Editing Weierstrass elliptic function (section)

== Laurent expansion ==
Let <math>r:=\min\{{|\lambda}|:0\neq\lambda\in\Lambda\}</math>. Then for <math>0<|z|<r</math> the <math>\wp</math>-function has the following [[Laurent series|Laurent expansion]]
<math display="block">\wp(z)=\frac1{z^2}+\sum_{n=1}^\infin (2n+1)G_{2n+2}z^{2n} </math>
where
<math display="block">G_n=\sum_{0\neq\lambda\in\Lambda}\lambda^{-n}</math> for <math>n \geq 3</math> are so called [[Eisenstein series]].<ref name=":1" />

==Differential equation==
Set <math>g_2=60G_4</math> and <math>g_3=140G_6</math>. Then the <math>\wp</math>-function satisfies the differential equation<ref name=":1" />
<math display="block"> \wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3.</math>
This relation can be verified by forming a linear combination of powers of <math>\wp</math> and <math>\wp'</math> to eliminate the pole at <math>z=0</math>. This yields an entire elliptic function that has to be constant by [[Liouville's theorem (complex analysis)|Liouville's theorem]].<ref name=":1" />

===Invariants===
[[Image:Gee three real.jpeg|thumb|The real part of the invariant ''g''<sub>3</sub> as a function of the square of the nome ''q'' on the unit disk.]]
[[Image:Gee three imag.jpeg|thumb|The imaginary part of the invariant ''g''<sub>3</sub> as a function of the square of the nome ''q'' on the unit disk.]]

The coefficients of the above differential equation <math>g_2</math> and <math>g_3</math> are known as the ''invariants''. Because they depend on the lattice <math>\Lambda</math> they can be viewed as functions in <math>\omega_1</math> and <math>\omega_2</math>.

The series expansion suggests that <math>g_2</math> and <math>g_3</math> are [[homogeneous function]]s of degree <math>-4</math> and <math>-6</math>. That is<ref name=":0">{{Cite book|last=Apostol, Tom M.|url=https://www.worldcat.org/oclc/2121639|title=Modular functions and Dirichlet series in number theory|date=1976|publisher=Springer-Verlag|isbn=0-387-90185-X|location=New York| pages=14| oclc=2121639}}</ref>
<math display="block">g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)</math>
<math display="block">g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2)</math> for <math>\lambda \neq 0</math>.

If <math>\omega_1</math> and <math>\omega_2</math> are chosen in such a way that <math>\operatorname{Im}\left( \tfrac{\omega_2}{\omega_1} \right)>0 </math>, <math>g_2</math> and <math>g_3</math> can be interpreted as functions on the [[upper half-plane]] <math>\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}</math>.

Let <math>\tau=\tfrac{\omega_2}{\omega_1}</math>. One has:<ref name=":2">{{citation|title=Modular functions and Dirichlet series in number theory|date=1976|at=p.&nbsp;14|publication-place=New York|publisher=Springer-Verlag|language=German|isbn=0-387-90185-X| surname1=Apostol, Tom M.}}</ref>
<math display="block">g_2(1,\tau)=\omega_1^4g_2(\omega_1,\omega_2),</math>
<math display="block">g_3(1,\tau)=\omega_1^6 g_3(\omega_1,\omega_2).</math>
That means ''g''<sub>2</sub> and ''g''<sub>3</sub> are only scaled by doing this. Set
<math display="block">g_2(\tau):=g_2(1,\tau) </math> and <math display="block">g_3(\tau):=g_3(1,\tau).</math>
As functions of <math>\tau\in\mathbb{H}</math>, <math>g_2</math> and <math>g_3</math> are so called [[Modular form|modular forms.]]

The [[Fourier series]] for <math>g_2</math> and <math>g_3</math> are given as follows:<ref>{{Cite book|last=Apostol, Tom M.|url=https://www.worldcat.org/oclc/20262861|title=Modular functions and Dirichlet series in number theory|date=1990| publisher=Springer-Verlag|isbn=0-387-97127-0|edition=2nd|location=New York|pages=20|oclc=20262861}}</ref>
<math display="block">g_2(\tau)=\frac43\pi^4 \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] </math>
<math display="block">g_3(\tau)=\frac{8}{27}\pi^6 \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] </math>
where
<math display="block">\sigma_m(k):=\sum_{d\mid{k}}d^m</math>
is the [[divisor function]] and <math>q=e^{\pi i\tau}</math> is the [[Nome (mathematics)|nome]].

===Modular discriminant===
[[Image:Discriminant real part.jpeg|thumb|The real part of the discriminant as a function of the square of the nome ''q'' on the unit disk.]]

The ''modular discriminant'' <math>\Delta</math> is defined as the [[discriminant]] of the characteristic polynomial of the differential equation <math display="block"> \wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3</math> as follows:
<math display="block"> \Delta=g_2^3-27g_3^2. </math>
The discriminant is a modular form of weight <math>12</math>. That is, under the action of the [[modular group]], it transforms as
<math display="block">\Delta \left( \frac {a\tau+b} {c\tau+d}\right) = \left(c\tau+d\right)^{12} \Delta(\tau) </math>
where <math>a,b,d,c\in\mathbb{Z}</math> with <math>ad-bc = 1</math>.<ref>{{Cite book|last=Apostol | first = Tom M.| url=https://www.worldcat.org/oclc/2121639|title=Modular functions and Dirichlet series in number theory| date=1976| publisher=Springer-Verlag|isbn=0-387-90185-X|location=New York|pages=50|oclc=2121639}}</ref>

Note that <math>\Delta=(2\pi)^{12}\eta^{24}</math> where <math>\eta</math> is the [[Dedekind eta function]].<ref>{{Cite book| last=Chandrasekharan, K. (Komaravolu), 1920-|url=https://www.worldcat.org/oclc/12053023|title=Elliptic functions| date=1985| publisher=Springer-Verlag|isbn=0-387-15295-4|location=Berlin|pages=122|oclc=12053023}}</ref>

For the Fourier coefficients of <math>\Delta</math>, see [[Ramanujan tau function]].

===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>