分析Ⅱ（影印版）

內(nèi)容概要

本書是作者在巴黎第七大學(xué)講授分析課程數(shù)十年的結(jié)晶，其目的是闡明分析是什么，它是如何發(fā)展的。本書非常巧妙地將嚴(yán)格的數(shù)學(xué)與教學(xué)實(shí)際、歷史背景結(jié)合在一起，對(duì)主要結(jié)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(guò)程了作者在本書中較早地引入了一些較深的內(nèi)容，如在第一卷中介紹了拓?fù)淇臻g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(gòu)造。    本書第一卷的內(nèi)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(nèi)容包括Fourier級(jí)數(shù)和Fourier積分以及可以通過(guò)Fourier級(jí)數(shù)解釋的Weierstrass的解析函數(shù)理論。

書籍目錄

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 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

編輯推薦

　　“天元基金影印數(shù)學(xué)叢書”主要包含國(guó)外反映近代數(shù)學(xué)發(fā)展的純數(shù)學(xué)與應(yīng)用數(shù)學(xué)方面的優(yōu)秀書籍，天元基金邀請(qǐng)國(guó)內(nèi)各個(gè)方向的知名數(shù)學(xué)家參與選題的工作，經(jīng)專家遴選、推薦，由高等教育出版社影印出版?！斗治觥芬粫谝痪淼膬?nèi)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(nèi)容包括Fourier級(jí)數(shù)和Fourier積分以及可以通過(guò)Fourier級(jí)數(shù)解釋的Weierstrass的解析函數(shù)理論?！斗治觥房勺鳛楦吣昙?jí)本科生教材或參考書。

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