Definition:Incomplete Elliptic Integral of the Second Kind/Definition 2

Special Function

 * $\ds \map E {k, \phi} = \int \limits_0^x \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \rd v$

is the incomplete elliptic integral of the second kind, and is a function of the variables:
 * $k$, defined on the interval $0 < k < 1$
 * $x = \sin \phi$, where $\phi$ is defined on the interval $0 \le \phi \le \pi / 2$.

Also see

 * Equivalence of Definitions of Incomplete Elliptic Integral of the Second Kind