华东师范大学学报(自然科学版) ›› 2017, Vol. ›› Issue (4): 64-70.doi: 10.3969/j.issn.1000-5641.2017.04.006

• 数学 • 上一篇    下一篇

覆盖空间在多连通积分定理证明中的运用

胡光明1, 龙见仁2   

  1. 1. 北京航空航天大学 数学与系统科学学院, 北京 100191;
    2. 贵州师范大学 数学科学学院, 贵阳 550001
  • 收稿日期:2016-11-27 出版日期:2017-07-25 发布日期:2017-07-20
  • 通讯作者: 龙见仁,男,教授,研究方向为复分析.E-mail:longjianren2004@163.com. E-mail:longjianren2004@163.com
  • 作者简介:胡光明,男,博士研究生,研究方向为复分析及Teichmüller空间.E-mail:1186529024@qq.com.
  • 基金资助:
    贵州省科学技术基金(黔科合J字[2015]2112号);国家自然科学基金(11501142)

Application of the covering space in the complex integral of multiply connected domains

HU Guang-ming1, LONG Jian-ren2   

  1. 1. School of Mathematics and Systems Science, Beihang University, Beijing 100191, China;
    2. School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China
  • Received:2016-11-27 Online:2017-07-25 Published:2017-07-20

摘要: 众所周知,单连通区域上解析函数所确定的变上限积分是一个单值函数,然而对于多连通区域D上解析函数fz)的变上限积分Fz)=∫z0zfζFz)不仅依赖于zz0D内固定的一点),还依赖以下两点:(1)积分的路径;(2)函数fz)关于洞是否恰当.由此可以知道Fz)可能是一个多值函数.以上结果均可以在一般复变函数教材中找到,这里不再赘述.本文利用黎曼曲面的正则覆盖曲面知识,给出了解析函数fz)在多连通区域上积分的一种新诠释.

关键词: 多连通区域, 正则覆盖曲面, 覆盖变换群, 恰当微分, 交换群

Abstract: It is well known that the integral with variable upper limit of analytic function is a single value function in the simple connected domain, while the integral with variable upper limit of analytic function in the multiply connected domains is as following: z0zf(ζ), F(z) is not only dependent on the z (z0 is the fixed point in D), but also depends on the integral path and function f(z) being exact or not in every hole. Therefore F(z) is likely to be multiple valued function. In this paper, we give a new proof method about the integral of analytic function f(z) in the multiply connected domain by the regular covering surface.

Key words: multiply connected domains, regular covering surface, group of covering transformations, exact differential, Abelian group

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