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We study the Dyson-Schwinger equation for the quark propagator in Minkowskispace. In order to have analytical behaviour at the timelike axis of momentaunder control, we use the Stieltjes and the Hilbert transformation for theinteraction kernels and discuss the solution from the perspective of thesetransformations. In addition, a lattice fit for the gluon propagator andapproximation for quark-gluon vertex are employed, and within the model thequark propagator is obtained through the solution of Dyson-Schwinger equationin Minkowski space. The resulting propagators in all studied cases do not showup particle like pole and production thresholds. Instead of, the quarkpropagator satisfies Hilbert transformation and the associated dynamical massfunction becomes complex without a presence of particle like branch point.

Lattice data inspired but Minkowski space calculated QCD fundamental propagator