Definition:Elliptic Integral of the Second Kind/Incomplete

Special Function

Definition 1

$\displaystyle \map E {k, \phi} = \int \limits_0^\phi \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$

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 $0 \le \phi \le \pi / 2$.

Definition 2

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

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 $0 \le \phi \le \pi / 2$.

Note that:

$E \left({k, \dfrac \pi 2}\right) = E \left({k}\right)$

where $E \left({k}\right)$ denotes the complete elliptic integral of the second kind.

Also known as

Some sources omit the incomplete from the definition, calling this merely the elliptic integral of the second kind.