$k\cdot v = k_4$. We then write the renormalised quark propagator functions as $$A(p^2,\mu^2) = 1 + \frac{\Sigma_A(K,\kappa)}{m_R(\mu^2)},   \label{sigAdef}$$ $$B(p^2,\mu^2) = m_R(\mu^2) + \Sigma_B(K,\kappa),  \label{sigBdef}$$ where we have defined the independent momentum variable $$K = \frac{p^2 + m_R^2}{2im_R} = k_4 + \frac{k^2}{2im_R}, \label{Kdef}$$ and renormalisation point $$\kappa = \frac{\mu^2 + m_R^2}{2im_R},$$ in the complex $K$ plane. The change of dependent variable $p^2 \rightarrow K$ induced by the transformation Eq. (\[Kdef\]) is illustrated in Fig. \[fig1\]. In general, when working to zeroth order in $1/m_R$, one can use the approximation $K = k_4$ (an approximation which was used in ref. [@BL97]). An exception to this rule, relevant to models such as Eq. (\[GFRprop\]) for which the gluon propagator is not heavily ultraviolet damped, is in the denominator of the heavy quark propagator in the integrand in the quark DSE. This point will become clearer at the end of Section IIIB.

From Eqs. (\[genprop\]), (\[sigAdef\]) and (\[sigBdef\]) we have $$S(p,\mu) = \frac{1 + \gamma_4}{2} \frac{1}{iK + \Sigma(K,\kappa)} 
    + O\left(\frac{1}{m_R}\right) ,     \label{hprop}$$ where we have defined the heavy quark self energy $$\Sigma(K,\kappa) =  \Sigma_B(K,\kappa) -  \Sigma_A(K,\kappa).\label{Sigdef}$$ We find in general that the DSE leads to a single integral equation for the complex valued 