Hardy type derivations on fields of exponential logarithmic series

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. In [KM10], we studied derivations on K. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In [Kuh00], the first author described the exponential closureKEL of (K, l). Here we show how to extend such a log-compatible derivation onK toKEL.