关于多值解析函数的教学研究

复变函数论中的多值函数教学是一个难点.钟玉泉先生的教材在这一难点的处理方面是较成功的.他通过例题,介绍多值函数分成单值分支的方法,介绍求函数值的方法,并对这些方法进行了总结,得出的结论是:当给定初值后,只有通过连续变化才能得到其它点的函数值.这一点和传统的代入法求函数值完全不同.然而该教材就在这一总结之后,又用代入法求Arcsin2.我们认为这自我否定了刚刚建立起来的求值方法,扰乱了读者的思想.本文通过对Arcsinz分成单值解析分支的讨论,对求Arcsin2提出了新的教材处理方案,以期和读者商榷.

On the Teaching Study of the Multiple-valued Analytic Functions

In teaching complex analysis,one always faces difficulties in explaining the multiple-value functions.In his textbook,Professor Zhong Yuquan is quite successful in handling this problem.He presents the methods of dividing multiple-valued functions into single-valued branch and calculate the values of the functions through some examples.He summarizes these methods,and concludes that: once an original value is given,one can only get the values at other points through continuous change of variables.This is completely different from the traditional substitution method.However,after the summary he still uses the substitution method to calculate Arcsin2.We think that this defeats his own method just established,and confuses the readers. In this paper,we would like to share with the readers an alternative elementary method to calculate Arcsin2,through a discussion on dividing Arcsinz into single-valued branch.