Definition:Elliptic Integral

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Special Function

An elliptic integral is an integral in the form:

$\ds \int_0^x \map R {t, \sqrt {\map P t} } \rd t$

where:

$\map P t$ is a polynomial of degree $3$ or $4$
$\map R {t, \sqrt {\map P t} }$ is a rational function of $t$ and $\sqrt {\map P t}$.


There exist some special cases:


Elliptic Integral of the First Kind

Incomplete Elliptic Integral of the First Kind

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

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


Complete Elliptic Integral of the First Kind

$\ds \map K k = \int \limits_0^{\pi / 2} \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$

is the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.


Elliptic Integral of the Second Kind

Incomplete Elliptic Integral of the Second Kind

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

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


Complete Elliptic Integral of the Second Kind

$\ds \map E k = \int \limits_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$

is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.


Elliptic Integral of the Third Kind

Incomplete Elliptic Integral of the Third Kind

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

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

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


Complete Elliptic Integral of the Third Kind

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

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

$k$, defined on the interval $0 < k < 1$
$n \in \Z$


Also see

  • Results about elliptic integrals can be found here.


Historical Note

The elliptic integrals were called that because they were first encountered in the problem of determining the length of the perimeter of an ellipse.


Sources