A boundary-value fact used below

For the function

the required polynomial bounds hold near $Q=-1$, uniformly on compact real angular intervals. Indeed, $\mathrm{Re}\mathrm{arcosh}Q\ge c\mathrm{Im}Q$ on a sufficiently small upper-half-plane compact patch, the geometric denominators in (eq:Sgeom) are bounded below by a constant times $\mathrm{Re}\mathrm{arcosh}Q$, and $\sinh (\mathrm{arcosh}Q)$ has only a square-root zero. Angular differentiation changes the polynomial bound, but not its existence.

Consequently, near a point with $T\ne 0$, the covariance has the reduced form

The analytic change from $T$ to $Q$ is regular there, since ${\partial }_{T}Q=-T/(r{r}^{\prime })\ne 0$. This observation is particularly useful at an angular transition: no moving image indicator is differentiated.