Definition:Elliptic Integral of the First Kind/Incomplete

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Special Function

Definition 1

$\displaystyle F \left({k, \phi}\right) = \int \limits_0^\phi \frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi} }$

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

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

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


Note that:

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

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


Amplitude

The parameter $\phi = \operatorname{am} u$ of $u = F \left({k, \phi}\right)$ 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