I am reading Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables. Exercise 10 on p.77 is related to Schwarz reflection principle:
Let $D$ be a connected open set, which is symmetrical with respect to the real axis and has non-empty intersection $I$ with it. Any holomorphic function $f(z)$ in $D$ can be expressed uniquely in the form
$$f(z)=g(z)+ih(z)\ \text{ for all } z\in D,$$
where $g$ and $h$ are holomorphic functions in $D$ which take real values in $I$. Show that…
The actual statements that he wanted the readers to show are omitted here. What I don't understand is the existence of $g$ and $h$.
If I pick a point $x\in I$, then within a small disc centered at $x$ inside $D$, the function $f$ has a power series representation $\sum_{n=0}^\infty a_n(z-x)^n$. So, I guess we may take $g(z)=\sum_{n=0}^\infty\operatorname{Re}(a_n)(z-x)^n$ and $h$ can be defined similarly. But why do they have analytic continuations to the whole region $D$? Why do their analytic continuations remain real-valued on the whole of $I$?
Perhaps the existence of $g$ and $h$ are proved in some other way?
Best Answer
Hint: Consider $g(z)=\frac{f(z)+\overline{f(\overline{z})}}{2}$.