Elliptic integral of the second kind ($E(k,\varphi )={\int }_0^{\varphi }\sqrt{1-k^2{\sin }^2\theta }d\theta$)

For real modulus $0\le k\le 1$, the displayed integral is the incomplete elliptic integral of the second kind, with amplitude $\varphi$. The modulus convention matters: some sources instead use the parameter $m=k^2$. The arc length of an ellipse reduces to this integral, because a trigonometric parametrization gives a square root of $1-k^2{\sin }^2\theta$. At $k=0$ it reduces to $\varphi$; at $k=1$ the real integrand is $|\cos \theta |$.

For $0\le k\le 1$, 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 $a\ge b>0$ 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.