  Three-body decay functions in time-like parton branching are calculated using
the jet calculus to the next-to-leading logarithmic (NLL) order in perturbative
quantum chromodynamics (QCD). The phase space contributions from each of the
ladder diagrams and interference diagrams are presented. We correct part of the
results for the three-body decay functions calculated previously by two groups.
Employing our new results, the properties of the three-body decay functions in
the regions of soft partons are examined numerically. Furthermore, we examine
the contribution of the three-body decay functions modified by the restriction
resulting from the kinematical boundary of the phase space for two-body decay
in the parton shower model. This restriction leads to some problems for the
parton shower model. For this reason, we propose a new restriction introduced
by the kinematical boundary of the phase space for two-body decay.
