A <compact H-hull> is a bounded, relatively closed set $A\subset\mathbb H$ for which $\mathbb H\setminus A$ is <simply connected domain>. Its <mapping-out function of a compact H-hull> is the unique <conformal map> $g_A:\mathbb H\setminus A\to\mathbb H$ with <hydrodynamic normalization at infinity>
$$
g_A(z)=z+\frac{a}{z}+O(|z|^{-2}).
$$
The <half-plane capacity> is
$$
\boxed{\operatorname{hcap}(A)=a.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-203/1/a/solution