《分析2(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2(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、方法和推演過程了作者在《分析2(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2(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2
- 作者：戈德門特(Godement R.)
- 出版日期：2009年12月1日
- 語種：簡體中文
- ISBN：9787040279542
- 外文名：Analysis Ⅱ
- 出版社：高等教育出版社
- 頁數：443頁
- 開本：16
- 品牌：高等教育出版社
基本介紹,內容簡介,作者簡介,圖書目錄,
基本介紹
內容簡介
“天元基金影印數學叢書”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。《分析》一書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。《分析》可作為高年級本科生教材或參考書。
作者簡介
作者：(美國)戈德門特（Godement R.）
圖書目錄
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. Taylor's Formula
18 - Taylor's 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 f sign
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 - Kepler's 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 Stirling's formula
19 - Analytic continuation of the zeta function
VII - Harmonic Analysis and Holomcrphic Functions
1 - Cauchy's 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. Dirichlet's method
11 - Dirichlet's theorem
12 - Fejer's 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 d'Alembert-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 - Poisson's 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 - Jacobi's 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
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. Taylor's Formula
18 - Taylor's 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 f sign
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 - Kepler's 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 Stirling's formula
19 - Analytic continuation of the zeta function
VII - Harmonic Analysis and Holomcrphic Functions
1 - Cauchy's 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. Dirichlet's method
11 - Dirichlet's theorem
12 - Fejer's 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 d'Alembert-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 - Poisson's 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 - Jacobi's 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
