1、第一部分 数学分析Part 1 Mathematical Analysis引言 数学分析的萌芽,发生与发展,经历了一个漫长的时期.萌芽时期是从古希腊数学家欧多克斯(Eudoxus，约公元前 408-355)提出穷竭法和阿基米德(公元前 287-212)用穷竭法求出抛物线弓形的面积开始.公元 263 年，刘徽为九章算术作注时提出“割圆术”，以及 1328 年英国大主教布兰德.瓦丁(Bradwardine,1290-1349) 在牛津发表著作中给出类似于均匀变化率和非均匀变化率的概念，都是极限思想的成功运用.到 16 世纪中叶,数学分析正式进入了酝酿阶段.其中有两部著作当时有很大的影响：一是德国数
2、学家开普勒(Kepler 1575-1630)的新空间几何,另一部是意大利数学家卡瓦列利 (Covalieri, 1598-1647)的 不可分量几何.十七世纪上半叶开始到中叶是数学分析的奠基性工作时期.主要先驱有法国的帕斯卡(pascal,1623-1662)和费尔马(Fermat,1601-1665),英国的沃利斯 (Wallis,1616-1703)和巴罗(Barrow,1636-1677).十七世纪下半叶,牛顿(Newton,1642-1727)和莱布尼兹在总结前人工作的基础上给出了微积分.微积分诞生以后,曾就它是否严密及基础是否稳固爆发过一场大的争论.为此有许多数学家企图弥补出现的不
3、严密性，如英国数学家麦克劳林(Maclaurin,1698-1746),泰勒(1685-1731), 法国数学家达朗贝尔 (DAlembert，1717-1783).其中, 达朗贝尔曾试图将微积分的基础归结为极限，但遗憾的是,他并未沿着这条路走到底.与此同时,许多数学家在不严密的基础上对微积分创立了许多辉煌的成就.如瑞典数学家欧拉(Euler,1707-1783)以微积分为工具解决了大量的天文、物力、力学等问题,开创了微分方程、无穷级数、变分学等诸多新学科.1748 年出版了无穷小分析引理一书，这是世界上第一本完整的有系统的分析学.还有法国数学家拉格朗日(Lagrange ，1736-1833
4、),拉普拉斯(Laplace,1749-1827),勒让德(Legendre,1752-1813),傅立叶(Fourier,1768-1830)等在分析学方面都作了重大的贡献.但在微积分基础上仍没有找到解决的办法.进入十九世纪以后,分析学的不严密性到了非解决不可的地步.但那时还没有变量,极限的严格定义.不知道什么是连续，不知道什么是级数的收敛性.定积分的存在性都是含第一部分 数学分析2糊不清的,这可从挪威数学家阿贝尔(Abel N.H,1802-1829)在 1826 年所说的:“在高等分析中仅有很少几个定理是用逻辑上站得住脚的形式证明,人们到处发现从特殊跳到一般的不可靠的推论方法”这句话中看
5、出,为了解决分析的严密性问题,奥地利数学家波儿察诺(Bolzano,1781-1848),阿贝尔和柯西(1789-1857) 作了大量的工作.1821 年,法国理工大学教授柯西写了分析教程一书,将分析学奠定在极限的概念之上,把纷乱的概念理出了一个头绪.但是它的叙述仍然使用“无限趋近”之类的语言,仍不是严格的.因此遭到了一些数学家的反对,法国数学家维尔斯图拉斯(Veierstrass.k,1861-1897)就是其中之一.他认为变量无非是一个字母,用来表示区间的数.这一想法导致了变量 在x取值时, 在 取值的新方法.由此得到了如今),x(0)(xf )(,)00xff广泛使用的 语言.“因为分析
6、学使用的工具是极限,而极限又要用到实数.因此,分析学的严密性是建立在实数理论基础上的.而在这方面,柯西、法国数学家梅莱(Meray，1835-1911)、法国数学家海涅(1821-1881) 、德国数学家康托 (Contor,1845-1918)、戴德金(Dedekind， 1831-1916)等都为建立实数理论做出了贡献.十九世纪后半叶,数学分析在理论上有了很大进展,1870 年海涅提出了一致连续的概念.1895 年法国数学家波莱尔( Borel,1871-1956)给出了有限覆盖定理.1872 年维尔斯图拉斯给出了处处连续而不可微的例子.德国数学家黎曼(Riemann,1826-1866)
7、和法国数学家达布(Darboux,J.G,1842-1917) 分别于 1854 年和 1885 年给出了有界函数，可积性的定义和充要条件.这些概念和例子构成了现今数学分析教科书的主要内容.现在数学分析已根植于自然科学和社会科学的各学科分支之中.微积分作为数学分析的基础,不仅要为全部数学方法和算法工具提供方法论,同时还要为人们灌输逻辑思维方法. 目前数学分析的主要内容已是高校数学专业必修课和理工管等学科的基础课.数 学 分 析 已 形 成 四 大 块 结 构 :分 析 引 论 、 微 分 学 、 积 分 学 、 无 穷 级 数 与 广 义 积 分 .数学 分 析 的 立 论 数 域 是 实 数
8、 连 续 统 ,研 究 的 主 要 对 象 是 函 数 ,研 究 问 题 使 用 的 主 要 工 具 是 极限 .IntroductionThe germination, appearance and development of Mathematical Analysis went through a long period. The germination period started when the ancient Greek mathematician Eudoxus (about 408-355BC) put forward the method of exhaustion, by
9、 which Archimedes worked out the area of parabolic segment of a circle. The idea of limits is well put into: in 263BC , Liu Hui raised “Cyclotomic Method” in his work Nine Chapters of Mathmatical Art; 引言3In 1328, the British archbishop Bradwardine (1290-1349) gave the definition for average rate of
10、change and rate of change in his book published in Oxford. By the middle of the 16th century, the preparing period of Mathematical Analysis really started. Two famous works made great influence at that time. One was New Space Geometry by German mathematician Kepler (1575-1630) , another was Geometri
11、a Indivisibilibus Continuorum Nova Quadam Ratione Promoto by an Italian mathematician Covalierieri (1598-1647).Great foundation of Mathematical Analysis had been laid from the early 17th century into the middle of 17th century. Among the pioneers were Pascal(1623-1662) and Fermat(1601-1665) from Fra
12、nce Wallis(1616-1703) and Barrow(1636-1677) from UK.In the late 17th century, Newton (1642-1727) and Leibnitz founded Calculus based on the works of early mathematicians. Right after its birth, there was a heated debate over whether it was logically strict and fundamentally stable. Consequently, man
