# Definition:Elliptic Integral of the First Kind/Complete/Definition 2

$\displaystyle K \left({k}\right) = \int \limits_0^1 \frac {\mathrm d v} {\sqrt{\left({1 - v^2}\right) \left({1 - k^2 v^2}\right)} }$
is the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.