holomorphic extension of a function [closed]

hi,

I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{U} \rightarrow \mathbb{R}$. Can this function be holomorphically extended to $U$ (maybe if we shrink $U$) in a unique way? I would be very thankful for answers.

bruno

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closed as off-topic by Ricardo Andrade, Stefan Kohl, David White, Willie Wong, Boris BukhDec 2 '13 at 23:08

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Ricardo Andrade, Stefan Kohl, David White, Boris Bukh
If this question can be reworded to fit the rules in the help center, please edit the question.

Yes, of course, after we shrink $U$. A convergent Taylor series at a real point converges in some complex neighborhood of this point.
Added reply to your comment: you can apply identity theorem. Two real analytic functions coinciding on an open set of $R^n$ coincide in a complex neighborhood of this set.