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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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The curve will have to be contained in the intersection of two level sets of harmonic functions. Hence, it must be an analytic curve.

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.

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