Green's function for infinitely long cylinder

1. The problem statement, all variables and given/known data

Find the Green's function for the Dirichlet boundary conditions for the interior of an infinite cylinder of radius a.

2. Relevant equations

[tex] \nabla^2 G(x,x') = -4 \pi \delta(x-x') [/tex]

and in general, Green's functions are of the form

[tex] G(x,x') = \frac{1}{|x-x'|}+F(x,x') [/tex]

where F(x,x') satisfies the Laplace equation.

3. The attempt at a solution

All I know, really, is that we have the two relations above. I guess I feel like I know why we have a solution with the Green's functions - as in, I know what to do with them once I've defined them - but I'm at a loss as to how to define them.

Do I just take F(x,x') to be the general solution to the Laplace equation in polar coordinates and then add that to the 1/|x-x'| term?

And I'm not sure quite what to make the boundary conditions. They're Dirichlet, so we need to demand that G(x,x') = 0 for x' on the cylinder. We also need to demand that the potential doesn't blow up inside the cylinder. But this problem didn't specify what sort of surface the cylinder is (we can probably assume it's some sort of conductor, but there are no specifics) and there's no mention of any charge density anywhere. So, yeah, I'm a bit confused... can anyone help?
 

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