单复变函数 第2版（影印版）

　　 This book is intended as a textbook for a first course in the theory offunctions of one complex variable for students who are mathematicallymature enough to understand and execute arguments. The actual pre-requisites for reading this book are quite minimal; not much more than astiff course in basic calculus and a few facts about partial derivatives. Thetopics from advanced calculus that are used (e.g., Leibniz's rule for differ-entiating under the integral sign) are proved in detail.
　　

preface
i. the complex number system
1. the real numbers
2. the field of complex numbers
3. the complex plane
4. polar representation and roots of complex numbers
5. lines and half planes in the complex plane
6. the extended plane and its spherical representation
ii. metric spaces and the topology of c
1. definition and examples of metric spaces
2. connectedness
3. sequences and completeness
4. compactness
5. continuity
6. uniform convergence
iii. elementary properties and examples of analytic functions
1. power series
2. analytic functions
3. analytic functions as mappings, m6bius transformations
iv. complex integration

　　 This book is intended as a textbook for a first course in the theory offunctions of one complex variable for students who are mathematicallymature enough to understand and execute arguments. The actual pre-requisites for reading this book are quite minimal; not much more than astiff course in basic calculus and a few facts about partial derivatives. Thetopics from advanced calculus that are used (e.g., Leibniz's rule for differ-entiating under the integral sign) are proved in detail.
　　 Complex Variables is a subject which has something for all mathematicians.In addition to having applications to other parts of analysis, it can rightlyclaim to be an ancestor of many areas of mathematics (e.g., homotopy theory,manifolds). This view of Complex Analysis as "An Introduction to Mathe-matics" has influenced the writing and selection of subject matter for this book.The other guiding principle followed is that all definitions, theorems, etc.should be clearly and precisely stated. Proofs are given with the student inmind. Most are presented in detail and when this is not the case the reader istold precisely what is missing and asked to fill in the gap as an exercise. Theexercises are varied in their degree of difficulty. Some are meant to fix theideas of the section in the reader's mind and some extend the theory or giveapplications to other parts of mathematics. (Occasionally, terminology is usedin an exercise which is not defined--e.g., group, integral domain.)
　　 Chapters I through V and Sections VI.1 and VI.2 are basic. It is possibleto cover this material in a single semester only if a number of proofs areomitted. Except for the material at the beginning of Section VI.3 on convexfunctions, the rest of the book is independent of VI.3 and VI.4.
　　 Chapter VII initiates the student in the consideration of functions aspoints in a metric space. The results of the first three sections of this chapterare used repeatedly in the remainder of the book. Sections four and five needno defense; moreover, the Weierstrass Factorization Theorem is necessaryfor Chapter XI. Section six is an application Of the factorization theorem.The last two sections of Chapter VII are not needed in the rest of the bookalthough they are a part of classical mathematics which no one shouldcompletely disregard.
　　 The remaining chapters are independent topics and may be covered in any order desired.
　　 Runge's Theorem is the inspiration for much of the theory of Function Algebras. The proof presented in section VIII.1 is, however, the classical one involving "pole pushing". Section two applies Runge's Theorem to obtain a more general form of Cauchy's Theorem. The main results of sections three and four should be read by everyone, even if the proofs are not.
　　 Chapter IX studies analytic continuation and introduces the reader to analytic manifolds and covering spaces. Sections one through three can be considered as a unit and will give the reader a knowledge of analytic continuation without necessitating his going through all of Chapter IX.
　　 Chapter'X studies harmonic functions including a solution of the Dirichlet Problem and the introduction of Green's Function. If this can be called applied mathematics it is part of applied mathematics that everyone should know.
　　 Although they are independent, the last two chapters could have been combined into one entitled "Entire Functions". However, it is felt that Hadamard's Factorization Theorem and the Great Theorem of Picard are sufficiently different that each merits its own chapter. Also, neither result depends upon the other.
　　 With regard to Picard's Theorem it should be mentioned that another proof is available. The proof presented here uses only elementary arguments while the proof found in most other books uses the modular function.
　　 There are other topics that could have been covered. Some consideration was given to including chapters on some or all of the following: conformal mapping, functions on the disk, elliptic functions, applications of Hilbert space methods to complex functions. But the line had to be drawn somewhere and these topics were the victims. For those readers who would like to explore this material or to further investigate the topics covered in this book, the bibliography contains a number of appropriate entries.
　　 Most of the notation used is standard. The word "iff" is used in place of the phrase "if and only if", and the symbol is used to indicate the end of a proof. When a function (other than a path) is being discussed, Latin letters are used for the domain and Greek letters are used for the range.
　　 This book evolved from classes taught at Indiana University. I would like to thank the Department of Mathematics for making its resources available to me during its preparation. I would especially like to thank the students in my classes; it was actually their reaction to my course in Complex Variables that made me decide to take the plunge and write a book. Particular thanks should go to Marsha Meredith for pointing out several mistakes in an early draft, to Stephen Berman for gathering the material for several exercises on algebra, and to Larry Curnutt for assisting me with the final corrections of the manuscript. I must also thank Ceil Sheehan for typing the final draft of the manuscript under unusual circumstances.
　　 Finally, I must thank my wife to whom this book is dedicated. Herencouragement was the most valuable assistance I received.
　　 John B. Conway
　　 PREFACE FOR THE SECOND EDITION
　　 I have been very pleased with the success of my book. When it was apparent that the second printing was nearly sold out, Spfinger-Verlag asked me to prepare a list of corrections for a third printing. When I mentioned that I had some ideas for more substantial revisions, they reacted with characteristic enthusiasm.
　　 There are four major differences between the present edition and its predecessor. First, John Dixon's treatment of Cauchy's Theorem has been included. This has the advantage of providing a quick proof of the theorem in its full generality. Nevertheless, I have a strong attachment to the homotopic version that appeared in the first edition and have proved this form of Cauchy's Theorem as it was done there. This version is very geometric and quite easy to apply. Moreover, the notion of homotopy is needed for the later treatment of the monodromy theorem; hence, inclusion of this version yields benefits far in excess of the time needed to discuss it.
　　 Second, the proof of Runge's Theorem is new. The present proof is due to Sandy Grabiner and does not use "pole pushing". In a sense the "pole pushing" is buffed in the concept of uniform approximation and some ideas from Banach algebras. Nevertheless, it should be emphasized that the proof is entirely elementary in that it relies only on the material presented in this text.
　　 Next, an Appendix B has been added. This appendix contains some bibliographical material and a guide for further reading.