分析Ⅰ（影印版）

前言

為了更好地借鑒國外數(shù)學(xué)教育與研究的成功經(jīng)驗，促進(jìn)我國數(shù)學(xué)教育與研究事業(yè)的發(fā)展，提高高等學(xué)校數(shù)學(xué)教育教學(xué)質(zhì)量，本著“為我國熱愛數(shù)學(xué)的青年創(chuàng)造一個較好的學(xué)習(xí)數(shù)學(xué)的環(huán)境”這一宗旨，天元基金贊助出版“天元基金影印數(shù)學(xué)叢書”。該叢書主要包含國外反映近代數(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，天元基金邀請國內(nèi)各個方向的知名數(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。為了提高我國數(shù)學(xué)研究生教學(xué)的水平，暫把選書的目標(biāo)確定在研究生教材上。當(dāng)然，有的書也可作為高年級本科生教材或參考書，有的書則介于研究生教材與專著之間。歡迎各方專家、讀者對本叢書的選題、印刷、銷售等工作提出批評和建議。

內(nèi)容概要

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級數(shù)和Fourier積分以及可以通過Fourier級數(shù)解釋的Weierstrass的解析函數(shù)理論。    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)實際、歷史背景結(jié)合在一起，對主要結(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、方法和推演過程。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。

作者簡介

作者：（法國）戈德門特（Roger Godement）

書籍目錄

PrefaceI - 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 logiciansII - 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 WeierstrassIII- 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 functionsAppendix 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 spacesIV - 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 zIndex

章節(jié)摘錄

插圖：The concept of a set10 is a primitive concept in mathematics; one can no moreprovide a definition than Euclid could define mathematically what a point is.In my youth there were those who said that a set is "a collection of objects ofthe same nature"; apart from the vicious circle （what indeed is a "collection" ？a set？）, to talk of "nature" is empty and means nothing11. Certain denigratorsof the introduction of "modern math" into elementary education have beenscandalised to see that in some textbooks they have had the temerity to formthe union of a set of apples with a set of pears; never mind that a normalchild will tell you that this gives a set of fruits, or even of things, and if askedto count the number of elements of the union any moderately intelligent childcan explain to you that it does not matter that the first set consists of applesrather than oranges and the second of pears rather than dessert spoons; thefact that the Louvre Museum combines disparate collections - of pictures,sculptures, ceramics, gold work, mummies, etc. - has never troubled anyone.One calls this: to acquire the sense of abstraction.The logicians have in any case long since invented a radical method ofeliminating questions concerning the "nature" of mathematical objects orsets （the two terms are synonymous）. One can describe this in a figurativeway by saying that a set is a "primary" box containing "secondary" boxes,its elements, no two of which have identical contents, which in their turncontain "tertiary" boxes themselves containing... The Louvre is a collectionof collections （of paintings, sculptures, etc.）, the collection of paintings isitself a collection of paintings stolen by Bonaparte, Monge and Berthollet inItaly （we unfortunately had to return it in 1815）, bequeathed by ... privatecollectors, bought at sales, etc.

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