Using the Schwarz reflection principle to show a function is real and constant Let $f$ be analytic and bounded on $U = \{z \in \mathbb{C} : \text{Re}(z) > 0 \text{ and } \arg(z) \in (-\frac{\pi}{6}, \frac{\pi}{6})\}$. Suppose for every $r > 0$ $\lim_{\theta \to \pm \frac{\pi}{6}} f(re^{i\theta}) = g(r) \in \mathbb{R}.$ I want to show that $f$ is real and constant. I want to use the Schwarz reflection principle to extend this function to the whole complex plane but am not sure how to do so. Would I rotate the wedge so that I could reflect over the real axis and repeat this process?
 A: Your suggestion of rotating and applying Schwarz reflection is a good idea. The tricky point here is tracking the symmetries of our extensions to make the final reflection work. Rotating and applying Schwarz gives us an analytic function $\tilde{f}$ defined and bounded on $ \Omega = \{re^{i\theta} \mid r > 0, \theta \in (-\pi/3, \pi/3)\}$ and so that:

*

*$\tilde{f}(re^{i\theta}) = f(re^{i(\theta - \pi/6})$ for $\theta \in (0, \pi/3)$,

*$\tilde f (\bar{z}) $ = $\overline{\tilde{f}(z)}$ for all $z$ in $\Omega $.

In particular, for any $r > 0$, $\lim_{\theta \to \pm \pi/3} \tilde{f}(re^{i\theta})$ is real. So we can once more rotate and reflect, getting a bounded analytic function $h$ defined on $\Omega' = \{ re^{i\theta} \mid r > 0, \theta \in (-2\pi/3, 2\pi/3)\}$ and so that:

*

*$h(re^{i\theta} =  \tilde{f}(re^{i(\theta -\pi/3)})$ for $\theta \in (0, 2\pi/3)$,

*$h (\bar{z}) $ = $\overline{h(z)}$ for all $z$ in $\Omega'$.

We also observe that our process of reflection has forced $h$ to be real on the rays $\theta = \pi/3, 0, -\pi/3$, and that for all $r>0$, $\lim_{\theta \to 2\pi/3}h(re^{i\theta})$ is real. So by restricting $h$ to the half-plane $\Omega'' = \{re^{i\theta} \mid r > 0, \theta \in (-\pi/3, 2\pi/3) \}$, $h|_{\Omega''}$ satisfies the hypotheses of Schwarz reflection.
Rotating and reflecting gives us a bounded analytic function $\tilde{h}$ defined on $\mathbb{C}-\{0\}$. By the classification of singularities, $\tilde{h}$ can be extended to $\mathbb{C}$, so by Liouville, $\tilde{h}$ is constant. As $\tilde{h}$ agrees with $f$ on an open set after rotation, we conclude $f$ is constant. Realness follows easily from the hypotheses.