13、y mathematicians tried to remedy its loose foundation, among whom were the French mathematician DAlember(1717-1783) , who once tried to define the base of calculus to limit , but to our regret, abandoned the idea halfway.Meanwhile, many mathematicians had made great achievement on the loose calculus
14、. For example, Sweden mathematician Euler(1707-1783) , by using calculus as a tool, solved many problems in the fields of astronomy, physics and mechanics,and also founded many new subjects such as differential equation, infinite series and calculus of variations. And the first systematically integr
15、ated book on analysis, The Infinitesimal Analysis, was published in 1748. French mathematician Lagrange(1736-1833)， Laplace(1749-1827), Legendre (1752-1813), Fourier (1768-1830) also contributed a lot to Mathematical Analysis. But no efficient solution to the loose base of Mathematical Analysis had
16、been found.Stepping into the 19th century, the loose foundation of the Mathematical Analysis came up to the degree that it had to be solved. But there were no strict definition for limits, and the terms such as continuity and the convergence of series were unknown. The existence of definite integral
17、 is still not definite, which can be seen from the statement of the Norwegian mathematician Abel N.H.(1802-1829) in 1826, “only few proofs of the theorems in advanced analysis can logically hold water. Unreliable reasoning methods drawing conclusions of general cases from special ones can be found e
18、verywhere”. In order to solve the loose foundation of Mathematical Analysis, Austria mathematician Bolzano (1781-1848), Abel and Cauchy did great amount of work. In 1821, Prof. Cauchy of the Science and Engineering university in France wrote the book Analysis Course , in which Mathematical Analysis
19、was 第一部分 数学分析4defined on the concept of limit, thus got a major line out of the disorderly numerous concepts. But the language of his statement was still not strict enough to avoid the expressions such as “approach infinitely”, thus met the opposition of some mathematicians , among whom was the Fren
20、ch Veierstrass.K (1861-0897) who believed that the variable was not more than a letter, which is used to represent the number in an interval . This idea resulted in the new method that if belongs to the interval , then must be a number of x ),(0x)(xfthe interval . Hence todays widely used language c
21、ame )(,)(00xff “into being .Since the tool of Mathematical Analysis is limit , which is related to real numbers, the strictness of Mathematical Analysis is based on the real number theory, in which aspect, French Mathematician Meray (1835-1911),German mathematician Contor(1845-1918), Dedekind(1831-1
22、916) all made great contribution to the foundation of real number theory.In the late 19th century , Mathematical Analysis developed quickly theoretically: Heine put up the concept of uniform continuity. In 1895, Borel (1871-1956) gave the theorem of finite covering; In 1872, Weierstrass gave a funct
23、ion which is continuous at every point but not differentiable. German mathematician Riemann (1826-1866) and French mathematician Darboux J.G.(1842-1917) gave the definition of bounded function, integrability and its necessary and sufficient conditions respectively in 1854 and 1885. All of these made
24、 up the major content of Mathematical Analysis nowadays. At present, Mathematical Analysis is rooted in different subjects of natural science and social science. Calculus, as the base of Mathematical Analysis , not only supplies all mathematical methods and algorithms with methodology, but also cult
