从积不出来的积分到黎曼曲面理论

From Elliptic Integrals to the Theory of Riemann Surfaces

从介绍椭圆和双纽线弧长积分等积不出来的椭圆积分开始,给出数学家法尼亚诺和欧拉导出这些椭圆积分加法公式的计算过程,以及经典的阿贝尔定理的具体含义.讲解大数学家黎曼引入的黎曼曲面的重要概念,并且从几何的角度给出椭圆积分之所以积不出来的内在原因.介绍黎曼在黎曼曲面理论方面的重要工作及其对代数几何与现代数学发展的深远影响.

In this paper,we first introduce elliptic integrals in relation to the problems of finding the length of arcs of an ellipse or lemniscates,which could not be integrated in terms of the elementary functions,and follow by the processes of computation and the addition theorem made by Fagnano,Euler and Abel.We explain the meaning of the important notion of Riemann Surfaces and give the geometrical reason why elliptic integrals could not be integrated in terms of the elementary functions.Finally,we describe the important contributions of Riemann on the theory of Riemann Surfaces,which has a tremendous influence on the development of algebraic geometry and modern mathematics.