## 2 [PERSON]'s Function
\begin{array}{ll}A_{\theta}(\theta_{0} )\cos v\theta+B_{\theta}(\theta_{0})\ \sin v\theta&\quad 0<\theta<\theta_{0}<2\pi\\ C_{\theta}(\theta_{0})\cos v\theta+D_{\theta}(\theta_{0})\ \sin v\theta&\quad 0< \theta_{0}<\theta<2\pi\end{array} \right.. \tag{17}\]  
Applying the continuity principle to Equation (17) results in  
\[A_{\theta}(\theta_{0})=C_{\theta}(\theta_{0})\ \ \text{and}\ \ \ B_{\theta}( \theta_{0})=D_{\theta}(\theta_{0}). \tag{18}\]The second restriction for \(g_{\theta}(\theta;\theta_{0})\) is the symmetry between the point source and observation point, i.e.,  
\[g_{\theta}(\theta_{0};\theta)=g_{\theta}(\theta;\theta_{0}). \tag{19}\]  
Equation (19) leads to  
\[A_{\theta}(\theta)\cos v\theta_{0}+B_{\theta}(\theta)\ \sin v\theta_{0}=A_{ \theta}(\theta_{0})\cos v\theta+B_{\theta}(\theta_{0})\ \sin v\theta.\]  
To satisfy the conservation law,  
\[A_{\theta}(\theta_{0})=\cos v\theta_{0}\ \text{and}\ B_{\theta}(\theta_{0})= \sin v\theta_{0}. \tag{20}\]  
Equations (18) and (20) result in  
\[g_{\theta}(\theta;\theta_{0})=[\cos v\theta_{0}\cos v\theta+\sin v\theta_{0} \sin v\theta]=\cos[v(\theta-\theta_{0})].\]  
For a complete cylinder, \(g_{\theta}(\theta;\theta_{0})\) is periodic, with a period of \(2\pi\):  
\[\cos[v(2\pi+\theta-\theta_{0})]=\cos[v(\theta-\theta_{0})].\]  
Therefore,  
\[v=0,\pm 1,\pm 2,\ \cdots.\]  
The final form of Equation (17) is  
\[g_{\theta}(\theta;\theta_{0})=\cos[v(\theta-\theta_{0})]\ \ \ \ \ (v=0,\pm 1,\pm 2,\ \cdots). \tag{21}\]  
The solution of Equation (10) is a typical Bessel function, given by  
\[g_{r}(r;r_{0})=\left\{ \begin{array}{ll}A_{vr}(r_{0})I_{v}(\kappa_{z}r)&0<r<r _{0}<a\\ B_{vr}(r_{0})I_{v}(\kappa_{z}r)&0<r_{0}<r<a\end{array} \right. \tag{22}\]  
where \(I_{v}\) is the first kind of Bessel function of \(v\)-th order. Note that the second kind of Bessel function is excluded because it has a singularity at the origin, which is included in the domain of the cylinder. Applying the continuity principle to Equation (22), i.e., \(g_{r}(r;r_{0})|_{r=r_{0}^{+}}=g_{r}(r;r_{0})|_{r=r_{0}^{-}}\), results in  
\[A_{vr}(r_{0})I_{v}(\kappa_{z}r_{0}) = B_{vr}(r_{0})I_{v}(\kappa_{z}r_{0}).\]  
We find  
\[A_{vr}(r_{0}) = B_{vr}(r_{0}), \tag{23}\]  
where \(r_{0}=\sqrt{x_{0i}^{2}+x_{0j}^{2}}\). In addition, applying the symmetry principle to Equation (22) gives \(g_{r}(r_{0};r)=g_{r}(r;r_{0})\), with  
\[A_{vr}(r)I_{v}(\kappa_{z}r_{0}) = A_{vr}(r_{0})I_{v}(\kappa_{z}r).\]  
we obtain  
\[A_{vr}(r_{0})=J_{v}(\kappa_{z}r_{0}). \tag{24}\]  
Substituting Equations (23) and (24) into Equation (22) gives  
\[g_{r}(r;r_{0})=J_{v}(\kappa_{z}r_{0})J_{v}(\kappa_{z}r)\ \ (0<r<a). \tag{25}\]  
Applying the boundary condition, i.e.,  
\[g_{r}(r;r_{0})|_{r=a-r_{0}}=0,\qquad g_{r}(r;r_{0})|_{r=r_{0}}\
eq 0,\]to Equation (25) gives \(J_{v}[\kappa_{z}(a-r_{0})]=0\). Denoting the \(n\)-th root of the first kind of Bessel function of the \(v\)-th order as \(r_{vn}\), i.e., \(J_{v}(r_{vn})=0\), then \(r_{vn}=\kappa_{z}(a-r_{0})\)\((n=1,\,2,\,\cdots)\). Since the first root of the first kind of Bessel function is most significant, we select \(n=1\). \[\kappa_{z}=\frac{r_{v1}}{a-r_{0}}. \tag{26}\]  
Introducing a parameter, \(A_{v1}\), to Equation (25) gives  
\[g_{r}(r;r_{0})=A_{v1}\ J_{v}\bigg{(}\frac{r_{v1}}{a-r_{0}}r_{0}\bigg{)}J_{v} \bigg{(}\frac{r_{v1}}{a-r_{0}}r\bigg{)}\quad(0<r<a). \tag{27}\]  
From Equations (16), (21) and (27), the formula of the Green's function for the Kronecker delta function becomes  
\[g(r,\theta,z;r_{0},\theta_{0},z_{0})=A_{v1}\ J_{v}\bigg{(}\frac{r_{v1}}{a-r_{0 }}r_{0}\bigg{)}J_{v}\bigg{(}\frac{r_{v1}}{a-r_{0}}r\bigg{)}\cos[v(\theta- \theta_{0})]\] \[\times\left\{ \begin{array}{cc}-\frac{a-r_{0}}{2r_{0}}e^{-\frac{ r_{v1}}{a-r_{0}}(z_{0}-z)}&0<z<z_{0}<l\\ -\frac{a-r_{0}}{2r_{0}}e^{-\frac{r_{v1}}{a-r_{0}}(z-z_{0})}&0<z_{0}<z<l\end{array}  \right.. \tag{28}\]  
The next task is to find the constant \(A_{v1}\) in Equation (28). Equation (4) is rewritten as  
\[