分析Ⅰ（影印版）
2009-12
高等教育出版社
Roger Godement
431
450000
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为了更好地借鉴国外数学教育与研究的成功经验，促进我国数学教育与研究事业的发展，提高高等学校数学教育教学质量，本着“为我国热爱数学的青年创造一个较好的学习数学的环境”这一宗旨，天元基金赞助出版“天元基金影印数学丛书”。该丛书主要包含国外反映近代数学发展的纯数学与应用数学方面的优秀书籍，天元基金邀请国内各个方向的知名数学家参与选题的工作，经专家遴选、推荐，由高等教育出版社影印出版。为了提高我国数学研究生教学的水平，暂把选书的目标确定在研究生教材上。当然，有的书也可作为高年级本科生教材或参考书，有的书则介于研究生教材与专著之间。欢迎各方专家、读者对本丛书的选题、印刷、销售等工作提出批评和建议。
本书第一卷的内容包括集合与函数、离散变量的收敛性、连续变量的收敛性、幂函数、指数函数与三角函数；第二卷的内容包括Fourier级数和Fourier积分以及可以通过Fourier级数解释的Weierstrass的解析函数理论。 本书是作者在巴黎第七大学讲授分析课程数十年的结晶，其目的是阐明分析是什么，它是如何发展的。本书非常巧妙地将严格的数学与教学实际、历史背景结合在一起，对主要结论常常给出各种可能的探索途径，以使读者理解基本概念、方法和推演过程。作者在本书中较早地引入了一些较深的内容，如在第一卷中介绍了拓扑空间的概念，在第二卷中介绍了Lebesgue理论的基本定理和Weierstrass椭圆函数的构造。
作者：（法国）戈德门特（Roger Godement）
PrefaceI - Sets and Functions §1. Set Theory 1 - Membership, equality, empty set 2 - The set defined by a relation. Intersections and unions 3 - Whole numbers. Infinite sets 4 - Ordered pairs, Cartesian products, sets of subsets 5 - Functions, maps, correspondences 6 - Injections, surjections, bijections 7 - Equipotent sets. Countable sets 8 - The different types of infinity 9 - Ordinals and cardinals §2. The logic of logiciansII - Convergence: Discrete variables §1. Convergent sequences and series 0 - Introduction: what is a real number? 1 - Algebraic operations and the order relation: axioms of R 2 - Inequalities and intervals 3 - Local or asymptotic properties 4 - The concept of limit. Continuity and differentiability 5 - Convergent sequences: definition and examples 6 - The language of series 7 - The marvels of the harmonic series 8 - Algebraic operations on limits §2. Absolutely convergent series 9 - Increasing sequences. Upper bound of a set of real number 10 - The function log x. Roots of a positive number 11 - What is an integral? 12 - Series with positive terms 13 - Alternating series 14 - Classical absolutely convergent series 15 - Unconditional convergence: general case 16 - Comparison relations. Criteria of Cauchy and d'Alembert 17 - Infinite limits 18 - Unconditional convergence: associativity §3. First concepts of analytic functions 19 - The Taylor series 20 - The principle of analytic continuation 21 - The function cot x and the series ∑ 1/n2k 22 - Multiplication of series. Composition of analytic functions. Formal series 23 - The elliptic functions of WeierstrassIII- Convergence: Continuous variables §1. The intermediate value theorem 1 - Limit values of a function. Open and closed sets 2 - Continuous functions 3 - Right and left limits of a monotone function 4 - The intermediate value theorem §2. Uniform convergence 5 - Limits of continuous functions 6 - A slip up of Cauchy's 7 - The uniform metric 8 - Series of continuous functions. Normal convergence §3. Bolzano-Weierstrass and Cauchy's criterion 9 - Nested intervals, Bolzano-Weierstrass, compact sets 10 - Cauchy's general convergence criterion 11 - Cauchy's criterion for series: examples 12 - Limits of limits 13 - Passing to the limit in a series of functions §4. Differentiable functions 14 - Derivatives of a function 15 - Rules for calculating derivatives 16 - The mean value theorem 17 - Sequences and series of differentiable functions 18 - Extensions to unconditional convergence §5. Differentiable functions of several variables 19 - Partial derivatives and differentials 20 - Differentiability of functions of class C1 21 - Differentiation of composite functions 22 - Limits of differentiable functions 23 - Interchanging the order of differentiation 