poles in the quark propagator [@RWK92]. It has furthermore been conjectured that the propagator $S(p)$ could be an entire function in the complex $p^2$ plane [@M86; @BRW92]. Such a scenario would, for instance, avoid certain unpleasant consequences which can result when modelling mesons via the Bethe-Salpeter equation which samples the quark propagator over a region of the complex plane.

Determining the analytic structure of fermion propagators in QCD [@MH92; @SC92] or other confining theories [@M94; @AB96] by the direct solution of model DSEs is not easy. It appears that the pole or branch cut structure obtained in any particular model is heavily dependent on the approximations employed. In general, two aspects of the quark DSE must be approximated: the quark-gluon vertex, and the gluon propagator. In this paper, we shall look at both these aspects within the heavy quark sector. In existing numerical studies in the light quark sector which produce propagators with conjugate singularities [@MH92; @SC92], the quark gluon vertex has usually been approximated by the bare vertex (the so called rainbow approximation). In ref. [@BRW92], however, it was shown that an entire function propagator can be obtained if the vertex function is modelled by a more sophisticated form 