Marley Meyers

## Answered question

2022-10-24

Is it true that if a holomorphic function in the unit disk converges uniformly to the 0 function some connected arc of the unit circle, this function is globally null?
If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)

### Answer & Explanation

Jimena Torres

Beginner2022-10-25Added 20 answers

Let the (open) arc on which the boundary values of $f$ vanish be $A$. Since the boundary values of $f$ on $A$ are real, by the Schwarz reflection principle we know that the function
$g\left(z\right)=\left\{\begin{array}{ll}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}f\left(z\right)& ,|z|<1\\ \phantom{\rule{1em}{0ex}}0& ,z\in A\\ \overline{f\left(1/\overline{z}\right)}& ,|z|>1\end{array}$
is holomorphic on the connected open set $\mathbb{D}\cup A\cup \left(\mathbb{C}\setminus \overline{\mathbb{D}}\right)$. Since $g$ vanishes on a non-discrete set, the identity theorem yields $g\equiv 0$, in particular $f\equiv 0$ follows.

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