Editing Legendre form

{{short description|Canonical set of three elliptic integrals}}
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-major 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-first1=И. С. |author-last1=Градштейн |author-link1=Израиль Соломонович Градштейн |author-first2=И. М. |author-last2=Рыжик |author-link2=Иосиф Моисеевич Рыжик |editor-first1=Ю. В. |editor-last1=Геронимус |editor-link1=Юрий Венеаминович Геронимус |editor-first2=М. Ю́. |editor-last2=Цейтлин |editor-link2=Михаил Ю́льевич Цейтлин |script-title=ru:Таблицы интегралов, сумм, рядов и произведений |title=Tablitsy integralov, summ, rjadov i proizvedenii |trans-title=Tables of Integrals, Sums, Series, and Products |language=Russian |edition=5 |date=1971 |publisher=[[Nauka (publisher)|Nauka]] |location=Moscow |lccn=78876185 |title-link=Gradshteyn and Ryzhik |chapter=8.1: Special Functions: Elliptic Integrals and Functions}}</ref> as well as [[Numerical Recipes]].<ref name="numerical_recipes">{{cite book
  | author1 = William H. Press
  | author2 = Saul A. Teukolsky
  | author3 = William T. Vetterling
  | author4 = 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 = [https://archive.org/details/numericalrecipes0865unse/page/261 261–271]
  | isbn = 0-521-43108-5
  | chapter-url = https://archive.org/details/numericalrecipes0865unse/page/261
  }}
</ref> The choice of sign is that of [[Abramowitz and Stegun]]<ref name="abramowitz_stegun">{{AS ref|17: Elliptic Integrals|589||589–628|first1=Louis Melville |last1=Milne-Thomson|link1=Louis Melville Milne-Thomson}}</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 '''[[Landen's transformation|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> approaching 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]]