Analytic Function Theory
EINAR HILLEYale University
A new book, even a textbook, is supposed to have a message. Perhaps mine does have a message or two.
It is my hope that students of this book may come to respect the historical continuity of the subject. Analytic function theory is the result of a long development involving the contributions of many workers. Many theorems have names attached to them, names of men who were famous in their days. The student should know something about the historical development of the subject and about the men who took a major part in this development. He should understand mathematics as a living organism, a growing body of learning, fed by the efforts of many workers. If possible, he should acquire a sense of veneration for those who built up the structure that he is studying. This is the reason for the many historical footnotes.
There is also conceptual continuity: analytic function theory is just a part of the structure of mathematics. As the latter grows and changes, so does function theory. An abstract and postulational approach to mathematics is gradually penetrating into all fields, and it affects the instruction in mathematics at all levels. A modern treatise on function theory has to take this fact into account, both in building up the subject and in fitting it into the larger frame of mathematics. This book represents an effort to integrate the theory of analytic functions with modern analysis as a whole and, in particular, to present it as a branch of functional analysis, to which it gives concrete illustrations, problems, and motivation. Hence the emphasis on structural aspects such as linearity of the sets and of the operations under consideration. The algebraic aspects of the theory have been stressed whenever possible. Certain topological concepts, such as the notions of neighbourhood, distance, length, and metric space, have also been stressed. On the other hand, this is not a textbook in topology: if intuition helps, an appeal is made to intuition.
The author believes that complex integration is the proper basis of function theory. The Cauchy integral is a much more pliable and versatile tool than the power series when it comes to doing things in function theory. But before the student can really grasp integrals of analytic functions, he should have at his disposal a large number of such functions, and here the power series is invaluable as a source. The power series also leads to important connections with real analysis, and it is indispensable for the problem of analytic continuation. The emphasis has to lie on diversity rather than on purity of method: the more methods the student can learn, the better he will be equipped. For this reason some of the important theorems, such as the maximum principle and the inverse function theorem, have been treated by several different methods.
These general considerations have led to the following arrangement of the subject matter of Volume I: After a preliminary study of number systems, the geometry of the complex plane is developed, and simple functions such as linear fractions, powers, and roots are studied. The main theory begins in Chapter 4 with the definition of holomorphic functions, the Cauchy-Riemann equations, inverse functions, and the elements of conformal mapping. This is followed by a chapter on power series and one on the elementary transcendental functions. The systematic study of holomorphic functions occupies the last three chapters, devoted to complex integration, representation theorems, and the calculus of residues. Supplementary material on point sets, polygons, and Riemann and Riemann-Stieltjes integration is to be found in three Appendixes. There is a brief Bibliography at the end of the volume, and suggestions for collateral reading are appended to the various chapters. An explanation of the symbols used precedes Chapter 1.
A word about the numbering is in order. Section 7.3 is the third section of Chapter 7; the theorems in this section are numbered 7.3.1 through 7.3.4. Lemmas and definitions are numbered in the same way.
A student who intends to use this book should have had a good course in advanced calculus. Familiarity with abstract mathematical reasoning and some skill in manipulating identities, integrals, and series are the main prerequisites. The book is to a large extent autonomous, and the student will find most of the factual information that he needs incorporated in the text. This means that there are considerable parts of the book which a well-prepared student can omit or use for reference only. This applies in particular to Chapters 1 and 5. It should be realized that Chapters 4, 5, 7, 8, and 9 form the core of the book, and that the rest is ancillary material. In this connection it should also be remembered that the present volume is preparatory for a second volume, which, it is hoped, will follow fairly soon.
With suitable omissions, the present volume can be used as a text for a one-term introductory course; in fact, the author has used a preliminary draft for this purpose. It should be possible to cover everything in one year at a leisurely tempo. The second volume will provide additional material for a course on a somewhat higher level.
It is my pleasant duty to bring thanks to the friends who have helped with this undertaking. In the first place I am deeply indebted to Dr Ernest C Schlesinger, who read the whole manuscript in detail and suggested numerous corrections and improvements. Thanks are due also to Professors Garrett Birkhoff, C T lonescu-Tulcea, Shizuo Kakutani, and Angus E Taylor for constructive criticism. Finally, I wish to thank Ginn and Company for the honour they have shown me by letting my book inaugurate their new series "Introductions to Higher Mathematics" as well as for sympathetic consideration of an author's whims and wishes.
New Haven, Connecticut