= Ellipse perimeter expansion near a circle
{title2=$L=2\pi(1-t/4-3t^2/64+O(t^3)),\quad t=1-x^2$}
For an <ellipse> with semiaxes $1,x$, the perimeter integral has integrand $\sqrt{1-t\cos^2\theta}$. The uniformly convergent <binomial series> near $t=0$ permits termwise integration. Integrals of $\cos^2\theta$ and $\cos^4\theta$ over a full period give the displayed first corrections. The same coefficients follow from the <Frobenius method> for the transformed perimeter equation.
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