= Solution
An <analytic continuation> of $f_1$ from $\mathcal D_1$ to $\mathcal D_2$ is a holomorphic function $f_2$ on $\mathcal D_2$ whose restriction to the nonempty overlap $\mathcal D_1\cap\mathcal D_2$ agrees with $f_1$.
Suppose $f_2$ and $\widetilde f_2$ are two such continuations. Both equal $f_1$ on $\mathcal D_1\cap\mathcal D_2$, so
$$
f_2=\widetilde f_2
$$
on that nonempty open set. Since $\mathcal D_2$ is <connected space>[connected], the <identity theorem> gives
$$
\boxed{f_2=\widetilde f_2\text{ throughout }\mathcal D_2}.
$$
\b[Thus analytic continuation to $\mathcal D_2$ is unique.]
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