Hardy type derivations on fields of exponential logarithmic series

Abstract We consider the valued field K : = R ( ( Γ ) ) of formal series (with real coefficients and monomials in a totally ordered multiplicative group Γ). We investigate how to endow K with a logarithm l, which satisfies some natural properties such as commuting with infinite products of monomials. We studied derivations on K (Kuhlmann and Matusinski, in press [KM10] ). Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyze sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In Kuhlmann (2000) [Kuh00] , the first author described the exponential closure K EL of ( K , l ) . Here we show how to extend such a log-compatible derivation on K to K EL .