= Highest-weight complex-coordinate orbital wavefunction
{title2=$L^2[(x+iy)^nf(r)]=\hbar^2n(n+1)(x+iy)^nf(r)$}
For a differentiable radial factor and integer $n\geq0$, the coordinate polynomial obeys $L_3\psi=n\hbar\psi$ and $L_+\psi=0$. The identity $L^2=L_-L_++L_3^2+\hbar L_3$ gives the displayed angular <eigenvalue>. Lowering gives $L_-\psi=-2n\hbar z(x+iy)^{n-1}f(r)$, a nonzero <eigenfunction> with the same angular <eigenvalue> when $n>0$ and the original state is nonzero. At $n=0$ lowering gives the zero function, which is not an <eigenfunction>. Negative powers satisfy local differential identities off the axis but are not regular angular states.
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