In American universities a course covering roughly the material in this book is ordinarily given in the first graduate year. The way of presenting the material differs widely: in some schools the emphasis is on teaching a certain indispensable amount of classical function theory; in others the course is used to confront the student, for the first time, with mathematical rigor. In Harvard, for instance, the course is also traditionally used to review advanced calculus with complete rigour in view.
The author's ambition has been to write a text which is at once concise and rigorous, teachable and readable. Such a goal cannot be reached, it can only be approximated, and the author is aware of many shortcomings. No attempt has been made to make the book self-contained. On the contrary, a basic knowledge of real numbers and calculus, including the definition and properties of definite integrals, is taken for granted. Questions concerning limits and continuity are reviewed in connection with their application to complex numbers, and an effort is made not to rely on results which in elementary teaching are commonly derived in a sloppy or insufficient manner. If the teacher decides that real numbers or the definition of integral should be included in his course, there are a dozen or so reliable texts that he can consult. The author has omitted these topics mainly to keep down the bulk of this volume.
Even apart from the starting point, the writer of a textbook has a difficult task in deciding what to include and what to omit. The present author has wished to give the reader a solid foundation in classical complex-function theory by emphasizing the general principles on which it rests. He believes that a person who is thoroughly inculcated with the fundamental methods will not experience any new difficulty if he wishes to go on to a specialised topic in function theory. Nevertheless, it is with great regret that the author has omitted, for instance, the theory of elliptic functions. One of the main reasons is that it is hardly possible to improve on the beautiful treatment in E T Copson's book ("An Introduction to the Theory of Functions of a Complex Variable," London, 1935).
In the opposite direction some topics have been included which are usually not felt to be part of elementary function theory. Such is the case with the theory of subharmonic function and Perron's method for solving the Dirichlet problem, which are certainly as elementary as they are important.
The book begins with an elementary discussion of complex numbers and ends up on a note of sophistication with the theory of multiple-valued analytic functions. In between, the progress is gradual. From his venerated teacher, Ernst Lindelöf, the author has learned to postpone the use of complex integration until the student is entirely familiar with the mapping properties of analytic functions. Geometric visualization is a source of knowledge as well as a didactic tool whose value cannot be disputed.
There are many other acknowledgments to be made. For instance, the appearance of Carathéodory's "Funktionentheorie" has, of course, not been without influence on the final form of this manuscript, which was half-finished at the time. Above all, the author has adopted without significant change Emil Artin's splendid idea of basing homology theory on the notion of winding number. This approach makes it possible to present a complete and rigorous proof of Cauchy's theorem and all its immediate applications with a minimum amount of topology. Of course, to by-pass topology is no merit in itself, but in a book on function theory it is highly desirable to concentrate on that part of topology which is truly basic in the study of analytic functions. For the same reason no proof is included of the Jordan curve theorem, which, to the author's knowledge, is never needed in function theory.
The exercises in the book are to be taken as samples. The author has not had the inclination to relieve the teacher from making up more and better exercises; it is for him to decide what methods should be drilled, what alternative proofs the student should be asked to give, and what ingenuity he should be given the opportunity to show. It is to be hoped that no teacher will follow this book page by page, for nothing could be more deadening. A text is a guide for the teacher which saves him from the necessity of making up a detailed plan in advance, but the continuous contact with his class makes him the authority on desirable deviations and cuts.
One more point: the author makes abundant and unblushing use of the words clearly, obviously, evidently, etc. They are not used to blur the picture. On the contrary, they test the reader's understanding, for if he does not agree that the omitted reasoning is clear, obvious, and evident, he had better turn back a few pages and make a fresh start. There are also a few places, easily spotted, in which a voluntary gap serves the purpose of saving half a page of unconstructive and dull reasoning.
Lars V Ahlfors
Winchester, Massachusetts, USA