Complete elliptic integral of the first kind

Summary

The complete elliptic integral of the first kind is the function $K(k) = \int_0^{\pi/2} d\theta / \sqrt{1 - k^2\sin^2\theta}$ of the modulus $k$. It is a quarter of the real period of the Jacobi elliptic functions $\operatorname{sn}$ and $\operatorname{cn}$. It gives the exact period of a pendulum and can be computed very fast with the arithmetic–geometric mean.