Category:Legendre's Standard Elliptic Integrals
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This category contains results about Legendre's Standard Elliptic Integrals.
Definitions specific to this category can be found in Definitions/Legendre's Standard Elliptic Integrals.
Legendre's standard elliptic integrals are the following elliptic integrals:
Elliptic Integral of the First Kind
- $\ds \map F {k, \phi} = \int \limits_0^x \frac {\d v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$
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 $0 \le \phi \le \pi / 2$.
Elliptic Integral of the Second Kind
- $\ds \map E {k, \phi} = \int \limits_0^x \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \rd v$
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 $0 \le \phi \le \pi / 2$.
Elliptic Integral of the Third Kind
- $\ds \map \Pi {k, n, \phi} = \int \limits_0^x \frac {\d v} {\paren {1 + n v^2} \sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$
is the incomplete elliptic integral of the third kind, and is a function of the variables:
Subcategories
This category has the following 3 subcategories, out of 3 total.