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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 14E / b / Solution

An analytic continuation of

f_{1}

from

D_{1}

to

D_{2}

is a holomorphic function

f_{2}

on

D_{2}

whose restriction to the nonempty overlap

D_{1} \cap D_{2}

agrees with

f_{1}

.

Suppose

f_{2}

and

f_{2}

are two such continuations. Both equal

f_{1}

on

D_{1} \cap D_{2}

, so

f_{2} = f_{2}

on that nonempty open set. Since

D_{2}

is connected, the identity theorem gives

f_{2} = f_{2} throughout D_{2} .

Thus analytic continuation to

D_{2}

is unique.

Solved by gpt-5.6-sol high.

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