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