24 - Implicit functionsAppendix to Chapter III 1 - Cartesian spaces and general metric spaces 2 - Open and closed sets 3 - Limits and Cauchy's criterion in a metric space; complete spaces 4 - Continuous functions 5 - Absolutely convergent series in a Banach space 6 - Continuous linear maps 7 - Compact spaces 8 - Topological spacesIV - Powers, Exponentials, Logarithms, Trigonometric Functions §1. Direct construction 1 - Rational exponents 2 - Definition of real powers 3 - The calculus of real exponents 4 - Logarithms to base a. Power functions 5 - Asymptotic behaviour 6 - Characterisations of the exponential, power and logarithmic functions 7 - Derivatives of the exponential functions: direct method 8 - Derivatives of exponential functions, powers and logarithms §2. Series expansions 9 - The number e. Napierian logarithms 10 - Exponential and logarithmic series: direct method 11 - Newton's binomial series 12 - The power series for the logarithm 13 - The exponential function as a limit 14 - Imaginary exponentials and trigonometric functions 15 - Euler's relation chez Euler 16 - Hyperbolic functions §3. Infinite products 17 - Absolutely convergent infinite products 18 - The infinite product for the sine function 19 - Expansion of an infinite product in series 20 - Strange identities §4. The topology of the functions Arg(z) and Log zIndex
插图：The concept of a set10 is a primitive concept in mathematics; one can no moreprovide a definition than Euclid could define mathematically what a point is.In my youth there were those who said that a set is "a collection of objects ofthe same nature"; apart from the vicious circle （what indeed is a "collection" ？a set？）, to talk of "nature" is empty and means nothing11. Certain denigratorsof the introduction of "modern math" into elementary education have beenscandalised to see that in some textbooks they have had the temerity to formthe union of a set of apples with a set of pears; never mind that a normalchild will tell you that this gives a set of fruits, or even of things, and if askedto count the number of elements of the union any moderately intelligent childcan explain to you that it does not matter that the first set consists of applesrather than oranges and the second of pears rather than dessert spoons; thefact that the Louvre Museum combines disparate collections - of pictures,sculptures, ceramics, gold work, mummies, etc. - has never troubled anyone.One calls this: to acquire the sense of abstraction.The logicians have in any case long since invented a radical method ofeliminating questions concerning the "nature" of mathematical objects orsets （the two terms are synonymous）. One can describe this in a figurativeway by saying that a set is a "primary" box containing "secondary" boxes,its elements, no two of which have identical contents, which in their turncontain "tertiary" boxes themselves containing... The Louvre is a collectionof collections （of paintings, sculptures, etc.）, the collection of paintings isitself a collection of paintings stolen by Bonaparte, Monge and Berthollet inItaly （we unfortunately had to return it in 1815）, bequeathed by ... privatecollectors, bought at sales, etc.
《分析1(影印版)》：天元基金影印数学丛书
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有的书内容很多，结果每个主题讲得都不深入；有的书涵盖的内容并不广，但是每个主题都讲得很透彻。这本书属于后一种。刚读了第一章，发现和所有其他分析的书都不一样，讲得非常深入，非常有道理，也非常有趣，如同看小说或与某人聊天，比如你会在这本书中发现这样的玩笑：每个年过五百岁的老头都有夜夜梅开三度的本事 － 这用来说明，你可以赋予零集中的元素任何性质而不会有逻辑上的错误－也许只有法国人才会写出如此浪漫的数学书。总的来说，本书观点很高，又绝对不失严谨，强烈推荐。