Definition:Elliptic Integral of the First Kind/Incomplete

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Special Function

Definition 1

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

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


Definition 2

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

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


Completion

$\map F {k, \dfrac \pi 2} = \map K k$

where $\map K k$ denotes the complete elliptic integral of the first kind.


Amplitude

The parameter $\phi$ of $u = \map F {k, \phi}$ is called the amplitude of $u$.


Also known as

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


Also see




  • Results about the incomplete elliptic integral of the first kind can be found here.