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

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Special Function

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

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

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

Also see