Category:Definitions/Legendre's Standard Elliptic Integrals

This category contains definitions related to Legendre's Standard Elliptic Integrals.
Related results can be found in Category:Legendre's Standard Elliptic Integrals.

Legendre's standard elliptic integrals are the following elliptic integrals:

Elliptic Integral of the First Kind

$\ds \map F {k, \phi} = \int \limits_0^x \frac {\d v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

is the incomplete elliptic integral of the first 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 $0 \le \phi \le \pi / 2$.

Elliptic Integral of the Second Kind

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

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

Elliptic Integral of the Third Kind

$\ds \map \Pi {k, n, \phi} = \int \limits_0^x \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

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

$k$, defined on the interval $0 < k < 1$

$n \in \Z$

$x = \sin \phi$, where $\phi$ is defined on the interval $0 \le \phi \le \pi / 2$.