The inverse and the composition in the set of formal Laurent series

Abstract: The aim of this article is to investigate the issues of multiplicative inverses and composition in the set of formal Laurent series. We show the lack of uniqueness of inverse series to formal Laurent series. Some necessary and sufficient conditions for the existence of inverse series to a certain type formal Laurent series are provided. Moreover, we define a general composition of formal Laurent series and investigate the Right Distributive Law and the Chain Rule for formal Laurent series. Finally, we provide a sufficient condition for the boundary convergence of formal Laurent series.