# Electric octupole in cartesian coordinates

I am new to this: Does anyone know to define different electric octupole moments in cartesian coordinates? I am looking for expressions that look like this (for electric dipoles).

Given \begin{align} Y_1^{\pm 1}(\theta,\varphi)&= \pm C_{1} \left(\frac{x\pm iy}{r}\right) \end{align} why not start from the expression for the $$\ell=3$$ spherical harmonics in Cartesian coordinates and combine them to get real expressions, v.g. \begin{align} Y_3^3(\theta,\varphi)+ Y_3^{-3}(\theta,\phi)\sim (x-iy)^3+(x+iy)^3 = 2x^3-6 xy^2 \end{align} and so forth?
• I’m not sure what you mean. In general a polynomial like $z^3$ will expand in a sum of multipoles but the expansion is unique. – ZeroTheHero Dec 3 '19 at 13:48