论文标题
沃尔什（Walsh
Walsh's conformal map onto lemniscatic domains for polynomial pre-images I
论文作者
论文摘要
我们将沃尔什（Walsh）从紧凑型集合$ e \ subseteq \ mathbb {c} $的外观上考虑沃尔什（Walsh）的共形图。如果简单地连接$ e $，则允许的域是一个圆圈的外部，而如果$ e $具有多个组件，则Lemniscatic域是一般性lemaniseation的外部，由$ e $的对数能力和广义lemanisiscate的指数和中心确定。对于一般$ e $，我们根据$ e^c $的绿色功能来表征指数。在$ e $的其他对称条件下，我们还找到了Lemniscatic域中的中心。对于多项式预图像，$ e = p^{ - 1}（ω）$的简单连接的无限紧凑型套件$ω$，我们明确确定lemsiscation域中的指数，并得出一组方程式以确定lemsiscatic域的中心。最后，我们提出了几个示例，在这些例子中，我们明确地获取了Lemniscatic域的指数和中心以及保形图。
We consider Walsh's conformal map from the exterior of a compact set $E \subseteq \mathbb{C}$ onto a lemniscatic domain. If $E$ is simply connected, the lemniscatic domain is the exterior of a circle, while if $E$ has several components, the lemniscatic domain is the exterior of a generalized lemniscate and is determined by the logarithmic capacity of $E$ and by the exponents and centers of the generalized lemniscate. For general $E$, we characterize the exponents in terms of the Green's function of $E^c$. Under additional symmetry conditions on $E$, we also locate the centers of the lemniscatic domain. For polynomial pre-images $E = P^{-1}(Ω)$ of a simply-connected infinite compact set $Ω$, we explicitly determine the exponents in the lemniscatic domain and derive a set of equations to determine the centers of the lemniscatic domain. Finally, we present several examples where we explicitly obtain the exponents and centers of the lemniscatic domain, as well as the conformal map.
