An empty beer can is bounded by the surfaces z = 0, z = h, and ρ = R. By slamming it against his forehead, a frustrated football fan dents the can into the shape shown below. Our interest is the interior Dirichlet Green function of the dented can.

(a) Show that suitable choices for the allowed values of γ and β in the sums makes the ansatz

satisfy the boundary conditions at z = 0, z = h, φ = 0, and φ = 2π/p, and thus reduces the defining equation for the Green function to a one-dimensional differential equation for gαβ (ρ, ρ').

(b) Complete the solution for GD(r, r').

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