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:
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 name, 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.