A conformal map from an upper-half-plane domain has hydrodynamic normalization when $g(z)-z\to0$ at infinity. For a compact H-hull this normalization makes its <mapping-out function of a compact H-hull> unique.

Hydrodynamic normalization at infinity

A <compact H-hull> is a bounded relatively closed set $A\subset\mathbb H$ for which $\mathbb H\setminus A$ is a <simply connected domain>. The <Riemann mapping theorem> and <hydrodynamic normalization at infinity> give a unique <mapping-out function of a compact H-hull> $g_A:\mathbb H\setminus A\to\mathbb H$ with
$$
g_A(z)=z+\frac{a}{z}+O(|z|^{-2}).
$$
The coefficient $a\geq0$ is the <half-plane capacity> $\operatorname{hcap}(A)$.

For $\lambda>0$ and $x\in\mathbb R$, uniqueness of the normalized map gives
$$
g_{\lambda A+x}(z)=x+\lambda g_A\!\left(\frac{z-x}{\lambda}\right).
$$
Substituting the expansion of $g_A$ yields
$$
g_{\lambda A+x}(z)=z+\frac{\lambda^2a}{z-x}+O(|z|^{-2})=z+\frac{\lambda^2a}{z}+O(|z|^{-2}).
$$
Therefore the <scaling and translation of half-plane capacity> is
$$
\operatorname{hcap}(\lambda A+x)=\lambda^2\operatorname{hcap}(A).
$$

Solution

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

{title2=$g_A$}

The mapping-out function is the unique <conformal map> $g_A:\mathbb H\setminus A\to\mathbb H$ with <hydrodynamic normalization at infinity>
$$
g_A(z)=z+\frac{a}{z}+O(|z|^{-2}).
$$

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