“新时代国外优秀本科数学教材评价研究”课题（第5.6章）

作者简介：Walter Rudin（1921-2010），澳裔美国数学家，威斯康辛大学麦迪逊分校的数学教授。除了对复杂和调和分析的贡献，Rudin还以他的数学分析教科书而闻名：《Principles of Mathematical Analysis》、《Real and Complex Analysis》和《Functional Analysis》。Rudin的分析教材在全世界的数学教育中也很有影响力，已经被翻译成包括俄语、中文、和西班牙语在内的13种语言。

书评：

This book is full of interesting things, mostly proofs. The chapter on Banach algebras is a gem; this subject combines algebra, analysis, and topology, and the exposition shows clearly how the three areas work together. Walter Rudin (1921–2010) wrote the book in 1966 to show that real and complex analysis should be studied together rather than as two subjects, and to give a a modern treatment. Fifty years later it is still modern.

The first third of the book is devoted to measure and integration. The presentation is based on measures on abstract spaces with σσ-algebras. It includes brief introductions to Hilbert space and Banach spaces, with material that will be used in the complex-variables proofs later. This beginning section is the only part of the book that deals with spaces more general than the real line and the complex plane, however it’s not any harder than it would be if we stuck to the real line. This includes a chapter on differentiation (of measures) and a chapter on product spaces (i.e., the Fubini theorem). The rest of the book is about analysis on the complex plane. It starts with a short chapter on Fourier transforms, then presents a course in complex variables that is traditional in terms of the theorems proved, but has very slick proofs using what has gone before. The traditional part ends with the little Picard theorem. The last quarter of the book consists of several short chapters on advanced topics in complex analysis; these include HpHp spaces, Banach algebras, holomorphic Fourier transforms, and a characterization of functions that are the uniform limit of polynomials (Mergelyan’s theorem).

The approach is not very concrete; there are very few worked examples (many of the exercises do deal with specific functions). The book does not have the detailed chapters that we are used to on evaluating series and integrals and on special functions. But it is also not very abstract; it truly is mostly complex analysis, not general spaces. The proofs are informed by the more general viewpoint, and there is a strong functional-analysis flavor. For example, much use is made of the Hahn-Banach Theorem and some use of the Urysohn lemma and Tietze extension theorem.

The book has been widely criticized for lacking motivation, and this criticism is accurate. You don’t absolutely need a lot of background to read the book, but it is a collection of beautiful proofs without much context. For example, the discussion of spectra comes out of nowhere and is very mysterious unless you are well-acquainted with linear algebra and eigenvalues.

The book was aimed at first-year graduates and has been used successfully in many first-year graduate courses, and I think that is still about the right level for it. Undergraduates interested in the subject matter would be better served, before they tackle this book, by a more traditional complex analysis book such as Bak & Newman’s Complex Analysis or Ahlfors’s more advanced Complex Analysis, and by one of the many good introductions to Lebesgue integration (I like Boas’s A Primer of Real Functions).

本书十分有趣，尤其是证明部分。有关Banach代数的章节像一颗璀璨夺目的宝石；这部分结合了代数、分析与拓扑，同时也很好地解释了这三个领域是如何互相作用的。Walter Rudin于1966年写成本书，意在表明实分析与复分析应当一起学习，而不是把它们当成两门课，同时希望更现代化地阐述它们。五十年过后，本书仍未过时。

本书的前三分之一是关于测度与积分的。表述基于伴有σ代数的抽象空间上的测度。这部分包括了对Hilbert空间与Banach空间的简要介绍，也包含了之后会在复分析部分的证明中用到的内容。这也是本书中唯一处理比实轴与复平面更抽象的空间的部分，然而如果我们只讨论实轴的情况，这部分也不算很难。这部分包括了关于（测度的）微分的一章与关于乘积空间（即Fubini定理）的一章。本书的其余部分是关于复平面上的分析。它以关于傅里叶变换的简短的一章开始，再介绍复分析。虽然这部分定理的证明十分传统，但结合本书之前所叙述的内容也能得到非常巧妙的证明。传统的部分以Picard小定理结束。本书的最后四分之一包含有关复分析更深入内容的十分简短的几章；包括Hardy空间，Banach代数，全纯Fourier变换，和对多项式一致收敛极限的刻画（Mergelyan定理）。

本书的教学方法并不是很具体；实例非常少（然而很多习题则需要处理特定的函数）。同时本书也不像其他教材，并不包含详细讲述级数与积分和一些特殊的函数的章节。但它并没有过于抽象；它本质上还是复分析，并不对（与复平面相比）更一般的空间加以过多的讨论。书中的证明被以一种更普适的角度给出，并伴有很强的泛函分析的味道。例如Hahn-Banach定理的广泛应用以及Urysohn引理与Tietze延拓定理的使用。

