Null-vector cancellation of radial moments (${\int }_{|x|<1}(x_1+i{x}_2{)}^{n}f(|x|)d^{3}x=0(n\ge 1)$)

Assume the radial moment integral exists. Rotation through $\varphi$ about the third axis multiplies $x_1+i{x}_2$ by $e^{i\varphi }$ and preserves the region, volume element and radial weight. The integral equals $e^{in\varphi }$ times itself for every angle, so it vanishes. For degree two and four this also follows by contracting the isotropic sphere moments with the complex null vector $(1,i,0)$, whose bilinear square is zero. Odd moments vanish under inversion.