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Analytical Properties of Power Series on Levi-Civita Fields

Abstract

A detailed study of power series on the Levi-Civita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and re-expandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are infinitely often differentiable.

K. Shamseddine, M. Berz, Annales Mathematiques Blaise Pascal 12 (2005) 309-329

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