第一部分第一部分 数学分析数学分析PartPart 1 1 Mathematical Analysis引言引言 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(D’Alembert，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),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δ)δ,x(x00)(xf))(,)((00xfxf广泛使用的语言.““因为分析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)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 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 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 Geometria 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 France 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, many mathematicians tried to remedy its loose foundation, among whom were the French mathematician D’Alember(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. 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 integrated 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 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 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. Un。