= Poisson equation with a spherical harmonic source
{c}
{title2=$\Delta(r^pY_\ell)=[p(p+1)-\ell(\ell+1)]r^{p-2}Y_\ell$}
Let $Y_\ell$ be a <spherical harmonic> of degree $\ell$ on the unit <sphere>. The <Laplacian> in <spherical coordinates> separates its angular part from a radial power, giving the displayed identity for $r>0$. Thus a <Poisson equation> $\Delta\Phi=r^qY_\ell$ has the particular solution $\Phi=r^{q+2}Y_\ell/[(q+2)(q+3)-\ell(\ell+1)]$ when the denominator is nonzero. Add $Ar^\ell Y_\ell+Br^{-\ell-1}Y_\ell$ to impose the radial <boundary conditions>. If the denominator vanishes at $p=q+2$, differentiating the displayed identity with respect to $p$ gives instead $\Delta(r^p\log r\,Y_\ell)=(2p+1)r^{p-2}Y_\ell$. Regularity at the origin and decay at infinity must be checked separately; the radial identity alone is local away from zero.
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