A function $\displaystyle f(z)$ does have a laurent series about the origin, with real $\displaystyle a_n$ coefficients. Show that $\displaystyle \bar{f}(z) = f(\bar{z})$
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So that means that f can be written as
$\displaystyle f(z)=\sum_{n=-\infty}^{\infty}a_n z^n,a_n \in \mathbb{R}$
Now write out
$\displaystyle \overline{f(z)}=\overline{\sum_{n=-\infty}^{\infty}a_n z^n}=...$
$\displaystyle f(\bar{z})=\sum_{n=-\infty}^{\infty}a_n \bar{z}^n=...$
Use some properties of complex numbers and simplify