单复变函数(第2卷)(英文版)

　　《单复变函数(第2卷)(英文版)》是springer《数学研究生教材》第 159卷，系世界著名教学家j. b.coway编写的《单复变函数》之续集，本卷在第1卷的基础上讨论了单复变函数论中的一些专门问题。目次：基本理论回顾；单连通区域共形等价；有限连通区域共形等价；解析复盖映射；比勃拉赫猜想的debranges证明；基本概念分析；调合函数概论；圆盘哈代空间；平面势论。
　　 读者对象：数学专业的研究生和科研人员。

《单复变函数(第2卷)(英文版)》
preface
13 return to basics
1 regions and curves
2 derivatives and other recollections
3 harmonic conjugates and primitives
4 analytic arcs and the reflection principle
5 boundary values for bounded analytic functions
14 conformal equivalence for simply connected regions
1 elementary properties and examples
2 crosscuts
3 prime ends
4 impressions of a prime end
5 boundary values of riemann maps.
6 the area theorem.
7 disk mappings: the class $
15 conformal equivalence for finitely connected regions
1 analysis on a finitely connected region.
2 conformal equivalence with an analytic jordan region
3 boundary values for a conformed equivalence between finitely connected jordan regions

　　This is the sequel to my book Functions of One Complex Variable I, and probably a good opportunity to express my appreciation to the mathemat-ical community for its reception of that work. In retrospect, writing that book was a crazy venture.
　　As a graduate student I had had one of the worst learning experiences of my career when I took complex analysis; a truly bad teacher. As a non-tenured assistant professor, the department allowed me to teach the graduate course in complex analysis. They thought I knew the material; I wanted to learn it. I adopted a standard text and shortly after beginning to prepare my lectures I became dissatisfied. All the books in print had virtues; but I was educated as a modern analyst, not a classical one, and they failed to satisfy me.
　　This set a pattern for me in learning new mathematics after I had become a mathematician. Some topics I found satisfactorily treated in some sources; some I read in many books and then recast in my own style. There is also the matter of philosophy and point of view. Going from a certain mathematical vantage point to another is thought by many as being independent of the path; certainly true if your only objective is getting there. But getting there is often half the fun and often there is twice the value in the journey if the path is properly chosen.
　　One thing led to another and I started to put notes together that formed
　　chapters and these evolved into a book. This now impresses me as crazy
　　partly because I would never advise any non-tenured faculty member to begin such a project; I have, in fact, discouraged some from doing it. On the other hand writing that book gave me immense satisfaction and its re-ception, which has exceeded my grandest expectations, makes that decision to write a book seem like the wisest I ever made. Perhaps I lucked out by being born when I was and finding myself without tenure in a time (and possibly a place) when junior faculty were given a lot of leeway and allowed to develop at a slower pacc something that someone with my background and temperament needed. It saddens me that such opportunities to develop are not so abundant today.
　　The topics in this volume are some of the parts of analytic function theory that I have found either useful for my work in operator theory or enjoyable in themselves; usually both. Many also fall into the category of topics that I have found difficult to dig out of the literature.
　　I have some difficulties with the presentation of certain topics in the literature. This last statement may reveal more about me than about the state of the literature, but certain notions have always disturbed me even though experts in classical function theory take them in stride. The best example of this is the concept of a multiple-valued function. I know there are ways to make the idea rigorous, but I usually find that with a little work it isn't necessary to even bring it up. Also the term multiple-valued function violates primordial instincts acquired in childhood where I was sternly taught that functions, by definition, cannot be multiple-valued.
　　The first volume was not written with the prospect of a second volume to follow. The reader will discover some topics that are redone here with more generality and originally could have been done at the same level of sophistication if the second volume had been envisioned at that time. But I have always thought that introductions should be kept unsophisticated.The first white wine would best be a Vouvray rather than a Chassagne-Montrachet.
