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I am self-studying complex analysis, and I am a little bit confused on notations.

Suppose that $f:U \to \mathbb C $ is a holomorphic function defined on an open subset of $\mathbb C^n $. I understand that every holomorphic extension of $f$ to a connected open set containing $U$ is uniquely determined by $f$. I understand also that there may be many maximal holomorphic extensions of $f$. However, sometimes I read things like "find THE domain of holomorphy of the locally-defined holomorphic function ..." and so on.

My question is: what is THE domain of analyticity of a function? If it is meant to be the domain of a maximal holomorphic extension on it, then why use the word "THE", since such maximal holomorphic extension is not unique? They refer to some particular maximal holomorphic extension?

Thank you.

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  • $\begingroup$ There are cases where there is a unique maximal domain $D\subset{\mathbb C}$ of holomorphy, e.g., if $f$ is holomorphic in the unit disk $D$ but cannot be extended over the boundary $\partial D$ at any point of $\partial D$. Consider $f(z):=\sum_{k\geq1} z^{k!}$. $\endgroup$
    – Christian Blatter
    Mar 9, 2016 at 12:28
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    $\begingroup$ @ChristianBlatter Thank you for the comment. However, the terminology I described is used also for functions that do not admit a unique maximal analytic extension. $\endgroup$
    – user321275
    Mar 9, 2016 at 13:05

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