Complete elliptic integral of the second kind

Summary

The complete elliptic integral of the second kind is the function $E(k) = \int_0^{\pi/2} \sqrt{1 - k^2\sin^2\theta}\,d\theta$ of the modulus $k$. It gives the perimeter of an ellipse: an ellipse with semi-major axis $a$ and eccentricity $e$ has perimeter $4aE(e)$. Together with the complete integral of the first kind it enters Legendre's relation.