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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

flag	asked Sep 22 at 13:44 bruno 29●4

Alexandre Eremenko · Accepted Answer · 2012-09-24 13:08:49Z UTC

6

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.

edited Sep 24 at 13:08

answered Sep 22 at 14:11

Alexandre Eremenko
10.6k●1●24●39

but how can one apply the identity theorem in tis case (in oreder to get uniqueness)? – bruno Sep 23 at 6:53

holomorphic extension of a function

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