(a) By finding appropriate solutions of the radial equation in part b of Problem 2.17, find the Green function for the interior Dirichlet problem of a cylinder of radius b [g_m(?, ?' = b) = 0. See (1.40)]. First find the series expansion akin to the free-space Green function of Problem 2.17. Then show that it can be written in closed form as

(b) Show that the solution of the Laplace equation with the potential given as ?(b, ?) on the cylinder can be expressed as Poisson's integral of Problem 2.12.

(c) What changes are necessary for the Green function for the exterior problem (b

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p°p2 + b- 2pp'b? cos(o- o') G = In b?(p? + p'2 - 2pp' cos(o - o')) or (b² - p?)(b - p?) + b² |p - p'2| G = In b² \p - p'P