  We calculate the dressed gluon and ghost propagators of Landau gauge
Yang-Mills theory in the complex momentum plane from their Dyson-Schwinger
equations. To this end, we develop techniques for a direct calculation such
that no mathematically ill-posed inverse problem needs to be solved. We provide
a detailed account of the employed ray technique and discuss a range of tools
to monitor the stability of the numerical calculation. Within a truncation
employing model ansaetze for the three-point vertices and neglecting effects
due to four-point functions, we find a singularity in the gluon propagator in
the second quadrant of the complex $p^2$-plane. Although the location of this
singularity turns out to be strongly dependent on the model for the three-gluon
vertex, it occurs always at complex momenta for the range of models considered.
