分析Ⅱ（影印版）

书籍目录

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 distributionAppendix to Chapter V - Introduction to the Lebesgue TheoryVI - 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 functionVII - 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 distributionsPostface. Science, technology, armsIndexTable of Contents of Volume I

编辑推荐

　　“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。

作者简介

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。

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。

图书封面

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