Definition:Elliptic Integral of the First Kind/Complete

Special Function
The integral:
 * $\displaystyle \int \limits_0^{\pi / 2} {\frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi}}}$

is known as the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.

It is denoted $K \left({k}\right)$.

Also see

 * Definition:Incomplete Elliptic Integral of the First Kind


 * Definition:Incomplete Elliptic Integral of the Second Kind
 * Definition:Complete Elliptic Integral of the Second Kind


 * Definition:Incomplete Elliptic Integral of the Third Kind
 * Definition:Complete Elliptic Integral of the Third Kind