Schwarz Reflection Principle on a unit disk

Question

Suppose $f$ is a analytic function defined on $\bar{D}(0;1)$ and has real value on the boundary. I'm trying to show $f$ can be extended to entire plane by $$g(z) = \begin{cases}f(z) &, \lvert z\rvert \leqslant 1\\ \frac{1}{\overline{f(\overline{z}^{-1})}}, &, \lvert z\rvert > 1\end{cases} $$ I tried to use $\gamma(z)=e^{iz}$, which sends real axis to unit circle, for Schwarz Reflection Principle. But I did not get the result. May I get a help?

Answer 1

You can write various forms of the reflection principle, it seems to me that the formula that you are writing is correct for a map from unit disk to C which maps a unit circle to a unit circle (and not the real axis). In the case of a map from unit disk which maps the boundary to the real axis the formula is $$ g(z) = \begin{cases} f(z) \quad |z|\leq 1, \\ \overline{f(\frac{1}{\bar{z}})} \quad |z|>1 \end{cases} $$ which is what TrialAndError has.

Answer 2

Assuming that $f$ is real on the boundary, then, for $|z|=1$, \begin{align} \sum_{n=0}^{\infty}f_{n}z^{n} & = \overline{\sum_{n=0}^{\infty}f_{n}z^{n}} \\ & = \sum_{n=0}^{\infty}\overline{f_{n}z^{n}} \\ & = \sum_{n=0}^{\infty}\overline{f_n}\frac{1}{z^{n}}. \end{align}