Legendre expansion of thin-disk gravity ($\Phi =-2G{\sum }_{k\ge 0}{\alpha }_{2k}(R^{-2k-1}{I}_{2k}+R^{2k}{J}_{2k})$)

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.

Integrating a circular ring's Newtonian gravitational potential over azimuth gives $d\Phi =-4Gs\Sigma (s)K(k)ds/(r+s)$, with modulus $k=2\sqrt{rs}/(r+s)$. Differentiating using the complete elliptic integral of the first kind derivative gives the displayed circular speed kernel, where $E$ is the complete elliptic integral of the second kind. For an interior point of a smooth disk use a Cauchy principal value; beyond the outer radius the integral is ordinary. A discontinuous nonzero surface-density edge in an exactly zero-thickness disk has a logarithmically singular in-plane force at that edge.