The calculation of the two-loop corrections to the three jet production
rate and to event shapes in electron-positron annihilation requires
the computation of a number of up to now unknown two-loop four-point
master
integrals with one off-shell and three on-shell legs. In this paper,
we compute those master integrals which correspond to planar
topologies by solving differential equations in the external
invariants which are fulfilled by the master integrals.
We obtain the master integrals
as expansions in $\e=(4-d)/2$, where $d$ is the
space-time dimension. The results are expressed in terms of
newly introduced two-dimensional harmonic polylogarithms, whose
properties are shortly discussed. For all two-dimensional
harmonic polylogarithms appearing in the divergent parts of the
integrals, expressions in terms of
Nielsen's polylogarithms are given.