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The question is whether it is possible to find two distinct harmonic functions on the unit disk $\mathbb{D}=\{z: \ |z|<1 \}$ such that they agree on $\mathbb{D} \cap \mathbb{R}$. If yes, please give an example.

The first thing that came in mind is that any harmonic function in a simply connected region is the real part of some analytic function, so I tried using identity theorem for analytic functions, but that did not lead me to anywhere. In fact, by the mean value property, for any harmonic function, the zeros are non-isolated.

Now I'm stuck and do not know how to proceed, any hints would be appreciated.

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$y$ and $0$ are harmonic functions on $\mathbb C$ (considered as $\mathbb R^2$ with coordinates $x, y$) that agree on the real axis. More generally, take the imaginary part of any analytic function that is real on the real axis.

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  • $\begingroup$ Ah, there was an error in my answer somewhere, nice. $\endgroup$ – James S. Cook Jul 29 '15 at 2:11

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