《分析1》是2009年高等教育出版社出版的圖書，作者是戈德門特(Roger Godement)。
基本介紹
- 中文名：分析1
- 外文名：Analysis I
- 作者：戈德門特(Roger Godement)
- 語言：英語
- 出版時間：2009年12月1日
- 出版社：高等教育出版社
- 頁數：430 頁
- ISBN：9787040279559
- 定價：34.20
- 開本：16 開
內容簡介,圖書目錄,作者簡介,
內容簡介
《分析1(影印版)》：天元基金影印數學叢書
《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。
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
作者簡介
作者：（法國）戈德門特（Roger Godement）
