共焦曲线与解析函数

CONFOCAL CURVES AND ANALYTIC FUNCTIONS

一曲線的迷向切線的有限交點稱为這曲線的焦點,焦點可虛可實,本文中的焦點係指實焦點而言。作者前在平面曲線的焦點一文中曾證明:一曲線的焦點經過解析函數變換以後,一般情形,为其影曲線的焦點,本文的主要部分是証明:一平面中全部直線除兩種特殊直線外,經過解析函數變換以後變为共焦曲線組。

In the main part of this paper, we prove the following theorem: Let f (w) be a single-valued function analytic in any finite region of the w-plane except for isolated singular points. Under the transformation z= f (w), if the image of a straight line in the w-plane has a focus, then f' (w) has at least one zero. Conversely if f'(w) has zeros, then under this transformation, all the straight lines of the w-plane are transformed into coniocal curves with the images si the zeros of f' (w) as common foci. except for the following two sets of lines: (i) the lines passing through any zero of f' (w); (ii) the perpendicular bisector of the line joining any two zeros of f' (w). They are called special lines of the transformation.