to be the infrared dominant form Eq. (\[delpr\]), gives the differential equation $$\Sigma(k_4) = \frac{3\mu^2}{16i} \frac{d}{dk_4}\ln[ik_4 + \Sigma(k_4)],$$ which, together with the boundary condition $\sigma_{\rm Q}(k_4) 
\rightarrow 0$ as $k_4 \rightarrow -i\infty$, admits the solution $$\sigma_{\rm Q}(k_4) = \beta\left[\sqrt{\pi} e^{-\beta^2k_4^2} 
              + 2iF(-\beta k_4) \right],  \label{bcsoln}$$ where $\sigma_Q$ is defined by Eq. (\[sgdef\]), $\beta = 2\sqrt{2}/\mu\sqrt{3}$ and $$F(z) = e^{-z^2}\int_0^z e^{t^2} dt = \frac{i\sqrt{\pi}}{2} e^{-z^2}
             {\rm erf}(-iz),$$ is Dawson’s integral. We note that this solution is an entire function of $k_4$, which, as pointed out earlier, is a desirable feature of a quark propagator. This comes as no surprise, as it simply the heavy quark limit of the model considered in ref. [@BRW92], in which it was demonstrated that the combination of Ball-Chiu vertex and infrared dominant gluon propagator leads to an entire function propagator for all values of the bare current quark mass.

It is of interest to determine to what extent this analytic structure is a feature of the Ball-Chiu vertex, and to what extent it is a feature of the infrared dominant gluon propagator. If the infrared dominant propagator is replaced by the gaussian smeared form Eq. (\[gauss\]), we obtain the integral equation $$\begin{aligned}
\lefteqn{\Sigma(k_4) = \frac{\mu^2}{2\alpha^2\sqrt{\pi}} 
   \int_{-\infty}^\infty 