【作者】 陆源；

【导师】 郭世荣；

【作者基本信息】 内蒙古师范大学 ， 科学技术史， 2009， 硕士

【摘要】 本文以原始文献分析法为研究方法,以研读阿贝尔的法文论文为主,着重考察18-19世纪椭圆积分向椭圆函数转变的过程。论文分为三章。第一章在18世纪微积分进一步发展的背景下,分析欧拉、勒让德、伯努利家族等数学家对椭圆积分进行的研究工作,指出勒让德将椭圆积分发展成三种标准表达形式以及欧拉、伯努利家族得到的椭圆积分加法定理,成为19世纪阿贝尔等人进一步研究椭圆积分和发展椭圆函数论的理论基础。第二章详细梳理阿贝尔建立和发展椭圆函数论的思想和方法,认为他在试图对椭圆积分进行求解无果的前提下,转变思想,采取与三角函数类比的方式,从反函数的角度入手,由椭圆积分诱导出椭圆函数的概念。另一方面,阿贝尔对R的阶大于4的进行积分,提出了比椭圆积分更广泛的阿贝尔积分,同时进一步推广欧拉的椭圆积分加法定理,得到了现在被称为“阿贝尔大定理”的加法定理,从而创建了定性分析阿贝尔积分的关键性定理.第三章通过分析雅可比、外尔斯特拉斯、黎曼等人在椭圆函数论方面的成绩,指出由于受到阿贝尔反演思想的影响,上述数学家给出了更一般、简捷的椭圆函数理论形式,并且反演了超椭圆积分,推动了近代复分析函数领域的成型与发展.文章回溯18-19世纪椭圆函数发展历程,指出阿贝尔在欧拉、勒让德、伯努利家族等数学家工作的基础上,创立椭圆函数理论,由此影响了19世纪特殊函数论的发展,对复变函数论,甚至是近代代数几何学的发展产生深远影响.更多还原

【Abstract】 This dissertation regards analytic approach of the primitive literature as the research approach and relies mainly on studying Abel’s French thesis carefully and investigates the course that the oval total mark transforms into oval function in the 18-19th century emphatically.The thesis is divided into three chapters. Chapter one, under the background of fact that calculus was further developed in the 18th century, analyzes Euler(1707-1783), Legendre (1752～1833), Bernoulli family mathematicians etc. Research works to elliptic integral points out that three standard expression forms Adrien-Marie Legendre has developed elliptic integral into and oval integration addition theory Euler and Bernoulli family has received have become theoretical foundation Abel etc. further studied elliptic integral and developed oval theory of function in the 19th century.Chapter two combs thought and method in detail with which Abel has set up and developed oval theory of functions . On the one hand ,I think Abel ,on the premise of attempting to solve elliptic integral and having no result, has changed the thought and taken analogy way with the trigonometric function and proceeded with angle of inverse function and induced the concept of oval function by elliptic integral. On the other hand, Abel has integrated whose rank is greater than 4 and put forward Abel integral more extensive than elliptic integral and further popularized Euler’s elliptic integral addition theorem at the same time and received the addition theorem known as " Abel’s great theorem " now and thus established key theorem to qualitatively analysize Abel integral.Chapter three , by analyzing Jacobi(1804-1851), Weierstrass, Riemann etc.’s achievements in oval theory of functions, points out ,because of influence by inversion of Abel’s thinking, mathematicians described above have given more general, more simple oval function theory form and performed the ultra oval total mark instead and promoted shaping and development of Complex Analysis function field in modern times.This article tracks back the development course of oval function in the 18-19 century and points out Abel, based on the works of Euler, Legendre, Bernoulli family mathematicians have founded the theory of elliptic functions and thus influenced the development of the special functions theory in the 19th century and has great influence on the development of Complex function theory and even modern algebra geometry .更多还原