An asymptotic (long-distance) electrostatic potential has the form

(a) Use a traceless Cartesian multipole expansion to show that no localized charge distribution exists which can produce an asymptotic potential of this form.

(b) Repeat part (a) using a primitive Cartesian multipole expansion.

(c) Show that a suitable ρ(r) can be found which produces the given potential if we remove the restriction that the charge distribution is localized.

p(r, 0, 0) = A r Bx + p.5 + higher-order terms.