= Complete elliptic integral of the second kind
{title2=$E(k)=E(k,\pi/2)$}
For $0\leq k\leq1$, define $E(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2\theta}\,d\theta$. An <ellipse> with <semi-major and semi-minor axes> $a\geq b>0$ and <eccentricity of a conic> $e=\sqrt{1-b^2/a^2}$ has perimeter $4aE(e)$. Differentiating its parametrization gives the speed $a\sqrt{1-e^2\sin^2\theta}$, and fourfold symmetry reduces the full <arc length> to a quadrant. The <binomial series> gives $E(k)=(\pi/2)(1-k^2/4-3k^4/64+O(k^6))$ near zero.
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