I am trying to solve the following BVP within an annular region of radii $r_1$, and $r_2$ : $$ \begin{cases} \nabla^2u=f\\ u(r_1) = p\\ u(r_2) = q \end{cases} $$ If we define an auxiliary problem in terms of Green's function as $$ \begin{cases} \nabla^2G=\delta^2(r-r')\\ G(r_1) = 0 \\ G(r_2) = 0\\ \end{cases} $$ We have the solution of u (as given by Green's identities as) $$\DeclareMathOperator{\Dm}{\operatorname{d}\!} u = \oint u \frac{\partial G}{\partial n} \Dm S + \int Gf \Dm V \tag{Eqn. A} $$ How do I proceed to obtain the form of the Green's function ?
I understand that G for a finite boundary problem is done by superposition :
$G = G_{Freespace} + G_{Homogeneous}$
From my little searching I found that, $G_{Freespace} = Aln(r-r')$, and $G_{Homogeneous}=a_0 + a_nr^ncos(n\phi) + b_nr^nsin(n\phi)$
However, the expected solution ( from a paper) that I am seeking is of the form:
$2\pi G=H_0 (r_1,r_2) + \sum_{n=1}^{\infty} H_n(r_1,r_2) cos(n(\phi_1 - \phi_2))$
The form of $H_0$ and $H_n$ are given in the attachment below.
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How do I obtain the solution above from the problem posed ?