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 real $|k|<1$, define $E(\varphi ,k)={\int }_0^{\varphi }\sqrt{1-k^2{\sin }^{2}x}dx$ and $F(\varphi ,k)={\int }_0^{\varphi }(1-k^2{\sin }^{2}x{)}^{-1/2}dx$. Smoothness of the integrands permits differentiation under the integral sign. Subtracting the two integrands gives $E-F=-k^2{\int }_0^{\varphi }{\sin }^{2}x/\sqrt{1-k^2{\sin }^{2}x}dx$, which is $k{\partial }_{k}E$. This proves the displayed identity for $k\ne 0$. At $k=0$, both sides extend continuously with value zero. The variable $k$ is the modulus; derivatives with respect to the parameter $m=k^2$ instead obey ${\partial }_{m}E=(E-F)/(2m)$, with removable value $-\frac{1}{2}{\int }_0^{\varphi }{\sin }^{2}xdx$ at zero.

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.