= Elliptic-integral circular-speed kernel of a thin disk
{title2=$V_d^2(r)=2G\operatorname{PV}\int s\Sigma(s)[K/(r+s)+E/(r-s)]\,ds$}
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.
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