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

Solution

Im g_{A} (z_{n}) \geq ϵ > 0

. The images cannot tend to infinity, because the inverse mapping-out function satisfies

g_{A}^{- 1} (w) = w + o (1)

at infinity while

z_{n}

remains bounded. A further subsequence therefore converges to some

w \in H

. Continuity of

g_{A}^{- 1}

inside

H

would give

z = g_{A}^{- 1} (w) \in H ∖ A,

contradicting

z \in A

. Thus

Im g_{A} (z_{n}) \to 0

Convergence of the real parts can fail. For the vertical slit

A = {i y : 0 < y \leq 1}, g_{A} (z) = z^{2} + 1,

the two sides of the slit at

z = i y

0 < y < 1

, map to the two boundary values

\pm 1 - y^{2}

. Alternating sequences approaching

i y

from the two sides make

g_{A} (z_{n})

alternate between neighborhoods of these distinct limits.

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