Definition:Elliptic Integral of the Second Kind/Incomplete
< Definition:Elliptic Integral of the Second Kind(Redirected from Definition:Incomplete Elliptic Integral of the Second Kind)
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Special Function
Definition 1
- $\displaystyle E \left({k, \phi}\right) = \int \limits_0^\phi \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi$
is the incomplete elliptic integral of the second kind, and is a function of the variables:
Definition 2
- $\displaystyle E \left({k, \phi}\right) = \int \limits_0^x \dfrac {\sqrt{1 - k^2 v^2} } {\sqrt{1 - v^2}} \, \mathrm d v$
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 $0 \le \phi \le \pi / 2$.
Note that:
- $E \left({k, \dfrac \pi 2}\right) = E \left({k}\right)$
where $E \left({k}\right)$ denotes the complete elliptic integral of the second kind.
Also known as
Some sources omit the incomplete from the definition, calling this merely the elliptic integral of the second kind.