= Homogeneous harmonic inversion
{title2=$\Delta(r^{-(2n+1)}H_n)=r^{-(2n+1)}\Delta H_n$}
A degree-$n$ <homogeneous polynomial> satisfies <Euler homogeneous function theorem> $\mathbf r\cdot\nabla H_n=nH_n$. The <product rule> gives $\Delta(r^sH_n)=r^s\Delta H_n+s(2n+s+1)r^{s-2}H_n$ in three dimensions. Taking $s=-(2n+1)$ gives the displayed identity and the equivalence of harmonicity on $r>0$. The origin is excluded; even $1/r$ has a distributional <Dirac delta function> source there.
Back to article page
