function $\Sigma(K,\kappa)$. The form given by Eq.(\[hprop\]) represents the heavy quark propagator in the dominant region near the the bare propagator mass pole $p^2 = -m_R^2$. Obtaining an integral equation for $\Sigma(K)$ involves the change of integration $\int d^4q \rightarrow \int d^4k' = 
\int_{-\infty}^\infty dk'_4 \int_0^\infty d\left|{\bf k}\right|
\left|{\bf k}\right|^2$ induced by the change of variable Eq. (\[kmudef\]). For this change of integration to be valid, the propagator, and hence the functions $A(q^2)/[q^2A(q^2)^2 + B(q^2)^2]$ and $B(q^2)/[q^2A(q^2)^2 + B(q^2)^2]$, must be analytic over the shaded region in Fig. \[fig1\], $${\rm Re}\,(q^2) > -m_R^2 + \frac{\left({\rm Im}\,(q^2)\right)^2}{4m_R^2}.$$ Equivalently, the function defined by $$\sigma_{\rm Q}(K,\kappa) = \frac{1}{iK + \Sigma(K,\kappa)}, \label{sgdef}$$ must be analytic over the shaded region ${\rm Im}\, K < 0$.

The confinement criterion that $S(p)$ should be free from timelike poles on the negative real $p^2$ axis translates in the heavy quark case to a requirement that $\sigma_{\rm Q}$ should be free from poles on the imaginary $K$ axis. The stronger conjecture [@BRW92], that the quark propagator should be an entire function of $p^2$ translates in the heavy quark formalism to a conjecture that $\sigma_{\rm Q}$ should be an entire function of $K$.

From Eq. (\[hprop\]), we have that the 