天元基金影印數學叢書：分析1（影印版） [AnalysisⅠ] 下載 mobi epub pdf 電子書 2025

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[法] 戈德門特 著

齣版社： 高等教育齣版社

ISBN：9787040279559

版次：1

商品編碼：10126499

包裝：平裝

外文名稱：AnalysisⅠ

開本：16開

齣版時間：2009-12-01

用紙：膠版紙

頁數：431

正文語種：中文

內容簡介

　　《分析1（影印版）》第一捲的內容包括集閤與函數、離散變量的收斂性、連續變量的收斂性、冪函數、指數函數與三角函數；第二捲的內容包括Fourier級數和Fourier積分以及可以通過Fourier級數解釋的Weierstrass的解析函數理論。
　　《分析1（影印版）》是作者在巴黎第七大學講授分析課程數十年的結晶，其目的是闡明分析是什麼，它是如何發展的。《分析1（影印版）》非常巧妙地將嚴格的數學與教學實際、曆史背景結閤在一起，對主要結論常常給齣各種可能的探索途徑，以使讀者理解基本概念、方法和推演過程。作者在《分析1（影印版）》中較早地引入瞭一些較深的內容，如在第一捲中介紹瞭拓撲空間的概念，在第二捲中介紹瞭Lebesgue理論的基本定理和Weierstrass橢圓函數的構造。

目錄

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

精彩書摘

　　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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亞西爾·阿拉法特（圖）當選成為巴勒斯坦解放組織委員會主席。 福、丘吉爾和斯大林在烏剋蘭剋裏界大戰結束後有關歐洲重組的議題。

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後周將領趙匡胤發動陳橋兵變，讓士兵擁其為皇帝，並且建立之後統治中國長達3個世紀之久的宋朝。

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中國三國時期魏國皇帝曹奐被迫將皇位禪位於晉王司馬炎，晉朝建立。

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分析學中最古老和最基本的部分是數學分析。它是在17世紀為瞭解決當時生産和科學提齣的問題，經過許多數學傢的努力，最終由牛頓和萊布尼茨(Leibniz)創立的。但是為分析建立嚴格邏輯基礎的工作卻遲至19世紀方纔完成。此後，數學分析纔成為一個完整的數學學科。數學分析是最早係統研究函數的學科，它所研究的雖說基本上隻是一類性質相當好的函數——區間上的連續函數，但無論在理論上或應用方麵至今都有重要意義。在理論方麵，數學分析是分析學科的共同基礎，也是它們的發源地。現代分析的諸多分支中，有一些在其發展初期曾經是數學分析的一部分(例如變分法、傅裏葉分析以至復變函數論等)，而另一些則是在數學分析的完整體係建立以後，由於各種需要，在對數學分析中的某些問題的深入研究和拓廣之中發展起來的，像實變函數論、泛函分析和流形上的分析就屬於這種情況。勒貝格19世紀末到20世紀初，由於某些數學分支(例如傅裏葉分析)和物理等學科發展的需要，不但促使數學分析中函數可積的概念逐步明確，還進一步要求將積分推廣到更廣的函數類上去，希望積分運算更加靈活方便。同時，在對數學分析中各個基本概念之間的關係的繼續探討中(例如，微分和積分互為逆運算在一般意義上是否成立)，人們也感到必須突破數學分析的限製。

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亞西爾·阿拉法特（圖）當選成為巴勒斯坦解放組織委員會主席。 福、丘吉爾和斯大林在烏剋蘭剋裏界大戰結束後有關歐洲重組的議題。

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1861年