We apply domain functionals to study the conformal capacity of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.

翻译：我们运用域功能研究冷凝器(G,E)的符合能力,即$G美元是复杂平面上一个简单的连接域,$E是美元的一个紧凑子集。由于前后不一,我们的主要工具是双曲几何和功能,如双曲周界($E$)。基于采用快速多极方法的新计算算法与分析技术相结合。例如,在整个过程中,利用计算实验来显示既定不平等的清晰度。对于已知分析解决方案的模型问题,则观测到非常精确的计算。