Even and odd functions in Laurent decomposition I have the following problem and have no idea how to approach it, could anyone give me any hint about it? Thanks!
Suppose that $f(z) = f_0(z) + f_1(z)$ is the Laurent decomposition of an analytic function $f(z)$ on the annulus $\{A < |z| < B \}$. Show that if $f(z) $ is an even function, then $f_0(z)$ and $f_1(z)$ are even functions.

Continuing with Martin R's answer, we have that $\lim_{z \to \infty} f_1(z) = 0$, so $f_1$ is bounded. 
Also, since $f_0$ is analytic, and hence continuous, on the compact set $|z| < B$, $f_0$ is bounded. Therefore, $h(z)$ is bounded and analytic on the entire plane. Liouville's theorem implies that $h(z)$ is identically zero.
Therefore, both $f_0$ and $f_1$ are even.
