Do all coefficients in a Laurent series need to be real? Suppose we are given $ f(x)$ holomorphic in $ \Bbb C \setminus\{0\}$ that has Laurent series around $0$ and $ f(x)\in \Bbb R $  for all $ x \in \Bbb R $, $x\ne0$.
Does it imply that all coefficients in the Laurent series need to be real?
How can I prove it? Any hints would be appreciated.

Yes, it is true. Let $\sum_{n=-\infty}^\infty a_nz^n$ be that Laurent series. Then$$\overline{f\left(\overline z\right)}=\sum_{n=-\infty}^\infty\overline{a_n}z^n.$$Then, for real numbers $x$ you have$$\sum_{n=-\infty}^\infty a_nx^n=f(x)=\overline{f(x)}=\sum_{n=-\infty}^\infty\overline{a_n}x^n.$$Therefore, $(\forall n\in\mathbb{Z}):a_n=\overline{a_n}$, which means that every $a_n$ is real.
