Editing Weierstrass elliptic function (section)

===The constants ''e''<sub>1</sub>, ''e''<sub>2</sub> and ''e''<sub>3</sub>===
<math>e_1</math>, <math>e_2</math> and <math>e_3</math> are usually used to denote the values of the <math>\wp</math>-function at the half-periods.
<math display="block">e_1\equiv\wp\left(\frac{\omega_1}{2}\right)</math>
<math display="block">e_2\equiv\wp\left(\frac{\omega_2}{2}\right)</math>
<math display="block">e_3\equiv\wp\left(\frac{\omega_1+\omega_2}{2}\right)</math>
They are pairwise distinct and only depend on the lattice <math>\Lambda</math> and not on its generators.<ref>{{citation| first=Rolf | last = Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin| at=p.&nbsp;270|isbn=978-3-540-32058-6|date=2006|language=German}}</ref>

<math>e_1</math>, <math>e_2</math> and <math>e_3</math> are the roots of the cubic polynomial <math>4\wp(z)^3-g_2\wp(z)-g_3</math> and are related by the equation:
<math display="block">e_1+e_2+e_3=0.</math>
Because those roots are distinct the discriminant <math>\Delta</math> does not vanish on the upper half plane.<ref>{{citation| first=Tom M. |last = Apostol|title=Modular functions and Dirichlet series in number theory|publisher=Springer-Verlag|publication-place=New York|at=p.&nbsp;13|isbn=0-387-90185-X|date=1976|language=German}}</ref> Now we can rewrite the differential equation:
<math display="block">\wp'^2(z)=4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3).</math>
That means the half-periods are zeros of <math>\wp'</math>.

The invariants <math>g_2</math> and <math>g_3</math> can be expressed in terms of these constants in the following way:<ref>{{citation|surname1=K. Chandrasekharan|title=Elliptic functions|publisher=Springer-Verlag|publication-place=Berlin|at=p.&nbsp;33| isbn=0-387-15295-4|date=1985|language=German}}</ref>
<math display="block">g_2 = -4 (e_1 e_2 + e_1 e_3 + e_2 e_3)</math>
<math display="block">g_3 = 4 e_1 e_2 e_3</math>
<math>e_1</math>, <math>e_2</math> and <math>e_3</math> are related to the [[modular lambda function]]:
<math display="block">\lambda (\tau)=\frac{e_3-e_2}{e_1-e_2},\quad \tau=\frac{\omega_2}{\omega_1}.</math>