Definition:Legendre's Standard Elliptic Integrals
Jump to navigation
Jump to search
Special Function
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:
- $k$, defined on the interval $0 < k < 1$
- $n \in \Z$
- $x = \sin \phi$, where $\phi$ is defined on the interval $0 \le \phi \le \pi / 2$.
Also see
- Results about Legendre's standard elliptic integrals can be found here.
Source of Name
This entry was named for Adrien-Marie Legendre.
Historical Note
The elliptic integrals were called that because they were first encountered in the problem of determining the length of the perimeter of an ellipse.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): elliptic integral
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): elliptic integral