Complete elliptic integral of the second kind

Remarks

Example: $E(0) = \pi/2$, and $E(1) = \int_0^{\pi/2} \cos\theta \, d\theta = 1$. The ellipse $x = a\sin\theta$, $y = b\cos\theta$ with $0 < b \le a$ has perimeter $4aE(e)$, where $e = \sqrt{1 - b^2/a^2}$ is its eccentricity, because its arc length element is $a\sqrt{1 - e^2\sin^2\theta}\,d\theta$. Non-example: for real $k > 1$ the integrand $\sqrt{1 - k^2\sin^2\theta}$ is not real for $\theta$ near $\pi/2$, so the formula does not define a real number there. Series: $E(k) = \frac{\pi}{2}\left(1 - \sum_{n=1}^{\infty} \left(\frac{(2n)!}{2^{2n}(n!)^2}\right)^2 \frac{k^{2n}}{2n - 1}\right)$. Some authors write $E(m)$ with the parameter $m = k^2$.