复变函数z^(1/n)的多值性问题及其教学意义  

Multiple Value Problem of Complex Function z^(1/n) and Its Teaching Significance

作　　者：赵书霞 高磊 田宇[2] 王彦洁 ZHAO Shu- Xia;GAO Lei;TIAN Yu;WANG Yan-Jie(School of Physics and Opto-Electronic Engineering,Dalian University of Technology,Dalian,Liaoning 116023,China;Experiment Center,Dalian Institute of Science and Technology,Dalian,Liaoning 116052,China)

机构地区：[1]大连理工大学物理与光电工程学院,辽宁大连116023 [2]大连科技学院实验中心,辽宁大连116052

出　　处：《教育教学论坛》2017年第52期191-193,共3页Education And Teaching Forum

摘　　要：本文针对数学物理方法课程中复变函数z^(1/n)的多值性问题,采用两种方法证明了z^(1/n)只有n个不同的独立值,弥补了大多教材只给结论,忽略推倒的不足,帮助学生理清复变函数多值性与复数指数表示式中三角函数部分周期性的关系,为学生以后学习留数定理和孤立奇点等相关知识点扫清障碍,最后在笛卡尔坐标系里用矢量图像进一步展示了z^(1/n)的多值性.This paper is aimed at analyzing the multiple value property of complex function z n姨in the curriculum of mathematical physics methods.Two methods are devised to prove that totally n independent roots exist for the function z n姨,in order to make up the deficiency that most reference books just give this conclusion,but neglect the relevant proving process.This work can help students to clarify the relationship between the multiple value property of complex functions and the periodicity of trigonometric function part in the exponential expression of complex number.Meanwhile,it also paves the way for the students to learn new knowledge points,such as residue theorem,isolated singular points,and mathematical transformation etc.,in the future.Last,we still demonstrate this multi-value trait of z n姨function in the Cartesian coordinate systemby using vectors presentation of complex.

关 键 词：数学物理方法 复变函数 复数指数表达式 三角函数周期性 

分 类 号：O411.1[理学—理论物理]