Thread: F-Theory, Identity Theorem

1. F-Theory, Identity Theorem

Hi,

I'm a little stuck with this exercise.

Let $f:\mathbb{C}\to \mathbb{C}$ be a holomorphic function which satisfies $f(z)\in\mathbb{R}\quad\forall z\in\mathbb{R}$. I want to show that
$f(\overline{z}) = \overline{f(z)}$.

We do have a non-discrete set $\mathbb{R}$ on which this identity holds, so if both functions were holomorphic, the proof was complete using the identity theorem.
But why are the functions $z\mapsto f(\overline{z})$ and $z\mapsto \overline{f(z)}$ holomorphic?

Thank you!

2. If $f(z)$ il holomorphic $\forall z\in\mathbb{C}$, it will be true also for $z=0$, so that $f(z)$ can be written as…

$f(z)=\sum_{n=0}^{\infty} a_{n}\cdot z^{n}$ (1)

But $f(z) \in\mathbb{R}$ $\forall z\in\mathbb{R}$, so that all the $a_{n}$ in (1) are real. Since $\overline {z^n}= \overline {z}^{n}$ for (1) is $f(\overline {z})= \overline {f(z)}$

Kind regards

$\chi$ $\sigma$

3. Oh, thank you! So I was on the wrong train with wanting to use the theorem.