Editing Weierstrass elliptic function (section)

== Motivation ==
A [[Cubic_form|cubic]] of the form <math>C_{g_2,g_3}^\mathbb{C}=\{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x-g_3\} </math>, where <math>g_2,g_3\in\mathbb{C}</math> are complex numbers with <math>g_2^3-27g_3^2\neq0</math>, cannot be [[Rational_variety|rationally parameterized]].<ref name=":5" /> Yet one still wants to find a way to parameterize it.

For the [[quadric]] <math>K=\left\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\right\}</math>; the [[unit circle]], there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
<math display="block">\psi:\mathbb{R}/2\pi\mathbb{Z}\to K, \quad t\mapsto(\sin t,\cos t).</math>
Because of the periodicity of the sine and cosine <math>\mathbb{R}/2\pi\mathbb{Z}</math> is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of <math>C_{g_2,g_3}^\mathbb{C} </math> by means of the doubly periodic <math>\wp </math>-function (see in the section "Relation to elliptic curves"). This parameterization has the domain <math>\mathbb{C}/\Lambda </math>, which is topologically equivalent to a [[torus]].<ref>{{citation|surname1=Rolf Busam| title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;259| isbn=978-3-540-32058-6|date=2006|language=German}}</ref>

There is another analogy to the trigonometric functions. Consider the integral function
<math display="block">a(x)=\int_0^x\frac{dy}{\sqrt{1-y^2}} .</math>
It can be simplified by substituting <math>y=\sin t </math> and <math>s=\arcsin x </math>:
<math display="block">a(x)=\int_0^s dt = s = \arcsin x .</math>
That means <math>a^{-1}(x) = \sin x </math>. So the sine function is an inverse function of an integral function.<ref>{{citation| surname1=Jeremy Gray|title=Real and the complex: a history of analysis in the 19th century|publication-place=Cham|at=p.&nbsp;71| isbn=978-3-319-23715-2|date=2015|language=German}}</ref>

Elliptic functions are the inverse functions of [[elliptic integral]]s. In particular, let:
<math display="block">u(z)=\int_z^\infin\frac{ds}{\sqrt{4s^3-g_2s-g_3}} .</math>
Then the extension of <math>u^{-1} </math> to the complex plane equals the <math>\wp </math>-function.<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;294|isbn=978-3-540-32058-6|date=2006|language=German}}</ref> This invertibility is used in [[complex analysis]] to provide a solution to certain [[Nonlinear_system#Nonlinear_differential_equations|nonlinear differential equations]] satisfying the [[Painlevé property]], i.e., those equations that admit [[Zeros and poles|poles]] as their only [[Movable_singularity|movable singularities]].<ref>{{cite book | last=Ablowitz | first=Mark J. | last2=Fokas | first2=Athanassios S. | title=Complex Variables: Introduction and Applications | publisher=Cambridge University Press | date=2003 | isbn=978-0-521-53429-1 | doi=10.1017/cbo9780511791246|page=185}}</ref>