# Holomorphic extension of function

Let $C$ be a simple closed curve in $\mathbb C$, does there exits a domain $\Omega$ containing $C$ and holomorphic function $f: \Omega \to \mathbb C$ such that $f(z)= \bar{z}$ on $C$.

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Assuming analyticity, you can parametrize the curve by the unit circle and extend this parametrization to a conformal map between neighborhoods. Now the problem is reduced to $C$ being the unit circle, and the prescribed values on the circle are given by a power series in terms of $z$, $\bar z$. Just replace $\bar z$ with $1/z$ and you have a holomorphic extension in the form of a Laurent series.