## Appendix A Green's Function  
In cylindrical \((r,\theta,z)\) coordinates, the relationship between the Green's and the delta functions is expressed as follows:  
\[\
abla^{2}g(r,\theta,z;r_{0},\theta_{0},z_{0})=\frac{\delta(r-r_{0})\delta( \theta-\theta_{0})\delta(z-z_{0})}{r}, \tag{10}\]  
where the Green's function is given by the following:  
\[g(r,\theta,z;r_{0},\theta_{0},z_{0})=A_{v}\,g_{r}(r;r_{0})g_{\theta}(\theta; \theta_{0})g_{z}(z;z_{0}), \tag{11}\]  
where \(A_{v}\) is a coupling constant, which will be specified later by an azimuthal constant \(v\ (=0,\ \pm 1,\ \pm 2,\cdots)\). At any point in the cylindrical domain other than the PS locating point, the values of the delta functions are zero:  
\[\
abla^{2}[g_{r}(r;r_{0})g_{\theta}(\theta;\theta_{0})g_{z}(z;z_{0})]=0,\]  
and Equation (10) can be rewritten as follows:  
\[\frac{r^{2}}{g_{r}}\frac{\partial^{2}g_{r}}{\partial r^{2}}+\frac{r}{g_{r}} \frac{\partial g_{r}}{\partial r}+\frac{1}{g_{\theta}}\frac{\partial^{2}g_{ \theta}}{\partial\theta^{2}}+\frac{r^{2}}{g_{z}}\frac{\partial^{2}g_{z}}{ \partial z^{2}}=0. \tag{12}\]  
Introducing \(\kappa_{z}^{2}\) for the real roots of \(g_{z}\) and \(v\) as an azimuthal constant for \(g_{\theta}\) into Equation (12) gives three individual differential equations for \(g_{z}\), \(g_{\theta}\) and \(g_{r}\). \[\frac{1}{g_{z}}\frac{\partial^{2}g_{z}}{\partial z^{2}}=\kappa_{z}^{2}, \tag{13}\]  
\[\frac{1}{g_{\theta}}\frac{\partial^{2}g_{\theta}}{\partial\theta^{2}}=-v^{2},\]  
\[r^{2}\frac{\partial^{2}g_{r}}{\partial r^{2}}+r\frac{\partial g_{r}}{\partial r }+\Big{(}\kappa_{z}^{2}r^{2}-v^{2}\Big{)}g_{r}=0.\]  
The solutions of the three functions can be obtained by applying the conditions for the delta function, such as the continuity, the discontinuity and/or the symmetry principles around the PS. \[g_{z}(z;z_{0})= \begin{cases}-\frac{1}{2\kappa_{z}}e^{-\kappa_{z}(z_{0}-z)}&0 <z<z_{0}<l\\ -\frac{1}{2\kappa_{z}}e^{-\kappa_{z}(z-z_{0})}&0<z_{0}<z<l\end{cases} , \tag{14}\]  
\[g_{\theta}(\theta;\theta_{0})=\cos[v(\theta-\theta_{0})]\quad(v=0,\pm 1,\pm 2, \cdots), \tag{15}\]  
\[g_{r}(r;r_{0})=J_{v}(\kappa_{z}r_{0})J_{v}(\kappa_{z}r)\quad(b<r<a). \tag{16}\]  
The coefficient \(A_{v}\) can be obtained by substituting Equations (14)-(16) into Equation (10) and integrating over a given cylindrical domain (\(0\leq z\leq l,\ 0\leq\theta\leq 2\pi,b\leq r\leq a\)). Since \(\int_{0}^{l}\delta(z-z_{0})dz=1\), and  
\[\left.\frac{\partial g_{z}}{\partial z}\right|_{0}^{z=l}=-\frac{e^{-\kappa_{z} z_{0}}}{2}e^{kz}\bigg{|}_{0}^{z_{0}}+\left.\frac{e^{\kappa_{z}z_{0}}}{2}e^{- \kappa_{z}z}\right|_{z_{0}}^{l}=\frac{1}{2}\Big{[}e^{-\kappa_{z}z_{0}}+e^{- \kappa_{z}(l-z_{0})}-2\Big{]}, \tag{17}\]then  
\[ \begin{array}{c}A_{v}\int_{0}^{l}\Big{\{}\Big{\{}\frac{g\sigma\kappa_{2}}{r} \frac{\partial}{\partial r}\Big{(}r\frac{\partial\sigma_{2}r}{\partial r}\Big{)} +\frac{g\sigma\kappa_{2}}{r^{2}}\frac{\partial^{2}g\partial}{\partial\theta^{2 }}+g_{t}\xi g_{\theta}\frac{\partial^{2}g_{\xi}}{\partial z^{2}}\Big{\}}dz\\ =A_{v}g_{r}g_{\theta}\int_{0}^{l}\frac{\partial^{2}g_{\xi}}{\partial z^{2}}dz\\ =\frac{A_{v}}{2}\Big{[}e^{-\kappa_{2}z_{0}}+e^{-\kappa_{2}(l-z_{0})}-2\Big{]} g\tau g_{\theta}\end{array}  \tag{109}\]  
Similarly, since \(\int_{0}^{2\pi}\delta(\theta-\theta_{0})d\theta=1\),  
\[\int_{0}^{2\pi}g_{\theta}d\theta=\int_{0}^{2\pi}\cos[v(\theta-\theta_{0})]d \theta= \begin{cases}2\pi\text{ if }\theta=\theta_{0}\\ 0\text{ otherwise}.\end{cases}  \tag{110}\]  
From Equations (101), (109) and (110), we obtained the following:  
\[A_{v}\pi g_{r}\Big{[}e^{-\kappa_{2}z_{0}}+e^{-\kappa_{z}(l-z_{0})}-2\Big{]}= \frac{\delta(r-r_{0})}{r}. \tag{111}\]  
Substituting Equation (107) into Equation (111) gives the following:  
\[\pi A_{v}J_{v}(\kappa_{2}r_{0})J_{v}(\kappa_{2}r)\Big{[}e^{-\kappa_{2}z_{0}}+ e^{-\kappa_{z}(l-z_{0})}-2\Big{]}=\frac{\delta(r-r_{0})}{r}. \tag{112}\]  
Multiplying both sides of Equation (112) by \(rJ_{v}(\kappa_{2}r)\) and integrating over \((b,a)\) gives the solution for \(\delta(r-r_{0})\) as follows:  
\[