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主 编 杨劲根
副主编 楼红卫 李振钱 郝群
编写人员（按汉语拼音为序）
陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林
1. 序言
2. 非数学专业的数学教材
3.数学分析和泛函分析
4.单复变函数
5.多复变函数
6.代数
7.数论
8.代数几何
9.拓扑与微分几何
10.偏微分方程
11.概率论
12.计算数学
13.其他
14.附录
复变函数论是数学系本科生的必修科，也是很多数学分支的研究生课程，其重要性是公认的。
粗看起来，数学分析中的导数和积分推广到复数上就是复分析了。但是复的可导函数，即解析函数，具有实的可导函数不具备的优美性质。它中的一些主要定理如柯西定理、柯西积分公式、刘维尔定理、留数公式等都有强大的威力。
由于单复变函数论的基本理论非常成熟，国外的优秀教材也比较定型，我们下面介绍的基本教材都可以作为首选。
学习复变函数前学生应有非常扎实的数学分析基础，特别是无穷级数和广义积分的收敛性。
书名：Complex Analysis, 3rd edition
作者：Lars V. Ahlfors
出版商：McGraw-Hill (1979)
页数：331
适用范围：大学数学系本科,数学专业研究生
预备知识：数学分析和线性代数
习题数量：小
习题难度： 中等
推荐强度：10
书评： 出自数学大家之手的这本书已经公认为单复变函数论的经典著作， 既可以选取部分内容作为我国综合性大学数学系本科生的复变函数论教材， 又可以用来作为大学高年级学生和研究生的选修课内容，同时它又是从事复分析研究的标准参考书。 有关单复变函数论的教材、参考书不下几十种，但是除了干巴巴的概念、 定理的正确叙述与严格证明之外提供大量解释性文字的书本并不多见，而在这些叙述中既没有多余的话， 又能使读者开阔视野并感受到作者深厚功力的更为少见，本书恰恰为其中的佼佼者。这本书已经出了三版。 在这第三版中大部分内容未作更动，叙述依然简洁而流畅，但是作者彻底改写了第八章， 以层论的观点描述Riemann面上整体解析函数的存在性，使经典的内容现代化。（张锦豪）
国外评论摘选
1) This book has been, since its first edition in 1953, the standard textbook for rigorously learning complex analysis, and not without a reason. The wonderful theory of this branch of mathematics is appropriately emphasized and thoroughly constructed, leading to more general and precise results than most textbooks. While the constant appearance of new texts on the field can only help appreciate the subject from a different perspective, few give you such a deep and serious treatment like this gem. Postscript: An earlier reviewer claims that Ahlfors never defines the set of complex numbers, while this is indeed done in the fourth through sixth pages in a much more analytical way than generally found elsewhere. It is quite possible to dislike this author's style or approach (or anybody's for that matter), but it would be difficult to charge Ahlfors with being sloppy with his writing.
2) How can anyone fail to read this book? The exposition is rigorous, coherent, precise without being either pedantic or overwhelming. A certain level of mathematical maturity is requisite, such as one might acquire in the course of digesting Rudin's Principles of Mathematical Analysis or Apostol's book. This is not a compendium of results and exercises for engineers or physicists, it is a concise introductory text in pure mathematics. In that sense it is too abstract and proof oriented for that aforementioned audience which would be better served by a text in mathematical methods. Even pure mathematics students would benefit from supplementing this book with more detailed, computationally oriented books such as Conway or Boas. It's unrealistic to expect to find everything in one text and to further expect it to remain cogent and approachable. Ahlfor's beautiful little book has justifiably remained a classic for four decades.
3) I'm not sure why the other reviews are so positive. The book is very thorough and rigorous I'm sure, but the explanations are terrible. Everyone I've talked to in my class agrees that it's extremely difficult to learn from if you don't already know complex analysis, because the definitions and order of treatment are very un-intuitive. Example: residue at $a$ is defined as the number $R$ that makes $f-R/(z-a)$ the derivative of a single-valued analytic function in $0<|z-a|<\delta;$ why didn't he even mention that it's the coefficient of $1/(z-a)$ in the Taylor expansion? And he didn't even give any examples of specific residues. I ended up using a mathematical methods for physics book; it was the only way I could develop any kind of intuition for the subject.
