= Legendre expansion of thin-disk gravity
{title2=$\Phi=-2G\sum_{k\geq0}\alpha_{2k}(R^{-2k-1}I_{2k}+R^{2k}J_{2k})$}
For a thin axisymmetric <astrophysical disk>, angular integration of the Newtonian kernel eliminates odd <Legendre polynomials>. Define $I_n=\int_0^R\Sigma(s)s^{n+1}ds$ and $J_n=\int_R^\infty\Sigma(s)s^{-n}ds$. The even coefficient is $\alpha_n=\pi[n!/(2^n((n/2)!)^2)]^2$. In differentiating the paired inner and outer contributions, their moving-boundary terms cancel. The resulting <galaxy rotation curve> includes an inward inner-ring contribution and an outward exterior-ring contribution. For smooth <surface density>, the force at the disk is interpreted with a symmetric <Cauchy principal value> or a vanishing-thickness regularization.
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