本书的阐述往往过于直接，缺乏对学习这些内容原因的讨论，因而广受批评，这种批评是准确的。你并不一定需要很多背景知识来阅读这本书，但它却包含了很多没有多少上下文的漂亮的证明。例如，关于谱的讨论像是无中生有的，非常神秘，除非你很熟悉线性代数和特征值。

这本书适用于一年级研究生，并成功应用于很多一年级研究生的课程中，同时我认为这确实是它的正确定位。对这本书感兴趣的本科生在开始学习这本书之前，最好先读一本更传统的复分析教材，如Bak与Newman的《Complex Analysis 》或Ahlfors的更高深的《Complex Analysis》，以及一本介绍Lebesgue积分的教材（我喜欢Boas的《A Primer of Real Functions》）。

点评人：Allen Stenger(Missouri Journal of Mathematical Sciences编辑，专注于数论和经典分析领域)

资料整理：孟文斌

翻译：赵旭彤

《Complex Analysis》

作者：Serge Lang

出版商：Springer

出版年：2003

ISBN：9780387985923

适用范围：研究生

推荐强度：9

作者简介：Serge Lang（1927-2005），法裔美国数学家、活动家，在耶鲁大学任教多年。他以数论方面的成就和他的数学教科书闻名，包括颇具影响力的《Algebra》。1960年，他获得弗兰克·尼尔森·科尔奖，是波旁派的一员。

书评：

The first and second editions of this book have been reviewed earlier: see Zbl 0366.30001 and 819.30001. The plan of the book has been retained. Part I (290 pages) contains the basic theory up to the calculus of residues, and basic results about harmonic functions. Part II (43 pages), entitled Geometric Function Theory, contains the Riemann mapping theorem, the Schwarz reflection principle, and a proof of Picard’s theorem via the module function. Part III (116 pages) presents several topics for the more advanced reader: Hadamard’s three circles theorem and applications, Weierstraß’s product theorem, elliptic functions, and the proof of the prime number theorem following Newman and Korevaar. In an Appendix (24 pages) useful material on difference equations, analytic differential equations, fixed points of linear transformations, and Cauchy’s formula for C∞C∞-functions is presented.

Compared with the third edition, some material on harmonic functions and vector fields has been added, and the wealth of examples and exercises has still been increased. These exercises, plus their solutions, have been collected in a separate book; see the following review.

We noticed that proof reading is not the strength of the author. Thus, as already present in the third edition, there are two sections VII, §4; VIII, §4; VIII, §5. The statement of the reflection principle is still not complete, and Fig. 12 on p. 222 is not correct. We also note that in the Bibliography important recent texts, even Springer Graduate Texts, are missing, and that there is a complete absence of historical remarks. In spite of this, it is a highly recommendable book for a two semester course on complex analysis.

本书第一版与第二版的书评已在早些时候给出，参照zbMATH 上编号为0366.30001与 819.30001的书评。而对本版来说，教学规划被被保留了下来。第一部分（290页）包括留数计算的基本理论，以及关于全纯函数的基本结论。第二部分（43页），几何函数论，包括Riemann映照定理，Schwarz反射原理，以及一个借由模函数证明Picard定理的方法。第三部分（116页）为更高学术水平的读者准备了一些主题：Hadamard三圆定理及其应用，Weierstrass分解定理，椭圆函数，与素数定理的证明（主要依据了Newman与Korevaar的成果）。在附录（24页）中，有关差分方程、解析微分方程、线性变换的不动点以及光滑函数的Cauchy公式的内容被给出。

与第三版相比，本版加入了与调和函数与向量场相关的内容，也增加了不少例题与习题。这些习题与答案一同被收录到另一本书中。

校对不是作者的强项。所以，也正如第三版那样，“第七章第4节”、“第八章第4节” 与“第八章第5节”都出现了两次。有关Schwarz反射原理的陈述并不完善，并且222页的图12是错误的。我们也注意到参考文献中的数学教材，甚至是Springer出版社的GTM系列，是错误的。除去这些因素，它是一本十分值得推荐的用于两学期课程的复分析教材。

评人：D.Gaier (德国数学家，吉森大学教授)

资料整理：孟文斌

翻译：赵旭彤

“新时代国外优秀本科数学教材评价研究”课题 （第5.6章）

“新时代国外优秀本科数学教材评价研究”课题（第5.6章）