r))$$ as $r \rightarrow \infty$. We deduce that $\Gamma_{NL}$ is a bounded function for all $r$. What is its order as a function of $R_e$? A straightforward estimate may be given by looking in a neighbourhood of the cylinder. It is straightforward to establish that $$\frac{\partial J}{\partial r}|_{r=1} = \left(\frac{2 K_1(\epsilon)}{K_0(\epsilon)}  \right)^2 \sim \frac{4}{\epsilon ^2 (-\log (\epsilon )+\log (2)-\gamma )^2}$$ where the latter expression applies as $\epsilon \rightarrow 0$. We deduce that in the [*immediate*]{} neighbourhood of the cylinder $$\Gamma_{NL} = O\left(\frac{R_e}{(\log\epsilon)^2}\right)$$ which is $o(R_e)$ provided only that $\epsilon \rightarrow 0$ as $R_e \rightarrow 0$. Although $J$ grows to a maximum as $r$ is increased away from unity, before reaching a maximum and then decaying for large $r$, some numerical experiments with [*Mathematica*]{} confirm that the maximum of $\epsilon^2 J$ slowly decreases as $\epsilon$ decreases to zero. So we can assert that $\Gamma_{NL}$ is well behaved and is $o(R_e)$. When we consider $\Gamma_{HA}$, it is easy to see that $g_1$ has a maximum on $r=1$, and that $$\Gamma_{HA}|_{r=1} = \epsilon^4 \sin\theta g_1(1) = O(\epsilon^2/\log\epsilon)$$ as $\epsilon \rightarrow 0$. So this term also behaves. If we desire that the Helmholtz artifact tends to zero faster than the non-linear term (which is 