Consider a potential problem in the half-space defined by z > 0, with Dirchlet boundary conditions on the plane z = 0 (and at infinity).
(a) Write down the appropriate Green function G(x, x')
(b) If the potential on the plane z = 0 is specific to be phi = V inside a circle of a radius a centered at the origin, and phi = 0 outside that circle, find an integral expression for the potential at the point P specified in terms of cylindrical coordinates (p, phi, z).
(c) Show that, along the axis of the circle (p = 0), the potential is given by:
(d) Show that at rage distances (p^2 + z^2 >> a^2) the potential can be expanded in a power series in (p^2 + z^2)^-1, and that the leading terms are:
