分析Ⅰ（影印版）

章节摘录

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.

前言

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。

内容概要

作者：（法国）戈德门特（Roger Godement）

书籍目录

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

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《分析1(影印版)》：天元基金影印数学丛书

作者简介

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。

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。

图书封面

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