? Can I use it? These questions are fully justified. The instructor of a more advanced branch of mathematics-such as the theory of complex variables-who is a mathematician should try to put himself into the position of his students who are prospective physicists or engineers. Before going into heavy definitions and lengthy proofs, the student wants to satisfy himself that the subject is interesting and useful enough to expend his time and effort on definitions and proofs. Having realized these points as I taught successive generations of students, I adapted my lectures to their standpoint. I evolved the following guidelines. Start from something that is familiar, useful, or challenging-from some connection with the world around us, from the prospect of some application, or from an intuitive idea. Do not be afraid of using colloquial language when it is more suggestive than the conventional precise terminology. In fact, do not introduce technical terms before the student can understand the need for them. Do not enter too early or too far into the heavy details of a proof. First, give a general idea or just the intuitive germ of the proof. Generally, realize that the natural way to learn is to learn by stages. First, we want to see an outline of the subject in order to perceive a concrete source or a possible use. Then, gradually, as we can see use, connections, and interest, we accept more willingly the responsibility of filling in the details. The ideas just stated influenced the organization of this book. Whenever the mathematical context offers a natural opportunity, there are a few words inserted about concrete phenomena or connected general ideas. Before the introduction of a formal definition, the intervening ideas may be previously discussed by examples or in more colloquial language. The proofs emphasize the main points and may leave to the student, now and then, more intermediate points than usual. The most notable departure from the usual is to be found, however, in the "Examples and Comments" that follow most sections and each chapter. There are, of course, the examples of the standard kind that offer an opportunity to practice what has been explained in the text. Yet there is what is not usual-a definite effort to let the student learn the subject by stages and by his own work. Some problems or comments ask the student to reconsider the definitions and proofs given in the text, directing his attention to more subtle points. Other problems introduce new material: a proof different from the one given in the text, or generalizations of (or analogues to) the facts considered, encouraging the student toward further study. Moreover, even the simpler problems, insofar as is possible, are arranged to give the student an opportunity to face a variety of research situations that will awaken his curiosity and initiative. I hope that this book is useful not only to future engineers and physicists but also to future mathematicians. Mathematical concepts and facts gain in vividness and clarity if they are well connected with the world around us and with general ideas, and if we obtain them by our own work through successive stages instead of in one lump. The course presented here has been taught several times at Stanford by me and by my friend and colleague, Gordon E. Latta, who shares my pedagogical ideas. I am grateful to him for sharing the writing which, because of other interests and duties, I was not able to do alone. We may have achieved less than we hoped for at various points and in various respects, yet we still think that this book is a modest concrete contribution to the widespread debate about the lines along which the instruction in the universities should evolve. George Polya Stanford, August 1974