Can Not Find Correct Green's Function

[t38]

We have long wire with constant charge density that is put inside a grounded metal housing with a shape of cylindrical section (a ≤ r ≤ b and 0 ≤ ϕ ≤ α). We need to find potential inside the box. 

 

Relevant equations are:

 

Δf=-(μ/ε0)*∂^2(r), where μ is linear charge density [As/m], ε0 is Vacuum permittivity, f is the potential (Δ is laplace operator) and ∂^2(r) is two-dimensional delta function.

 

f (a ≤ r ≤ b and 0 ≤ ϕ ≤ α) = 0. 

 

Green's function for ΔG =∂^2(r-r0),two dimensions and no boundary conditions: G(r,r0)=(1/2*pi)*ln|r-r0|

 

I am trying to solve the problem with green's function since it seems that solving this problem with separation of variables is too complicated.