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In | In [[mathematics]], the '''Legendre forms of [[elliptic integral]]s''' are a canonical set of three elliptic integrals to which all others may be reduced. [[Adrien-Marie Legendre|Legendre]] chose the name ''elliptic integrals'' because<ref>{{cite book | ||
| last = Gratton-Guinness | |||
| first = Ivor | |||
| title = The Fontana History of the Mathematical Sciences | |||
| publisher = Fontana Press | |||
| year = 1997 | |||
| pages = 308 | |||
| isbn = 0-00-686179-2}} | |||
</ref> the second kind gives the [[arc length]] of an [[ellipse]] of unit semi-minor axis and [[eccentricity (mathematics)|eccentricity]] <math>\scriptstyle{k}</math> (the ellipse being defined parametrically by <math>\scriptstyle{x = \sqrt{1 - k^{2}} \cos(t)}</math>, <math>\scriptstyle{y = \sin(t)}</math>). | |||
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the [[Carlson symmetric form]]s. A more detailed treatment of the Legendre forms is given in the main article on [[elliptic integral]]s. | |||
== Definition == | |||
The '''incomplete elliptic integral of the first kind''' is defined as, | |||
:<math>F(\phi,k) = \int_0^\phi \frac{1}{\sqrt{1 - k^2 \sin^2(t)}} dt,</math> | |||
the '''second kind''' as | |||
:<math>E(\phi,k) = \int_0^\phi \sqrt{1 - k^2 \sin^2(t)}\,dt,</math> | |||
and the '''third kind''' as | |||
:<math>\Pi(\phi,n,k) = \int_0^\phi \frac{1}{(1 - n \sin^2(t))\sqrt{1 - k^2 \sin^2(t)}}\,dt.</math> | |||
The argument ''n'' of the third kind of integral is known as the '''characteristic''', which in different notational conventions can appear as either the first, second or third argument of ''Π'' and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik<ref name=gradshteyn_ryzhik>{{cite book | |||
| author = Israil Solomonovich Gradshteyn, Iosif Moiseevich Ryzhik, | |||
| title = Tables of Series, Products and Integrals | |||
| language = Russian | |||
| location = Moscow | |||
| publisher = Nauka | |||
| year = 1971 | |||
| edition = 5 | |||
| chapter = Chap. 8.1 Special Functions: Elliptic Integrals and Functions}} | |||
</ref> as well as [[Numerical Recipes]].<ref name=numerical_recipes>{{cite book | |||
| author = William H. Press, Saul A. Teukolsky, William T. Vetterlimg and Brian P. Flannery | |||
| title = Numerical Recipes in C | |||
| publisher = Cambridge University Press | |||
| year = 1992 | |||
| edition = 2 | |||
| chapter = Chap. 6.11 Special Functions: Elliptic Integrals and Jacobian Functions | |||
| pages = 261–271 | |||
| isbn = 0-521-43108-5 | |||
| url = http://www.nrbook.com/a/bookcpdf/c6-11.pdf}} | |||
</ref> The choice of sign is that of [[Abramowitz and Stegun]]<ref name=abramowitz_stegun>{{cite book | |||
| author = Milton Abramowitz and Irene A. Stegun, | |||
| title = Handbook of Mathematical Functions | |||
| publisher = Dover | |||
| year = 1964 | |||
| chapter = Chap. 17 Elliptic Integrals | |||
| pages = 589–628 | |||
| isbn = 0-486-61272-4 | |||
| url = http://www.math.sfu.ca/~cbm/aands/page_589.htm}} | |||
</ref> as well as Gradshteyn and Ryzhik,<ref name=gradshteyn_ryzhik/> but corresponds to the <math>\scriptstyle{\Pi(\phi,-n,k)}</math> of [[Numerical Recipes]].<ref name=numerical_recipes/> | |||
The respective '''complete elliptic integrals''' are obtained by setting the '''amplitude''', <math>\scriptstyle{\phi}</math>, the upper limit of the integrals, to <math>\scriptstyle{\pi / 2}</math>. | |||
The Legendre form of an [[elliptic curve]] is given by | |||
:<math>y^2 = x(x - 1)(x - \lambda)</math> | |||
== Numerical Evaluation == | |||
The classic method of evaluation is by means of '''[[John Landen|Landen]]'s transformations'''. Descending Landen transformation decreases the '''modulus''' <math>\scriptstyle{k}</math> towards zero, while increasing the amplitude <math>\scriptstyle{\phi}</math>. Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of <math>\scriptstyle{k}</math>, zero or one, the integral is readily evaluated. | |||
Most modern authors recommend evaluation in terms of the [[Carlson symmetric form]]s, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by [[Boost C++ Libraries]], [[GNU Scientific Library]]<!--see gsl/specfunc/ellint.c--> and [[Numerical Recipes]].<ref name=numerical_recipes/> | |||
== References == | |||
<references /> | |||
== See also == | |||
* [[Carlson symmetric form]] | |||
{{DEFAULTSORT:Legendre Form}} | |||
[[Category:Special functions]] | |||
Revision as of 06:09, 28 February 2013
In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because[1] the second kind gives the arc length of an ellipse of unit semi-minor axis and eccentricity (the ellipse being defined parametrically by , ).
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
Definition
The incomplete elliptic integral of the first kind is defined as,
the second kind as
and the third kind as
The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik[2] as well as Numerical Recipes.[3] The choice of sign is that of Abramowitz and Stegun[4] as well as Gradshteyn and Ryzhik,[2] but corresponds to the of Numerical Recipes.[3]
The respective complete elliptic integrals are obtained by setting the amplitude, , the upper limit of the integrals, to .
The Legendre form of an elliptic curve is given by
Numerical Evaluation
The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus towards zero, while increasing the amplitude . Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of , zero or one, the integral is readily evaluated.
Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.[3]
References
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 2.0 2.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 3.0 3.1 3.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534