天元基金影印数学丛书：分析1（影印版） [AnalysisⅠ] 下载 mobi epub pdf 电子书 2024

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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

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

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比想象中好，比较清晰，能看

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给别人买的给别人买的

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美国南方7个州的代表在亚拉巴马州蒙哥马利召开会议，共同組成美利坚联盟国。

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1861年

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分析学中最古老和最基本的部分是数学分析。它是在17世纪为了解决当时生产和科学提出的问题，经过许多数学家的努力，最终由牛顿和莱布尼茨(Leibniz)创立的。但是为分析建立严格逻辑基础的工作却迟至19世纪方才完成。此后，数学分析才成为一个完整的数学学科。数学分析是最早系统研究函数的学科，它所研究的虽说基本上只是一类性质相当好的函数——区间上的连续函数，但无论在理论上或应用方面至今都有重要意义。在理论方面，数学分析是分析学科的共同基础，也是它们的发源地。现代分析的诸多分支中，有一些在其发展初期曾经是数学分析的一部分(例如变分法、傅里叶分析以至复变函数论等)，而另一些则是在数学分析的完整体系建立以后，由于各种需要，在对数学分析中的某些问题的深入研究和拓广之中发展起来的，像实变函数论、泛函分析和流形上的分析就属于这种情况。勒贝格19世纪末到20世纪初，由于某些数学分支(例如傅里叶分析)和物理等学科发展的需要，不但促使数学分析中函数可积的概念逐步明确，还进一步要求将积分推广到更广的函数类上去，希望积分运算更加灵活方便。同时，在对数学分析中各个基本概念之间的关系的继续探讨中(例如，微分和积分互为逆运算在一般意义上是否成立)，人们也感到必须突破数学分析的限制。

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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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　　按前言所说，这是一本给高年级本科和研究生一年级用的书，在学过基本微积分后就可以阅读，在我们学校是用作数学分析1、2之后，实变函数之前的衔接课程的教材，只讲二分之一左右的内容。而我的学习体会是，我真不是个学数学的料，这是我看的第一本、希望也是最后一本分析学的书：）

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美国南方7个州的代表在亚拉巴马州蒙哥马利召开会议，共同組成美利坚联盟国。

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1945年

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