  We study the Dyson-Schwinger equation for the quark propagator in Minkowski
space. In order to have analytical behaviour at the timelike axis of momenta
under control, we use the Stieltjes and the Hilbert transformation for the
interaction kernels and discuss the solution from the perspective of these
transformations. In addition, a lattice fit for the gluon propagator and
approximation for quark-gluon vertex are employed, and within the model the
quark propagator is obtained through the solution of Dyson-Schwinger equation
in Minkowski space. The resulting propagators in all studied cases do not show
up particle like pole and production thresholds. Instead of, the quark
propagator satisfies Hilbert transformation and the associated dynamical mass
function becomes complex without a presence of particle like branch point.
