天元基金影印数学丛书：分析1（影印版） [AnalysisⅠ] pdf epub mobi txt 电子书 下载 2025

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[法] 戈德门特 著

出版社： 高等教育出版社

ISBN：9787040279559

版次：1

商品编码：10126499

包装：平装

外文名称：AnalysisⅠ

开本：16开

出版时间：2009-12-01

用纸：胶版纸

页数：431

正文语种：中文

内容简介

　　《分析1（影印版）》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。
　　《分析1（影印版）》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。《分析1（影印版）》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《分析1（影印版）》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。

目录

Preface
I - 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 logicians

II - 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 dAlembert
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 Weierstrass

III- 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 Cauchys
7 - The uniform metric
8 - Series of continuous functions. Normal convergence
§3. Bolzano-Weierstrass and Cauchys criterion
9 - Nested intervals， Bolzano-Weierstrass， compact sets
10 - Cauchys general convergence criterion
11 - Cauchys 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 functions
Appendix to Chapter III
1 - Cartesian spaces and general metric spaces
2 - Open and closed sets
3 - Limits and Cauchys 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 spaces

IV - 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 - Newtons binomial series
12 - The power series for the logarithm
13 - The exponential function as a limit
14 - Imaginary exponentials and trigonometric functions
15 - Eulers 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 z
Index

精彩书摘

　　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.

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相对于目前北美教材，欧洲大陆的比较传统、起点较高，这与“新数学运动”在美受挫而在法国等国最终基本坚持下来有关。国内需要这种风格的教材，起点高但又叙述平实生动，是分析方面很好的参考书。

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第一卷主要介绍了集合与收敛，有些有趣的故事。

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266年

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1945年

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本书可作为高等院校数学院系、应用数学系本科生的教材，对青年教师本书也是一部很好的教学参考书。数学系的一位怪老头曾经教导我们，看数学书，第一要注重的是definition，第二是theorem，最后才是application。按这位牛人的说法，数学书应该都写成词典的样子，无怪乎他会对Rudin的这本推崇有加。按俺一个物理系的外行看来这本书也是相当不错的，简洁明了毫无废话，可以一下子解决拓扑、微分形式、测度论的入门问题，事实上，对于物理的同学来说，这应该是在一个阶段内所能用到的数学知识的极限了。

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很好的一本书，值得购买，还要努力学习啊。1998年热带风暴赫敏是1998年大西洋飓风季期间第八个获得命名的热带气旋，由9月5日非洲西海岸涌现的一股东风波发展而成。这股东风波向西穿越大西洋后进入加勒比海西北部，并与其它天气系统产生相互影响。由此产生的新系统于9月17日在墨西哥湾中部被宣布成为热带低气压。风暴蜿蜒且缓慢地向北行进，之后升级为热带风暴并在路易斯安那州实现登陆，之后迅速弱化，于9月20日回复为热带低气压。虽然赫敏本身的破坏性不大，但其影响与其他热带气旋相结合，造成了一些农业上的损失。2月4日：斯里蘭卡獨立日

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中国三国时期魏国皇帝曹奂被迫将皇位禅位于晋王司马炎，晋朝建立。

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