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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-203/1/c/i/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-05

For

x \in R

, the conformal map

h (z) = x + g_{A} (z - x)

H ∖ (A + x)

onto

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 gives

hcap (A + x) = hcap (A) .

This is the translation part of scaling and translation of half-plane capacity.

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