含零点弧的双解析函数在极点处的性质

The Properties of Bianalytic Functions with Zero Arc at a Pole

中文关键词:  含零点弧的双解析函数  极点  收敛于圆或直线  充分条件.

英文关键词:bianalytic functions with zero arc  pole  convergence to a circle or line  sufficient condition.

中文摘要:

      本文针对含零点弧的双解析函数$w(z)=\bar{z}\phi_1(z) \phi_2(z)$ 在极点$z=0$处的极限性质进行研究. 给出了存在以$z=0$为端点的弧$\gamma$, 使得$w(z)=0$对$\forall z\in\gamma\backslash\{0\}$成立的一些条件. 其次, 在此条件下证明了$w(z)$在$z\to 0$时的极限点集为圆或直线. 最后, 给出了两个数值例子验证了结果.

英文摘要:

      In this paper, the properties of bianalytic functions $w(z)=\bar{z}\phi_1(z) \phi_2(z)$ with zero arc at the pole $z=0$ are discussed. Some conditions under which there exists an arc $\gamma$, an end of which is $z=0$, such that $w(z)=0$ for $\forall z\in\gamma\backslash\{0\}$ are given. Secondly, that the limit set of $w(z)$ is a circle or line as $z\to 0$ is proved in this case. Finally, two numerical examples are given to illustrate our results.