分析

V - Differential and Integral Calculus

1. The Riemann Integral

1 - Upper and lower integrals of a bounded function

2 - Elementary properties of integrals

3 - Riemann sums. The integral notation

4 - Uniform limits of integrable functions

5 - Application to Fourier series and to power series

2. Integrability Conditions

6 - The Borel-Lebesgue Theorem

7 - Integrability of regulated or continuous functions

8 - Uniform continuity and its consequences

9 - Differentiation and integration under the f sign

10 - Semicontinuous functions

11 - Integration of semicontinuous functions

3. The "Fundamental Theorem" （FT）

12 - The fundamental theorem of the differential and integral calculus

13 - Extension of the fundamental theorem to regulated functions

14 - Convex functions; Holder and Minkowski inequalities

4. Integration by parts

15 - Integration by parts

16 - The square wave Fourier series

17- Wallis formula

5. Taylors Formula

18 - Taylors Formula

6. The change of variable formula

19 - Change of variable in an integral

20 - Integration of rational fractions

7. Generalised Riemann integrals

21 - Convergent integrals： examples and definitions

22 - Absolutely convergent integrals

23 - Passage to the limit under the fsign

24 - Series and integrals

25 - Differentiation under the f sign

26 - Integration under the f sign

8. Approximation Theorems

27 - How to make C a function which is not

28 - Approximation by polynomials

29 - Functions having given derivatives at a point

9. Radon measures in R or C

30 - Radon measures on a compact set

31 - Measures on a locally compact set

32 - The Stieltjes construction

33 - Application to double integrals

10. Schwartz distributions

34 - Definition and examples

35 - Derivatives of a distribution

Appendix to Chapter V - Introduction to the Lebesgue Theory

VI - Asymptotic Analysis

1. Truncated expansions

1 - Comparison relations

2 - Rules of calculation

3 - Truncated expansions

4 - Truncated expansion of a quotient

5 - Gauss convergence criterion

6 - The hypergeometric series

7 - Asymptotic study of the equation xex = t

8 - Asymptotics of the roots of sin x log x = 1

9 - Keplers equation

10 - Asymptotics of the Bessel functions

2. Summation formulae

11 - Cavalieri and the sums 1k + 2k + ... + nk

12 - Jakob Bernoulli

13 - The power series for cot z

14 - Euler and the power series for arctan x

15 - Euler， Maclaurin and their summation formula

16 - The Euler-Maclaurin formula with remainder

17 - Calculating an integral by the trapezoidal rule

18 - The sum 1 + 1/2 ... + l/n， the infinite product for the F function， and Stirlings formula

19 - Analytic continuation of the zeta function

VII - Harmonic Analysis and Holomcrphic Functions

1 - Cauchys integral formula for a circle

1. Analysis on the unit circle

2 - Functions and measures on the unit circle

3 - Fourier coefficients

4 - Convolution product on

5 - Dirac sequences in T

2. Elementary theorems on Fourier series

6 - Absolutely convergent Fourier series

7 - Hilbertian calculations

8 - The Parseval-Bessel equality

9 - Fourier series of differentiable functions

10 - Distributions on

3. Dirichlets method

11 - Dirichlets theorem

12 - Fejers theorem

13 - Uniformly convergent Fourier series

4. Analytic and holomorphic functions

14 - Analyticity of the holomorphic functions

15 - The maximum principle

16 - Functions analytic in an annulus. Singular points. Meromorphic functions

17 - Periodic holomorphic functions

18 - The theorems of Liouville and dAlembert-Gauss

19 - Limits of holomorphic functions

20 - Infinite products of holomorphic functions

5. Harmonic functions and Fourier series

21 - Analytic functions defined by a Cauchy integral

22 - Poissons function

23 - Applications to Fourier series

24 - Harmonic functions

25 - Limits of harmonic functions

26 - The Dirichlet problem for a disc

6. From Fourier series to integrals

27 - The Poisson summation formula

28 - Jacobis theta function

29 - Fundamental formulae for the Fourier transform

30 - Extensions of the inversion formula

31 - The Fourier transform and differentiation

32 - Tempered distributions

Postface. Science， technology， arms

Index

Table of Contents of Volume I

《天元基金影印数学丛书：分析2（影印版）》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、方法和推演过程了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。

　　《天元基金影印数学丛书：分析2（影印版）》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。

“天元基金影印数学丛书”主要包含国外反映近代数学发展的纯数学与应用数学方面的优秀书籍，天元基金邀请国内各个方向的知名数学家参与选题的工作，经专家遴选、推荐，由高等教育出版社影印出版。《分析》一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。《分析》可作为高年级本科生教材或参考书。