using a Simpson’s rule quadrature. We find that the derivative-like terms in Eq. (\[bcie\]) prevent a numerical solution for values of $\hat{\alpha}$ less than 1, as numerical noise in the function values becomes unstable with respect to iteration at small values of $k_4$. This problem is a general feature of numerical treatments of DSEs with Ball-Chiu-like vertices. Also plotted are the $\hat{\alpha} = 0$ analytic results Eqs. (\[baresoln\]) and (\[bcsoln\]). In all cases the self energy is characterised by a real part which peaks at zero and an imaginary part which peaks near the typical scale of the model $k_4 \sim \mu$. The self energy for negative real $k_4$ can be obtained from these results using the reflection property $\Sigma(-k_4^*) = 
\Sigma(k_4)^*$.

To solve for the heavy quark propagator away from the real $k_4$ axis we shift the contour of integration into the complex plane parallel to the real $k_4$ axis and again solve iteratively. We note that, to determine $\Sigma(k_4)$ for complex arguments, it is necessary to move the contour of integration to pass through the point $k_4$. This is because the radial part of the $d^3{\bf k}'$ integration, carried out in going from Eq. (\[baresde\]) to Eq. (\[bareie\]) 