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

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- Solution b

Solution

Uniqueness under hydrodynamic normalization at infinity gives

g_{r A} (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

H

and has expansion

z + O (1/ z)

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

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