In this paper, we define an analog of power series functions overR, whenRis replaced by K = k((x))τ, a field of generalized power series with coefficients in an ordered field k and exponents in an ordered abelian group τ. To this end for any power series S(Y)ε K[[Y]] and any y ε K, we define a notion of convergence of S(y). Thus to any power series S(Y) is associated a partial function S : K→ K. We show that these partial functions have a lot of similarities with analytic functions overR. Then we prove properties of zeros of such functions which extend properties of roots of polynomials over k((x))τ.