Uniqueness under <hydrodynamic normalization at infinity> gives
$$
g_{rA}(z)=r\,g_A(z/r)
$$
and
$$
g_{A+x}(z)=x+g_A(z-x).
$$
Indeed, each right-hand side maps the required complement conformally onto $\mathbb H$ and has expansion $z+O(1/z)$. This is the <scaling and translation of half-plane capacity>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-203/1/b/solution