Abstract.
We extend the notion of absolute convergence for real series in several variables to a notion of convergence for series in a power series field ℝ((t Γ)) with coefficients in ℝ. Subsequently, we define a natural notion of analytic function at a point of ℝ((t Γ))m. Then, given a real function f analytic on a open box I of ℝ m, we extend f to a function f ★ which is analytic on a subset of ℝ((t Γ))m containing I. We prove that the functions f ★ share with real analytic functions certain basic properties: they are , they have usual Taylor development, they satisfy the inverse function theorem and the implicit function theorem.
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Received: 5 October 2000 / Revised version: 19 June 2001 / Published online: 12 July 2002
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Mourgues, MH. Analytic functions over a field of power series. Arch. Math. Logic 41, 631–642 (2002). https://doi.org/10.1007/s001530100125
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DOI: https://doi.org/10.1007/s001530100125