Show that if the Laurent series $\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n$ represents an even function, then $a_{2n+1}=0$ for $n=0,\pm 1,\pm 2,\ldots$, and if it represents an odd function, then $a_{2n}=0$ for $n=0,\pm 1,\pm 2,\ldots$.
where
$a_n=\frac{1}{2\pi i}\int_C \frac{f(z)}{(z-z_0)^{n+1}}dz$
I know the fact that if $f(-z)=f(z)$ then $f$ is even. But I have difficulty applying this to show what i need to have.