书名：Introduction to complex analysis
作者： Kunihiko Kodaira
出版商： Cambridge University Press (1984)
页数：256
适用范围：大学数学系本科
预备知识：数学分析和线性代数
习题数量：无
推荐强度：9.8
书评： 日本岩波讲座的基础数学中由小平邦彦撰写过三本关于复分析的小册子，其中的I、II 分册被译成英文出版为本书。其内容与我国综合性大学的复变函数论课程基本相符。 本书体现了数学大家小平邦彦一贯的写书风格，起点低，过程详尽，深入浅出，流畅而易读。 本书以复可微（有连续导数）的条件引入全纯函数的概念，为后续的处理带来很大方便。 同时以Cauchy积分定理为主线，从简单到复杂，循序渐进地揭示了这个定理的成立与拓扑的关系。 以远较一般教科书为多的篇幅介绍了Riemann球，引入局部坐标齐次坐标等概念， 并顺理成章地接着用来导出分式线性变换的群伦性质。 本书处处体现了小平邦彦深厚的研究功力与广阔的视野。 对于希望将来在Riemann面、Teichmuller空间、多复变函数、复几何、 代数几何等方面进一步深造的有志者来说是一本不可多得的基础好书。（张锦豪）
书名：Functions of One Complex Variable
作者： John B. Conway
出版商： Springer-Verlag (1973)
页数：313
适用范围：大学数学系本科或数学专业研究生一年级
预备知识：数学分析和线性代数
习题数量：大
习题难度： 从易到难都有，大部分中等
推荐强度：9.6
书评： 本书虽然为大学生学习单复变函数论而写，但是内容十分丰富。 作者用整个一章介绍最大模原理，除了我国教材中通常出现的内容外， 还证明了Hadamard三圆定理与Phragmen-Lindelof定理。 对多复变函数的近代理论有深远影响的Runge定理、Mittag-Leffler定理、 Weierstrass定理等也给予详尽的介绍。本书还包含了大Picard定理等值分布理论的基础。 同时以解析函数芽层的现代观点描述了解析延拓这一重要现象，并引入Riemann面， 进一步再用复流形的现代概念进行提升，非常精彩。 本书内容远远超过我国综合大学复变函数论课程的需要， 所以同时可以用来作为大学高年级与研究生一年级的选课教材。 本书观点颇高，论述严谨，排版紧凑。 虽然作者声称只需基本微积分以及关于偏导数等少量预备知识即可阅读本书， 但由于介绍预备知识的叙述水平超过了一般的数学分析，因此初学者若没有一定的数学天赋则很难自学。 但毫无疑问，这是每一位学习或应用复变函数论者的极好参考书。（张锦豪）
国外评论摘选
1) We're using this book for my graduate level complex analysis course, and over all, I'm pleased with it. Aside from some goofy notation (i.e., an empty box to represent the empty set?), it's pretty well written. The pace of the text isn't too fast or too slow, and there are plenty of exercises of a varying degree of difficulty to help you learn the material.
2) An ideal text for a first-year graduate students in mathematics studying Compex Analysis. And this depend how the professor present the material. The exposition is complete and very clear, including a lot of optional material for the curious. which could be very useful to those preparing for a qualifying exam in analysis at the PhD level.
3) This book was the recommended textbook for a course in Complex Analysis I took at college. I had already done a 1st course on analysis, but that didn't help me too much. This book, littered with loads of proofs and lemmas, is a little too terse, and the author expects students to understand a lot on their own. Concepts in Complex Analysis need to be demonstrated using examples, and diagrams, if possible. Like for eg. the concept of branches in complex functions. The book starts of defining the complex logrithmic function. The author never says what a branch exactly is. He writes down a hell lot of proofs and expects the student to figure out that the complex logarithm is infact a multi-valued function, and that a branch is essentially a slice of this multivalued function. Similiar problems crop up when the author discusses fractional linear transforms. Instead of showing whats happening with simple diagrams, the author makes things look extremely complicated with his equations and theorems. This book makes learning complex analysis a very mechanical exercise, devoid of all fun.
书名：Complex Analysis, 3rd edition
作者： Serge Lang
出版商： Addison-Wesley (1993)
页数：321
适用范围：大学数学系本科或数学专业研究生一年级
预备知识：数学分析和线性代数
习题数量：中
习题难度： 中
推荐强度：9.7
书评： 本书第三版较之第一版增加了许多超出本科生学习的内容。全书分为三部分， 其第一部分与我国综合性大学的复变函数论教材大致相当，第二、第三部分为进一步学习的内容， 可供大学生高年级或研究生低年级的选修课之用。本书将Cauchy定理分为两部分介绍，从局部到整体， 从简单到复杂，使读者很容易接受。特别是在一般Cauchy定理的证明中借用了分析味更浓的Dixon证明， 避开了初学者理解拓扑内涵的困难。将对数函数的介绍与解析延拓的放在一起， 使读者从更一般的角度理解如何选取多值函数的单值支。 本书的另一亮点是介绍了Zeta函数并用来证明素数分布定理。作者是位著名的数学家， 学识广博，擅长撰写数学基础类教材。一些深刻的定理在他的处理下通俗易懂， 所以本书虽然述及到许多深入的复分析内容，读来也是毫无困难，值得向初学者推荐。（张锦豪）
国外评论摘选
1) A person with absolutely no knowledge of complex numbers could begin with page one of this book. However, I think that some exposure to analysis is helpful before finishing the first chapter, but not necessary. I found this book easier to read and understand than some real analysis books, yet it helped me further understand real analysis in the process. I'm sure this is due to mere repetition of some of those concepts over a different field. As the author mentions in his foreword, the first half of the book can be used as an undergraduate text (Jr/Sn years) and the second half can also, but I would NOT have enjoyed it in undergraduate studies. I found it worthy of a first course in complex numbers at the graduate level. I especially liked it after studying real numbers. The placement of the chapter subject matter can be altered (to some degree) to ones liking. I think Lang has provided good examples and problems. There's a solutions manual (by Rami Shakarchi) for this text somewhere.
2) if you want an introduction to complex analysis, I advise you to pass on this book, and read Churchill and Brown's introductory book. Having said this, part I of Lang's book will seem mostly review if you follow my advice. Part II, on Geometric Function Theory, is more advance material that is presented reasonably well.