《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、歷史背景結合在一起，對主要結論常常給出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)》中較早地引入了一些較深的內容，如在第一卷中介紹了拓撲空間的概念，在第二卷中介紹了Lebesgue理論的基本定理和Weierstrass橢圓函式的構造。 《分析1(影印版)》第一卷的內容包括集合與函式、離散變數的收斂性、連續變數的收斂性、冪函式、指數函式與三角函式；第二卷的內容包括Fourier級數和Fourier積分以及可以通過Fourier級數解釋的Weierstrass的解析函式理論。
基本介紹
- 書名：分析1
- 作者：戈德門特(Roger Godement)
- 出版日期：2009年12月1日
- 語種：英語
- ISBN：9787040279559
- 品牌：高等教育出版社
- 外文名：Analysis I
- 出版社：高等教育出版社
- 頁數：430頁
- 開本：16
- 定價：34.20
基本介紹,內容簡介,作者簡介,圖書目錄,文摘,序言,
基本介紹
內容簡介
《分析1(影印版)》：天元基金影印數學叢書
作者簡介
作者：（法國）戈德門特（Roger Godement）
圖書目錄
Preface
I-Sets and Functions
1. Set Theory
1-Membership, equality, empty set
2-The set defined by a relation. Intersections and un. ions
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?
I-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 numbers
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 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 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 functions
Appendix 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 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-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 z
Index
I-Sets and Functions
1. Set Theory
1-Membership, equality, empty set
2-The set defined by a relation. Intersections and un. ions
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?
I-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 numbers
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 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 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 functions
Appendix 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 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-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 z
Index
文摘
插圖：
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.
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.
序言
為了更好地借鑑國外數學教育與研究的成功經驗，促進我國數學教育與研究事業的發展，提高高等學校數學教育教學質量，本著“為我國熱愛數學的青年創造一個較好的學習數學的環境”這一宗旨，天元基金贊助出版“天元基金影印數學叢書”。
該叢書主要包含國外反映近代數學發展的純數學與套用數學方面的優秀書籍，天元基金邀請國內各個方向的知名數學家參與選題的工作，經專家遴選、推薦，由高等教育出版社影印出版。為了提高我國數學研究生教學的水平，暫把選書的目標確定在研究生教材上。當然，有的書也可作為高年級本科生教材或參考書，有的書則介於研究生教材與專著之間。
歡迎各方專家、讀者對本叢書的選題、印刷、銷售等工作提出批評和建議。
該叢書主要包含國外反映近代數學發展的純數學與套用數學方面的優秀書籍，天元基金邀請國內各個方向的知名數學家參與選題的工作，經專家遴選、推薦，由高等教育出版社影印出版。為了提高我國數學研究生教學的水平，暫把選書的目標確定在研究生教材上。當然，有的書也可作為高年級本科生教材或參考書，有的書則介於研究生教材與專著之間。
歡迎各方專家、讀者對本叢書的選題、印刷、銷售等工作提出批評和建議。
