To fix terminology that I'm not 100% sure is universal, let the ring of Laurant series about $0$ be $\mathbb{C}[[z]][z^{-1}]$, and the ring of formal Laurant series about $0$ be $\mathbb{C}[[z,z^{-1}]]$, so that that the former are parameterized by sequences of coefficients that are infinite to the right, while the latter are parameterized by doubly infinite sequences. In particular, no assumptions are made about convergence of either.
Suppose that $f$ is holomorphic in a punctured neighborhood of the origin, and that $f$ has an essential singularity at $0$. To what extent can we describe/recover $f$ from its formal Laurent series, and given a formal Laurent series, can we tell if it comes from a function?
To explain my difficulties, for $n\in \mathbb Z$, let $a_n=\frac{1}{2\pi i}\oint_C \frac{f(z)}{z^{n+1}}dz$, where $C$ is a simply closed curve containing $0$ but no other singularities of $f$. If $f$ is holomorphic, then on some neighborhood of $0$,$\sum a_n z^n$ converges and is equal to $f(z)$. If we have a pole at $0$, the same result is true if we replace neighborhood with punctured neighborhood. Every function yields a convergent Laurent series, and every convergent Laurent series defines a function.
However, we can't a priori do the same thing with essential singularities. While the same process yields a formal Laurent series, the series need not converge anywhere. For example, take $f(z)=g(z)+g(1/z)$ where $g$ is holomorphic at $0$ but has a pole inside the unit disc. Similarly, given a formal Laurant series that only converges on an annulus (centered about $0$), I see no obvious reason why continuations of the corresponding function should all agree in a punctured neighborhood of the origin, independent of path. Even more troubling, if we start with a function having an essential singularity at $0$, take the corresponding formal Laurent series, note that it converges on some annulus, and then take a continuation of this function to a (punctured) neighborhood of $0$, I see no reason why we should get our original function.
Are there theorems that resolve these potential difficulties, or at least shine a light on how big (or small) an issue this actually is? If so, a reference would be wonderful.