复分析（英文版·第3版）

    书籍
    数学书籍
本书的诞生还是半个世纪之前的事情，但是，深贯其中的严谨的学术风范以及针对不同时代所做出的切实改进使得它愈久弥新，成为复分析领域历经考验的一本经典教材。本书作者在数学分析领域声乐卓著，多次荣获国际大次，这也是本书始终保持旺盛的生命力的原因之一。本书适合用做数学专业本科高年级学生及研究生教材。

Lars V.Ahlfors生前是哈佛大学数学教授。他于1924年进入赫尔辛基大学学习，并在1930年于芬兰著名的士尔库大学获得士学位。期间他还师从著名数学家Nevanlinna共同进行研究工作。1936年荣获菲尔茨奖。第二次世界大战结束后，辗转到哈佛大学从事教学工作。他又于1968年和1

Preface
CHAPTER 1 COMPLEX NUMBERS
1 The Algebra of Complex Numbers
2 The Geometric Representation of Complex Unmbers
CHAPTER 2 COMPLEX FUNCTIONS
1 Introduction to the Concept of Analytic Function
2 Elementary Theory of Power Series
3 The Exponential and Trigonometric Functions
CHAPTER 3 ANALYTIC FUNCTIONS AS MAPPINGS
1 Elementary Point Set Topology
2 Conformality
3 Linear Transformations
4 Elementary Conformal Mappings
CHAPTER 4 COMPLEX INTEGRATION
1 Fundamental Theorems
2 Cauchy ’s Integral Formula
3 Local Properties of Analytical Functions
4 The General Form of Cauchy’s Theorem
5 The Calculus of Residues
6 Harmonic Functions

　　 Complex Analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. There is, nevertheless, need for a new edition, partly because of changes in current mathematical terminology, partly because of differences in student preparedness and aims.
　　 There are no radical innovations in the new edition. The author still believes strongly in a geometric approach to the basics, and for this reason the introductory chapters are virtually unchanged. In a few places,throughout the book, it was desirable to clarify certain points that experience has shown to have been a source of possible misunderstanding or difficulties. Misprints and minor errors that have come to my attention have been corrected. Otherwise, the main differences between the second and third editions can be summarized as follows:
　　 1. Notations and terminology have been modernized, but it did not seem necessary to change the style in any significant way.
　　 2. In Chapter 2 a brief section on the change of length and area under conformal mapping has been added. To some degree this infringes on the otherwise self-contained exposition, for it forces the reader to fall back on calculus for the definition and manipulation of double integrals. The disadvantage is minor.
　　 3. In Chapter 4 there is a new and simpler proof of the general form of Cauchy's theorem. It is due to A. F. Beardon, who has kindly permitted me to reproduce it. It complements but does not replace the old proof,which has been retained and improved.
　　 4. A short section on the Riemann zeta function has been included.
　　This always fascinates students, and the proof of the functional equation illustrates the use of residues in a less trivial situation than the mere computation of definite integrals.
　　 5. Large parts of Chapter 8 have been completely rewritten. The main purpose was to introduce the reader to the terminology of germs and sheaves while emphasizing all the classical concepts. It goes without saying that nothing beyond the basic notions of sheaf theory would have
　　been compatible with the elementary nature of the book.
　　 6. The author has successfully resisted the temptation to include Riemann surfaces as one-dimensional complex manifolds. The book would lose much of its usefulness if it went beyond its purpose of being no more than all introduction to the basic methods and results of complex
　　function theory in the plane.
　　 It is my pleasant duty to thank the many who have helped me by.pointing out misprints, weaknesses, and errors in the second edition.
　　I am particularly grateful to my colleague Lynn Loomis, who kindly let me share student reaction to a recent course based oil my book.
　　 Lars V. Ahlfors