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Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.

My question is, for what curves $\Gamma$ can we find a holomorphic function $G:\Omega\to\mathbb C$ with boundary values $G(z)\to e^{ik\alpha}$ as $z\to\alpha+if(\alpha)$ approaches $\Gamma$?

For example, does this work for $f(\alpha)=ce^{i\alpha}$, with $c$ small?

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  • $\begingroup$ I find the setup mildly confusing: initially $\alpha$ is just an abstract parameter, and then we're asked to "extend" to $\Omega$, as if $\alpha$ was taken from a subset of this region. $\endgroup$
    – Christian Remling
    Commented Nov 7, 2016 at 1:26
  • $\begingroup$ @ChristianRemling Sorry for the confusion. For an example maybe we can write $\Gamma$ as $\{\alpha+ce^{i\alpha}:\alpha\in \mathbb{R}\}$. Then $\Gamma$ is a graph. A point on $\Gamma$ is denoted by $z(\alpha)$. Define a function along $\Gamma$ by $G(z(\alpha))=e^{ik\alpha}$. Then $e^{ik\alpha}$ is boundary value of a holomorphic function in $\Omega$ means that there is a holomorphic function, still denote by $G$, such that $G(z)\rightarrow G(z(\alpha))$ as $z\rightarrow z(\alpha)$ for $z\in \Omega$, in appropriate sense. $\endgroup$
    – user54646
    Commented Nov 7, 2016 at 16:08
  • $\begingroup$ Thanks for the clarification. I've taken the liberty to edit your question along these lines, but feel free to return to the original version if you don't like this. (There is still the small discrepancy that the first two paragraphs suggest that $f$ should be real valued, but it isn't in your example.) $\endgroup$
    – Christian Remling
    Commented Nov 7, 2016 at 16:32
  • $\begingroup$ As for the actual question, I think you can just use the classical results on the unit disk since your region is conformally equivalent to the disk. $\endgroup$
    – Christian Remling
    Commented Nov 7, 2016 at 16:47

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