Can an analytic function on the annulus be decomposed into two: one analytic on a disk, and one on the complement of a disk?

Question

For $0<r<R$, I am wondering if a given function $f$ which is analytic on the 2D-annulus,

$$\{(z,w)\in \mathbb C^2 \;|\; r<|z|<R,\ r<|z|<R\},$$

can be decomposed as such: $f = f_1 + f_2$, where

• $f_1$ is analytic on the hyperdisk $\{|z|,|w|< R\}$, and
• $f_2$ is analytic on the hyperdisk-at-infinity, $\{|z|,|w|> r\}$.

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I suspect the answer is no:

Naively, one would expect the Laurent series of $f$ to contain monomials like $(z,w)↦ z^aw^{-b}$ ($a$, $b$ positive), for example;

but I'm unsure how to characterise those functions which can or can't be decomposed like this.