本书为《分析》第一卷。本书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。本书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。

本书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。

本书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。本书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。作者在本书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。

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 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