of Eq. (\[expandedns\])we call it $\Gamma$and we do so without any assumption as the the form of $\epsilon$. We have, changing variables in the Jacobian to polar coordinates: $$\Gamma = \epsilon^4 \psi^1 - R_e \frac{\epsilon^2}{r}\left[ \frac{\partial \psi_0}{\partial r}\frac{\partial \psi_1}{\partial \theta} -  \frac{\partial \psi_1}{\partial r}\frac{\partial \psi_0}{\partial \theta}  \right]$$ Recall that we set $\psi_i = g_i(r) \sin\theta$. Having imposed the boundary conditions, we have $$g_0 = r - \frac{1}{r}\left(1 + \frac{2 K_1(\epsilon)}{\epsilon K_0(\epsilon)}  \right), \ \ 
g_1 = \frac{2 K_1(\epsilon r)}{\epsilon K_0(\epsilon)}$$ It follows that $$\Gamma = \epsilon^4 \sin\theta g_1 - R_e \epsilon^2 \sin\theta \cos\theta J(\epsilon, r) = \Gamma_{HA} +\Gamma_{NL}$$ where $J$ is the reduced Jacobian: $$J = \frac{1}{r}\left(g_1 \frac{\partial g_0}{\partial r} - g_0 \frac{\partial g_1}{\partial r}\right)$$ and $\Gamma_{HA} =  \epsilon^4 \sin\theta g_1 $ denotes the ‘Helmholtz artifact’ introduced by our approach, and $\Gamma_{NL}$ denotes the non-linear Jacobian term. Some calculation with Bessel function identities leads to $$J(\epsilon, r) = \frac{2 \left(K_0(\epsilon ) K_2(r \epsilon )-\frac{1}{r^2}K_0(r \epsilon ) K_2(\epsilon)\right)}{K_0(\epsilon )^2}$$ from which it is manifest that $J(\epsilon,1) = 0$ and the non-linear terms vanish on the cylinder boundary irrespective of the choice of $\epsilon$. Furthermore, the asymptotic behaviour of the Bessel functions tells us that provided $\epsilon>0$, $$J(\epsilon, r) = O(\frac{1}{\sqrt{r}} \exp(-\epsilon 