= Spin-raising and lowering operators
{title2=$\eth,\ \overline{\eth}$}
On a spin-$s$ field $f$, the spin-raising and lowering operations are $\eth f=-\sin^s\theta(\partial_\theta+i\csc\theta\partial_\varphi)(\sin^{-s}\theta f)$ and $\overline{\eth}f=-\sin^{-s}\theta(\partial_\theta-i\csc\theta\partial_\varphi)(\sin^s\theta f)$. The intermediate spin changes after each application. For an axisymmetric scalar $f(\mu)$, either operation applied twice yields $(1-\mu^2)f''(\mu)$.
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