Spherical Harmonics may help. The field of any bounded body can be expressed as

where
Y_j^m(θ, φ) = e^imφP_j^m(cosθ) and
P_j^m(x) = (−1)^m(1 − x²)^m/2(D^mP_j)(x) where D is the derivative operator.

P₀(x) = 1.
P₁(x) = x.
(n+1)P_n+1(x) = (2n+1)xP_n(x) − nP_n−1(x).
?