25、ivate peoples thinking mode. Presently, the major content of Mathematical Analysis has already become the compulsory course for math majors, and the selective course for science, engineering and management majors.Mathematical Analysis has formed a structure comprised of four major parts: differentia
26、ls, integrals, infinite series and generalized integrals. Mathematical Analysis is founded on the continuum of real numbers, the subject for study in Mathematical Analysis is function.The major research tool in Mathematical Analysis is a limit . 第一章 实数集与函数Chapter 1 Set of the Real Numbers and Functi
27、ons初等数学中研究的主要对象基本上是常量，而在数学分析中我们研究的是变量.第一章 实数集与函数5变量的变化范围是实数集.变量之间的对应关系是函数.本章我们将介绍实数、函数、复合函数、初等函数的基本概念及它们的一些性质.The main object investigated in elementary mathematics is constant quantities, while it is variables that we investigate in mathematical analysis, changeable domain of variable is the set of
28、 real numbers. The correspondent relation between variables is called function. In this chapter, we will introduce some fundamental notions such as real number, functions, composite functions, elementary functions and some of their properties.单词和短语 Words and expressions实数及其性质 real number and its pro
29、perties有理数 rational numbers 无理数 irrational numbers 定义 definition 命题 proposition 加 plus 减 minus 乘 multiplied by； times 除 over; is to; divided by绝对值与不等式 absolute value and inequality 三角不等式 the triangle inequality反三角不等式 the reverse triangle inequality贝努利不等式 Bernoullis inequality确界原理 principles of supre
30、mum and infimum数集 set of the numbers区间 interval 开区间 open interval 闭区间 closed interval 半开区间 semi-open interval半闭区间 semi-closed interval有限区间 finite interval邻域 neighborhood去心邻域 deleted neighborhood 和 sum差 difference积 product 商 quotient 数轴 number axis; number line封闭性 closeness 稠密性 the density无限区间 infini
31、te interval 点的 -右（左）邻域 the right（left）-hand -neighborhood of a point 有界集 bounded set无界集 unbounded set 下确界 infimum (or greatest lower bound)有序完备集 order complete set有序的 ordered阿基米德性质 Archimedes property实数的完备性 completeness of real numbers 全序域 complete ordered field 完备性定理 completeness axiom实数的阿基米德性质 Arc
32、himedean property of real numbers戴德金分割 Dedekind cut第一部分 数学分析6戴德金性质 Dedekind property 完备性的戴德金性质 Dedekinds form of completeness property 函数的定义 definition of function定义域 domain of definition 值域 region 自变量 independent variable 因变量 dependent variable 单调性 monotonicity初等函数 elementary function 函数四则运算 algebr
33、a of functions五种基本初等函数 five kinds of basic elementary functions 常量函数 constant function幂函数 power function指数函数 exponential function对数函数 logarithmic function三角函数 trigonometric function反三角函数 inverse trigonometric function 反函数 inverse function复合函数 composite function映射 mapping 象 image 原象 inverse image 分段函
34、数 piecewise function 符号函数 sign function 有界函数 bounded function无上界函数 unbounded upper function(or not bounded above function )单调函数 monotone function 单调增函数 monotone increasing function 奇（偶）函数 odd(even)function严格单调函数 strictly monotonefunction周期函数 periodic function 最小正周期 minimum positive period绝对值函数 absol
35、ute value function最大整数函数 greatest integer function恒等函数 identity function 多项式函数 polynomial function 线性函数 linear function 二次函数 quadratic function有理函数 rational function 双曲正弦 hyperbolic sine 双曲余弦 hyperbolic consine 三角恒等式 trigonometric identity奇偶恒等式 odd-even identity余函数恒等式 cofunction identity 毕达哥拉斯恒等式 Py
36、thagorean identity xctgx2222s1e1osin加法恒等式 addition identity 倍角恒等式 double-angle identity半角恒等式 half-angle identity 积恒等式 product identity 和恒等式 sum identity 2cossin2isnyxyxco基本概念和性质：Basic concepts and Properties第一章 实数集与函数71.非空实数集 称为有上界（下界）的，如果存在数 ，使得对一切 ，都S )(LMSx有 （ . 数 称为 的一个上界（下界）. Mx)L)(SA nonempty
37、set of real number is said to be bounded above provided that there is a number having the property that ( for all in . Such number (x)xis called an upper bounded (a lower bounded) for . ) S2.集称为有界的，如果集既有上界又有下界. A set is said to be bounded if it is bounded above as well as below.3.如果集 的所有上界集合有最小元 ，则
