Convergent Laurent series on an annulus

Fix an annulus $A(a;r,R)$ (Annuli in the complex plane) and complex coefficients $c_n$ indexed by $n\in \mathbb{Z}$. The formal expression

${\sum }_{n\in \mathbb{Z}}{c}_n(z-a{)}^n={\sum }_{n\ge 0}{c}_n(z-a{)}^n+{\sum }_{m\ge 1}{c}_{-m}(z-a{)}^{-m}$

is a convergent Laurent series on $A(a;r,R)$ when, for every closed subannulus

$K_{\rho ,\sigma }:=\{z\in \mathbb{C}:\rho \le \mathrm{\mid }z-a\mathrm{\mid }\le \sigma \}\subseteq A(a;r,R)$

with $r<\rho \le \sigma <R$, both one-sided series on the right converge uniformly on $K_{\rho ,\sigma }$.

Its sum is the function $f:A(a;r,R)\to \mathbb{C}$ defined by that convergent value at each point of the annulus, and the numbers $c_n$ are its Laurent coefficients.

The definition is local-uniform rather than merely pointwise because Laurent series are used as holomorphic expansions: later proofs integrate them term by term on circles inside the annulus.

The split into nonnegative and negative powers is part of the definition. On a punctured disc or exterior domain, the same series may converge in one direction further than in the other, and the annulus records exactly where both pieces are simultaneously valid.