If $f$ has a Laurent expansion like this:
$$ f(z) = \sum_{n = -\infty} ^{\infty} a_n(z-z_0)^n$$
in the annulus $r < |z-z_0|<\infty$, then, I need to prove that the function $f$ can also be represented like this:
$$ f(z)= \sum_{n = -\infty} ^{\infty} \overline a_n z^n $$
in an annulus $\overline r < |z| < \infty$ for some $r$.
At first, I did not know what the variables with underlines meant. However, when reading my notes again, I found another reference as to when they were used. It lies in the proof of this theorem:
A function given by a power series at $z_0$ with $R$ as its radius of convergence has at least one singular point on the circle $|z-z_0| = R$
The proof involves letting $f$ be a Taylor series on a disc of convergence $K = {|z-z_0|<R}$, and considering each point on the circular boundary of the disc $\Gamma$ to be regular w.r.t. $(f,K)$. Then, we should take a union of all circles $K_a$ with $K$ such that
$$ \overline K = K \cup \bigcup_{a \in \Gamma} K_a $$
Would the $\overline a_n$ and $\overline r$ be defined in a similar way? If so, how? If not, how can I move forward with the proof?