Definition:Elliptic Integral of the First Kind/Incomplete/Definition 1

Special Function

$\displaystyle F \left({k, \phi}\right) = \int \limits_0^\phi \frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi} }$

is the incomplete elliptic integral of the first kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$
$\phi$, defined on the interval $0 \le \phi \le \pi / 2$.

Amplitude

The parameter $\phi = \operatorname{am} u$ of $u = F \left({k, \phi}\right)$ is called the amplitude of $u$.