Analytic functions over a field of power series

 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. Received: 5 October 2000 / Revised version: 19 June 2001 / Published online: 12 July 2002