satisfying appropriate boundary conditions on the cylinder and at infinity. To this end we write $$\psi_2(r, \theta) = \psi_{3}(r) \sin\theta \cos\theta + \psi_4(r) \sin \theta$$ and require that $$\nabla^4 (\psi_{3}(r) \sin\theta \cos\theta) = - \gamma_{NL}(\epsilon, r)\sin\theta \cos\theta$$ and $$\nabla^4 (\psi_{4}(r) \sin\theta) = - \gamma_{HA}(\epsilon, r)\sin\theta$$ We also require that $\psi_i$ vanishes as $r \rightarrow \infty$ and both $\psi_i$ and its first derivative vanish on $r=1$. To proceed further we recall the elementary form of the Laplacian in 2D cylindrical polar coordinates: $$\nabla^2 f(r, \theta) = \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2}  \frac{\partial^2 f}{\partial \theta^2}$$ So if $f$ is of the form $f = h(r) \sin(k \theta)$ then $$\nabla^2 f(r, \theta) = \sin(k \theta) \biggl(\frac{d^2 h}{d r^2} + \frac{1}{r} \frac{dh}{dr} - \frac{k^2}{r^2} h\biggr)$$ Define the $k$’the radial modal Laplacian operator as $${\cal L}_k = \frac{d^2 \ }{d r^2} + \frac{1}{r} \frac{d\ }{dr} - \frac{k^2}{r^2}$$ We now have a pair of ODEs in the form $${\cal L}_2^2 \psi_3(r) =  - \gamma_{NL}(\epsilon, r)$$ $${\cal L}_1^2 \psi_4(r) =  - \gamma_{HA}(\epsilon, r)$$ In a simpler problem we would now construct a pair of Green’s functions $G_i(r,x)$, $i=1,2$, with the properties that $${\cal L}_i^2 G_i(r,x) = \delta(r-x)$$ subject to, [*if it is 