关于辐角改变量算法的两点注记

Two notes on the methods to calculate the argument increment

确定多值函数的单值分支是复分析的教学难点之一,辐角改变量法是解决单值分支问题的主要方法。对于含有z-a的函数的辐角改变量,现行教材多采用平移坐标原点的"间接辐角改变量法"计算;对于含有a-z的函数的辐角改变量,则借助公式Δ_Carg(a-z)=Δ_Carg(z-a)转成"间接辐角改变量法"计算。文章通过构造反例表明,上述算法及公式均存在误区,即在考虑到割线的因素时,上述方法与公式未必成立。同时,分析了"直接辐角改变量法"与"间接辐角改变量法"的本质区别,得到"间接辐角改变量法"及上述公式成立的条件。作为应用,给出不能使用"间接辐角改变量法"计算单值分支的实例。上述注记与实例将有效地克服相关的教学难点。

How to determine a single-valued branch of multi-valued functions, which is a teaching difficult point in complex analysis, can be solved by the method to calculate the argument increment. When the expression of a multi-valued function has the factor z- a, the argument increment of arg（z- a） can be derived in the present textbooks by the indirect method of translation of origin of coordinates. When the expression of a multi-valued function has the factor z- a, the argument increment of arg（z- a） can be derived by the indirect method of calculation of argument increment by means of the formula Δ_Carg（a- z）= Δ_Carg（z- a）. By counterexamples, it is illustrated in this paper that there are still misunderstandings in the indirect method and the above formula for the calculation of a single-valued branch, namely, they may not be right when one considers the branch line factor. Meanwhile we analyze the essential difference of both indirect method and direct method for the calculation of argument increment, and establish two important conditions which make the indirect method and above formula hold.As an application, an example in which the indirect method does not hold is given. These notes and examples will overcome the difficulty in teaching effectively.