  In this work we improve existing calculations of radiative energy loss by
computing corrections that implement energy-momentum conservation, previously
only implemented a posteriori, in a rigorous way. Using the path-integral
formalism, we compute in-medium splittings allowing transverse motion of all
particles in the emission process, thus relaxing the assumption that only the
softest particle is permitted such movement. This work constitutes the
extension of the computation carried out for x$\rightarrow$1 in Phys. Lett.
B718 (2012) 160-168, to all values of x, the momentum fraction of the energy of
the parent parton carried by the emitted gluon. In order to accomplish a
general description of the whole in-medium showering process, in this work we
allow for arbitrary formation times for the emitted gluon. We provide general
expressions and their realisation in the path integral formalism within the
harmonic oscillator approximation.
