Uniqueness Theorem of Algebroidal Functions in an Angular Domain

DOI：10.3770/j.issn:1000-341X.2010.03.016

 作者 单位 刘慧芳 江西师范大学数学与信息科学学院, 江西 南昌 3300271; 2. 华南师范大学数学科学学院, 广东 广州 510631 孙道椿 华南师范大学数学科学学院, 广东 广州 510631

设~$W(z)$和~$M(z)$分别为~$v$值和~$k$值代数体函数,$\triangle(\theta)$为~$W(z)$(或~$M(z)$)关于复数~$b$的无穷级(或~$\rho(r)$级)聚线.我们证明了若在角域~$\Omega(\theta-\delta,\theta \delta)$内有~$\overline{E}(a_j,W(z))=\overline{E}(a_j,M(z))(j=1,\cdots,2v 2k 1)$(其中$b, a_j(j=1,\cdots,2v 2k 1)$为复常数),则~$W(z)\equiv M(z)$.同时我们也得到当~$\triangle(\theta)$为~$W(z)$(或~$M(z)$)的无穷级(或~$\rho(r)$级)Borel方向时,该结论也成立.

Let $W(z)$ and $M(z)$ be $v$-valued and $k$-valued algebroidal functions respectively, $\triangle(\theta)$ be a $b$-cluster line of order $\infty$ (or $\rho(r)$) of $W(z)$ (or $M(z)$). It is shown that $W(z)\equiv M(z)$ provided $\overline{E}(a_j,W(z))=\overline{E}(a_j,M(z))~(j=1,\ldots,2v 2k 1)$ holds in the angular domain $\Omega(\theta-\delta,\theta \delta)$, where $b,a_j~(j=1,\ldots,2v 2k 1)$ are complex constants. The same results are obtained for the case that $\triangle(\theta)$ is a Borel direction of order $\infty$ (or $\rho(r)$) of $W(z)$ (or $M(z)$).