Editing Legendre form (section)

== 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>