　　This volume is divided into two parts. The first part, consisting of Chap-ters 13 through 17, requires only what was learned in the first twelve chap-ters that make up Volume I. The reader of this material will notice, how-ever, that this is not strictly true. Some basic parts of analysis, such as the Cauchy-Schwarz Inequality, are used without apology. Sometimes re-sults whose proofs require more sophisticated analysis are stated and their proofs are postponed to the second half. Occasionally a proof is given that requires a bit more than Volume I and its advanced calculus prerequisite.The rest of the book assumes a complete understanding of measure and integration theory and a rather strong background in functional analysis.
　　Chapter 13 gathers together a few ideas that are needed later. Chapter 14, "Conformal Equivalence for Simply Connected Regions," begins with a study of prime ends and uses this to discuss boundary values of Riemann maps from the disk to a simply connected region. There are more direct ways to get to boundary values, but I find the theory of prime ends rich in mathematics. The chapter concludes with the Area Theorem and a study of the set $ of schlicht functions.
　　Chapter 15 studies conformal equivalence for finitely connected regions.I have avoided the usual extremal arguments and relied instead on the method of finding the mapping functions by solving systems of linear equa-tions. Chapter 16 treats analytic covering maps. This is an elegant topic that deserves wider understanding. It is also important for a study of Hardy spaces of arbitrary regions, a topic I originally intended to include in this volume but one that will have to await the advent of an additional volume.
　　Chapter 17, the last in the first part, gives a relatively self contained treatment of de Branges's proof of the Bieberbach conjecture. I follow the approach given by Fitzgerald and Pommerenke [1985]. It is self contained except for some facts about Legendre polynomials, which are stated and explained but not proved. Special thanks are owed to Steve Wright and Dov Aharonov for sharing their unpublished notes on de Branges's proof of the Bieberbach conjecture.
　　Chapter 18 begins the material that assumes a knowledge of measure theory and functional analysis. More information about Banach spaces is used here than the reader usually sees in a course that supplements the standard measure and integration course given in the first year of graduate study in an American university. When necessary, a reference will be given to Conway [1990]. This chapter covers a variety of topics that are used in the remainder of the book. It starts with the basics of Bergman spaces, some material about distributions, and a discourse on the Cauchy transform and an application of this to get another proof of Runge's Theorem. It concludes with an introduction to Fourier series.
　　Chapter 19 contains a rather complete exposition of harmonic functions on the plane. It covers about all you can do without discussing capacity,which is taken up in Chapter 21. The material on harmonic functions from Chapter 10 in Volume I is assumed, though there is a built-in review.
　　Chapter 20 is a rather standard treatment of Hardy spaces on the disk,though there are a few surprising nuggets here even for some experts.
　　Chapter 21 discusses some topics from potential theory in the plane. It explores logarithmic capacity and its relationship with harmonic measure and removable singularities for various spaces of harmonic and analytic functions. The fine topology and thinness are discussed and Wiener's cfi-terion for regularity of boundary points in the solution of the Dirichlet problem is proved.
　　This book has taken a long time to write. I've received a lot of assistance along the way. Parts of this book were first presented in a pubescent stage to a seminar I presented at Indiana University in 1981-82. In the sem-inar were Greg Adams, Kevin Clancey, Sandy Grabiner, Paul McGuire,Marc Raphael, and Bhushan Wadhwa, who made many suggestions as the year progressed. With such an audience, how could the material help but improve. Parts were also used in a course and a summer seminar at the University of Tennessee in 1992, where Jim Dudziak, Michael Gilbert, Beth Long, Jeff Nichols, and Jeff vanEeuwen pointed out several corrections and improvements. Nathan Feldman was also part of that seminar and besides corrections gave me several good exercises. Toward the end of the writing process I mailed the penultimate draft to some friends who read several chapters. Here Paul McGuire, Bill Ross, and Liming Yang were of great help. Finally, special thanks go to David Minda for a very careful read-lng of several chapters with many suggestions for additional references and exercises.
　　On the technical side, Stephanie Stacy and Shona Wolfenbarger worked diligently to convert the manuscript to TEX. Jinshui Qin drew the figures in the book. My son, Bligh, gave me help with the index and the bibliography.
　　In the final analysis the responsibility for the book is mine.