  The unified description of fragmentation function evolution from large to
small x which was developed for the vacuum in previous publications is now
generalized to the medium, and is studied for the case in which the complete
contribution from the largest class of soft gluon logarithms, the double
logarithms, are accounted for and with the fixed order part calculated to
leading order. In this approach it proves possible to choose the remaining
degrees of freedom related to the medium such that the distribution of produced
hadrons is suppressed at large momenta while the production of soft
radiation-induced charged hadrons at small momenta is enhanced, in agreement
with experiment. Just as for the vacuum, our approach does not require further
assumptions concerning fragmentation and is more complete than previous
computations of evolution in the medium.
