Uniqueness requires proof - it relies on a theorem/lemma having to do with if an analytic function is constant in a region, then it is constant everywhere. Uniq

Uniqueness requires proof - it relies on a theorem/lemma having to do with if an analytic function is constant in a region, then it is constant everywhere. Uniqueness then arises as if there are two analytic continuations that are the same in that region, then the difference is 0 there, and thus 0 everywhere, resulting in the two being the same. That is a non-trivial theorem/lemma though, and the proof is typically gone through at a graduate complex analysis course.