# Analytic extension in several variables

Let $D \subset \mathbb{C}^{n}$ be an open domain and $K \subset D$ be a compact subset such that $D - K$ is simply connected. Let furthermore $f$ be a analytic function defined on $D-K$. Is it possible to extend $f$ to the whole $D$? If yes, by which theorem. If not, what further assumption do I need in order to make it possible ($D$ beeing a polydisk, some convergence theorems etc....)? Or doesn't it work at all?

bill

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I removed the tag functional-analysis, which did not fit the question. Welcome to Math.SE! – user53153 Jan 7 '13 at 14:53

It's enough to assume that $D\setminus K$ is connected. Look up Hartogs' extension theorem, which you can find in any textbook on several complex variables. Wikipedia's version
Of course, you need to assume that $n > 1$. If $n=1$, it's very easy to find counterexamples, e.g. $f(z) = 1/(z-p)$, where $p\in K$.)
yes but my function $f$ is not holomorphic? is the theorem also true for non-holomorphic but analytic functions? – bill Jan 7 '13 at 6:32
@AD. Fine, but nobody uses $\mathbb{C}^n$ when they want to include $n=1$.... – mrf Jan 7 '13 at 7:01