Let $S_r=\{se^{i\theta}:0<s\leq r\}$. Since $S_r=rS_1$, the <scaling and translation of half-plane capacity> gives
$$
\operatorname{hcap}(S_r)=r^2\operatorname{hcap}(S_1)=ar^2.
$$
The <half-plane-capacity parameterization> requires $ar(t)^2=2t$, so
$$
r(t)=\sqrt{\frac{2t}{a}},
\qquad
\gamma(t)=\sqrt{\frac{2t}{a}}e^{i\theta}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-203/1/c/solution