 Elliptic-integral circular-speed kernel of a thin disk

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.