A generalization of the Laurent series

Abstract

We study conditions on a domainG in the extended complex plane$\widehat{\mathbb{C}}$ and a sequence$\Lambda = \left\{ {\lambda _k } \right\}$ of points in the complement$G^c = \widehat{\mathbb{C}}\backslash G$ ofG under which every functionf(z) holomorphic inG and vanishing atz=∞ (if ∞∈G) admits a representation (possibly, nonunique) by a series

$$f\left( z \right) = \mathop \sum \limits_{k = 1}^\infty \mathop \sum \limits_{j = 1}^\infty \frac{{a_{k,j} }}{{\left( {z - \lambda _k } \right)j}},$$

(1)

which is uniformly and absolutely convergent in the interior ofG. For the case in which$\bar G \subset {\mathbb{C}}$ or$G^c \subset {\mathbb{C}}$, this expansion exists if and only if the system$\{ K(\lambda _k$, of disks covers ∂G for each ε>0.