《分析1（影印版）》第一卷的內容包括集合與函數(shù)、離散變量的收斂性、連續(xù)變量的收斂性、冪函數(shù)、指數(shù)函數(shù)與三角函數(shù)；第二卷的內容包括Fourier級數(shù)和Fourier積分以及可以通過Fourier級數(shù)解釋的Weierstrass的解析函數(shù)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(shù)十年的結晶，其目的是闡明分析是什么，它是如何發(fā)展的。《分析1（影印版）》非常巧妙地將嚴格的數(shù)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橢圓函數(shù)的構造。
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
書單推薦
