= Null-vector cancellation of radial moments
{title2=$\int_{|x|<1}(x_1+ix_2)^n f(|x|)\,d^3x=0\quad(n\geq1)$}
Assume the radial moment integral exists. Rotation through $\phi$ about the third axis multiplies $x_1+ix_2$ by $e^{i\phi}$ and preserves the region, volume element and radial weight. The integral equals $e^{in\phi}$ 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.
Back to article page
