holomorphic extension of a function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:48:00Z http://mathoverflow.net/feeds/question/107839 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107839/holomorphic-extension-of-a-function holomorphic extension of a function bruno 2012-09-22T13:44:09Z 2012-09-24T13:08:49Z <p>hi,</p> <p>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.</p> <p>bruno</p> http://mathoverflow.net/questions/107839/holomorphic-extension-of-a-function/107842#107842 Answer by Alexandre Eremenko for holomorphic extension of a function Alexandre Eremenko 2012-09-22T14:11:11Z 2012-09-24T13:08:49Z <p>Yes, of course, after we shrink $U$. A convergent Taylor series at a real point converges in some complex neighborhood of this point.</p> <p>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.</p>