38、称为集 的上确界（或最小上界）. SMIf the set of all upper bounds of a set has the smallest members, say , then is SMcalled the supremum (or the least upper bound) of .S4.集 的上确界 有下列二个性质：(i) 是集 的上界，即对任意 ,有Sx(ii)设有比 小的数是 的上界，即对 ， ， ，使得Mx0y. yThe supremum M of a set S has the following two properties: (i) is the upper
39、 bound of ,i.e. . (ii)No numbers less than can be an upper bound i.e. for any positiveSxMnumber , however small, a number ，such that . Sy本章重点：因为在数学分析中一元函数微积分讨论问题的范围是实数. 而函数是数学分析研究的主要对象. 所以对于实数与函数必须掌握如下几点：1 为什么要学习实数？2 为什么要引入确界的概念？3 为什么要学习绝对值不等式？4 何谓函数？怎样确定函数的定义域？何谓函数的值域？5 映射与函数的区别是什么？6 何谓初等函数？7 掌握复合函
40、数概念，会将复合函数“分解”为基本的初等函数.Key points of this chapter:Because the scope for problem discussed in one-variable calculous is real numbers, but the main object investigated in mathematical analysis is function, thus, the following points must be mastered as for real numbers and functions:1Why are real numb
41、ers studied?2Why is the notion of supremum and infimum introduced?第一部分 数学分析83Why is absolute value inequality studied?4What is a function? How to determine the domain of definition of a function? What is the region of a function?5What is the difference between mapping and functions?6What is an eleme
42、ntary function?7Master the notion of composite functions and the method to “decompose” composite functions into basic elementary functions.第二章 数列极限Chapter 2 Limits of Sequences因为数学分析中研究问题的主要工具是极限，而实数序列是最简单也是最重要的函数之一. 事实上,一般函数的许多性质都能由所了解的数列得到. 所以在本章中我们将研究实数列的极限、收敛序列的性质、收敛序列的运算法则、数列极限存在的判别准则等.The majo
43、r research tool in Mathematical Analysis is limit; while sequences of real numbers are the most simple, but one of the most important functions. In fact, properties of general functions can be deduced from the understanding of sequences. Accordingly, in this chapter, we will study the limit of seque
44、nces of real numbers, properties of convergent sequence, operational rules of limits of sequences, criteria of existence of limits of sequences and so on .单词和短语 Words and expressions数列极限 the limits of sequences发散序列 divergent sequence 无穷小序列 infinitesimal sequence 收敛序列 convergent sequence唯一性定理 uniquen
45、ess theorem有界性定理 boundedness theorem保号性 inheriting order properties保不等式性 inheriting inequality子列 subsequence 严格递增 strictly increasing单调递增序列 monotone increasing sequence单调递减序列 monotone decreasing sequence 严格递减 strictly decreasing必要条件 necessary condition充分条件 sufficient condition夹逼定理 squeeze principle
46、基本概念和性质：Basic concepts and Properties 1. 收敛数列的和收敛于极限的和. 第二章 数列极限9The sum of convergent sequences convergences to the sum of the limits.2. 收敛数列的积收敛于极限的积. The product of convergent sequences convergences to the product of the limits.3. 收敛数列的商收敛于极限的商(分母的极限不为零) . The quotient of convergent sequences con
47、vergences to the quotient of the limits(provided that the limit of the denominator is not equal to zero).4定义域为全体正整数集 ,值域为实数集的函数称为数列，记为 或NRNf:， . )(nfA function whose domain is the set of positive integer numbers and range a set of real numbers is called a real sequence . Thus a real sequence is deno
48、ted symbolically asor ， . RNf:)(nf5设 为数列， 为常数. 若对任给的正数 ，总存在正整数 ，使得当n 0时，有 ，则称数列 收敛于 ，定数 称为数列 的极| nana限，并记作或 . anlim)(nLet be a sequence and be a constant number, is said to be converge to naand is called the limit of if for every ,there exists n0a positive integer , such that ,for all natural numbers ,and N|n Nis denoted by or . nli )(6单调序列收敛当且仅当数列有界. A monotone sequence converges if and only if it is bounded.本章重点：极限是研究数学分析的主要工具.故在数学分析中处于十分重要的地位.要求掌握：1 学会用数学语言描述极限2 深刻理解和熟练书写数列收敛和发散的 定义.“N3 用数列收敛定义证明下列数列的极
