Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 203 / 1 / c / i

For $x\in \mathbb{R}$, the conformal mapmaps $\mathbb{H}\setminus (A+x)$ onto $\mathbb{H}$ and has hydrodynamic normalization at infinity. Its Laurent series is $h(z)=z+a/z+O(z^{-2})$. Uniqueness of the mapping-out function of a compact H-hull therefore givesThis is the translation part of scaling and translation of half-plane capacity.