Editing Elliptic function (section)

==Weierstrass ℘-function==
{{Main|Weierstrass elliptic function}}

One of the most important elliptic functions is the Weierstrass <math>\wp</math>-function. For a given period lattice <math>\Lambda</math> it is defined by

: <math>\wp(z)=\frac1{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(z-\lambda)^2}-\frac1{\lambda^2}\right).</math>

It is constructed in such a way that it has a pole of order two at every lattice point. The term <math>-\frac1{\lambda^2}</math> is there to make the series convergent.

<math>\wp</math> is an even elliptic function; that is, <math>\wp(-z)=\wp(z)</math>.<ref name=":0">{{citation|surname1=K. Chandrasekharan|title=Elliptic functions|publisher=Springer-Verlag|publication-place=Berlin|at=p.&nbsp;28|isbn=0-387-15295-4|date=1985|language=German
}}</ref>

Its derivative

: <math>\wp'(z)=-2\sum_{\lambda\in\Lambda}\frac1{(z-\lambda)^3}</math>

is an odd function, i.e. <math>\wp'(-z)=-\wp'(z).</math><ref name=":0" />

One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice <math>\Lambda</math> can be expressed as a rational function in terms of <math>\wp</math> and <math>\wp'</math>.<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;275|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

The <math>\wp</math>-function satisfies the [[differential equation]]

: <math>\wp'(z)^2=4\wp(z)^3-g_2\wp(z)-g_3,</math>

where <math>g_2</math> and <math>g_3</math> are constants that depend on <math>\Lambda</math>. More precisely, <math>g_2(\omega_1,\omega_2)=60G_4(\omega_1,\omega_2)</math> and <math>g_3(\omega_1,\omega_2)=140G_6(\omega_1,\omega_2)</math>, where <math>G_4</math> and <math>G_6</math> are so called [[Eisenstein series]].<ref>{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;276|isbn=978-3-540-32058-6|date=2006|language=German
}}</ref>

In algebraic language, the field of elliptic functions is isomorphic to the field

: <math>\mathbb C(X)[Y]/(Y^2-4X^3+g_2X+g_3)</math>,

where the isomorphism maps <math>\wp</math> to <math>X</math> and <math>\wp'</math> to <math>Y</math>.<gallery>
File:Weierstrass-p-1.jpg|Weierstrass <math>\wp</math>-function with period lattice <math>\Lambda=\mathbb{Z}+e^{2\pi i/6}\mathbb{Z}  </math>
File:Weierstrass-dp-1.jpg|Derivative of the <math>\wp</math>-function
</gallery>