Definition:Elliptic Integral
Special Function
An elliptic integral is an integral in the form:
- $\ds \int_0^x \map R {t, \sqrt {\map P t} } \rd t$
where:
- $\map P t$ is a polynomial of degree $3$ or $4$
- $\map R {t, \sqrt {\map P t} }$ is a rational function of $t$ and $\sqrt {\map P t}$.
There exist some special cases:
Elliptic Integral of the First Kind
Incomplete Elliptic Integral of the First Kind
- $\ds \map F {k, \phi} = \int \limits_0^\phi \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$
is the incomplete elliptic integral of the first kind, and is a function of the variables:
Complete Elliptic Integral of the First Kind
- $\ds \map K k = \int \limits_0^{\pi / 2} \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$
is the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.
Elliptic Integral of the Second Kind
Incomplete Elliptic Integral of the Second Kind
- $\ds \map E {k, \phi} = \int \limits_0^\phi \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$
is the incomplete elliptic integral of the second kind, and is a function of the variables:
Complete Elliptic Integral of the Second Kind
- $\ds \map E k = \int \limits_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$
is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.
Elliptic Integral of the Third Kind
Incomplete Elliptic Integral of the Third Kind
- $\ds \map \Pi {k, n, \phi} = \int \limits_0^\phi \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt{1 - k^2 \sin^2 \phi} }$
is the incomplete elliptic integral of the third kind, and is a function of the variables:
- $k$, defined on the interval $0 < k < 1$
- $n \in \Z$
- $\phi$, defined on the interval $0 \le \phi \le \pi / 2$.
Complete Elliptic Integral of the Third Kind
- $\ds \map \Pi {k, n} = \int \limits_0^{\pi / 2} \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt {1 - k^2 \sin^2 \phi} }$
is the complete elliptic integral of the third kind, and is a function of the variables:
- $k$, defined on the interval $0 < k < 1$
- $n \in \Z$
Also see
- Results about elliptic integrals can be found here.
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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): elliptic integral
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($\text {1815}$ – $\text {1897}$)
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): elliptic integral