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)
Jump to navigation
Jump to search
Special Function
Definition 1
- $\ds \map E {k, \phi} = \int \limits_0^\phi \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$
is the incomplete elliptic integral of the second kind, and is a function of the variables:
Definition 2
- $\ds \map E {k, \phi} = \int \limits_0^x \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \rd v$
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 $0 \le \phi \le \pi / 2$.
Completion
- $\map E {k, \dfrac \pi 2} = \map E k$
where $\map E k$ denotes the complete elliptic integral of the second kind.
Also known as
Some sources omit the incomplete from the name, calling this merely the elliptic integral of the second kind.
Also see
- Results about the incomplete elliptic integral of the second kind can be found here.