Editing Weierstrass elliptic function

{{short description|Class of mathematical functions}}
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In [[mathematics]], the '''Weierstrass elliptic functions''' are [[elliptic function]]s that take a particularly simple form. They are named for [[Karl Weierstrass]]. This class of functions is also referred to as '''℘-functions''' and they are usually denoted by the symbol&nbsp;℘, a uniquely fancy [[Cursive|script]] ''p''. They play an important role in the theory of elliptic functions, i.e., [[meromorphic function]]s that are [[Doubly_periodic_function|doubly periodic]]. A ℘-function together with its derivative can be used to parameterize [[elliptic curve]]s and they generate the field of elliptic functions with respect to a given period lattice. 
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[[Image:Weierstrass p.svg|100px|Symbol for Weierstrass P function]]<div class="thumbcaption">
Symbol for Weierstrass <math>\wp</math>-function
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[[File:Modell der Weierstraßschen p-Funktion -Schilling, XIV, 7ab, 8 - 313, 314-.jpg|thumb|right|Model of Weierstrass <math>\wp</math>-function]]

== 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>

==Definition==
[[File:Weierstrass elliptic function P.png|thumb|200px|Visualization of the <math>\wp</math>-function with invariants <math>g_2=1+i</math> and <math>g_3=2-3i</math> in which white corresponds to a pole, black to a zero.]]

Let <math>\omega_1,\omega_2\in\mathbb{C}</math> be two [[complex number]]s that are [[Linear independence|linearly independent]] over <math>\mathbb{R}</math> and let <math>\Lambda:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2:=\{m\omega_1+n\omega_2: m,n\in\mathbb{Z}\}</math> be the [[period lattice]] generated by those numbers. Then the <math>\wp</math>-function is defined as follows:
:<math>\weierp(z,\omega_1,\omega_2):=\wp(z) = \frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right).</math>
This series converges locally [[Uniform absolute-convergence|uniformly absolutely]] in the [[complex torus]] <math>\mathbb{C} / \Lambda</math>.

It is common to use <math>1</math> and <math>\tau</math> in the [[upper half-plane]] <math>\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z) > 0\}</math> as [[Linear_span|generators]] of the [[Lattice_(group)|lattice]]. Dividing by <math display="inline">\omega_1</math> maps the lattice <math>\mathbb{Z}\omega_1+\mathbb{Z}\omega_2</math> isomorphically onto the lattice <math>\mathbb{Z}+\mathbb{Z}\tau</math> with <math display="inline">\tau=\tfrac{\omega_2}{\omega_1}</math>. Because <math>-\tau</math> can be substituted for <math>\tau</math>, without loss of generality we can assume <math>\tau\in\mathbb{H}</math>, and then define <math>\wp(z,\tau) := \wp(z, 1,\tau)</math>.  With that definition, we have <math>\wp(z,\omega_1,\omega_2) = \omega_1^{-2}\wp(z/\omega_1,\omega_2/\omega_1)</math>.

== Properties ==
* <math>\wp</math> is a [[meromorphic function]] with a pole of order 2 at each period <math>\lambda</math> in <math>\Lambda</math>.
* <math>\wp</math> is a [[homogeneous function]] in that:
::<math>\wp(\lambda z , \lambda\omega_{1}, \lambda\omega_{2}) = \lambda^{-2} \wp (z, \omega_{1},\omega_{2}).</math> 
* <math>\wp</math> is an even function. That means <math>\wp(z)=\wp(-z)</math> for all <math>z \in \mathbb{C} \setminus \Lambda</math>, which can be seen in the following way:
::<math>\begin{align}
\wp(-z) & =\frac{1}{(-z)^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(-z-\lambda)^2}-\frac{1}{\lambda^2}\right) \\
& =\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(z+\lambda)^2}-\frac{1}{\lambda^2}\right) \\
& =\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2}\right)=\wp(z).
\end{align}</math>
:The second last equality holds because <math>\{-\lambda:\lambda \in \Lambda\}=\Lambda</math>. Since the sum converges absolutely this rearrangement does not change the limit.
* The derivative of <math>\wp</math> is given by:<ref name=":1">{{citation|surname1=Apostol, Tom M.|title=Modular functions and Dirichlet series in number theory|publisher=Springer-Verlag|publication-place=New York|at=p.&nbsp;11|isbn=0-387-90185-X|date=1976| language=German}}</ref> <math display="block">\wp'(z)=-2\sum_{\lambda \in \Lambda}\frac1{(z-\lambda)^3}.</math>
* <math>\wp</math> and <math>\wp'</math> are [[Doubly periodic function|doubly periodic]] with the periods <math>\omega_1 </math> and <math>\omega_2</math>.<ref name=":1" /> This means: <math display="block">\begin{aligned}
\wp(z+\omega_1) &= \wp(z) = \wp(z+\omega_2),\ \textrm{and} \\[3mu]
\wp'(z+\omega_1) &= \wp'(z) = \wp'(z+\omega_2).
\end{aligned}</math> It follows that <math>\wp(z+\lambda)=\wp(z)</math> and <math>\wp'(z+\lambda)=\wp'(z)</math> for all <math>\lambda \in \Lambda</math>.

== 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>

==Relation to Jacobi's elliptic functions==

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of [[Jacobi's elliptic functions]].

The basic relations are:<ref>{{cite book | author = Korn GA, [[Theresa M. Korn|Korn TM]] | year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw–Hill | location = New York | pages = 721 | lccn = 59014456}}</ref>
<math display="block">
\wp(z) = e_3 + \frac{e_1 - e_3}{\operatorname{sn}^2 w}
= e_2 + ( e_1 - e_3 ) \frac{\operatorname{dn}^2 w}{\operatorname{sn}^2 w}
= e_1 + ( e_1 - e_3 ) \frac{\operatorname{cn}^2 w}{\operatorname{sn}^2 w}
</math>
where <math>e_1,e_2</math> and <math>e_3</math> are the three roots described above and where the modulus ''k'' of the Jacobi functions equals
<math display="block">k = \sqrt\frac{e_2 - e_3}{e_1 - e_3}</math>
and their argument ''w'' equals
<math display="block">w = z \sqrt{e_1 - e_3}.</math>

== Relation to Jacobi's theta functions ==
The function <math>\wp (z,\tau)=\wp (z,1,\omega_2/\omega_1)</math> can be represented by [[Theta function#Auxiliary functions|Jacobi's theta functions]]:
<math display="block">\wp (z,\tau)=\left(\pi \theta_2(0,q)\theta_3(0,q)\frac{\theta_4(\pi z,q)}{\theta_1(\pi z,q)}\right)^2-\frac{\pi^2}{3}\left(\theta_2^4(0,q)+\theta_3^4(0,q)\right)</math>
where <math>q=e^{\pi i\tau}</math> is the nome and <math>\tau</math> is the period ratio <math>(\tau\in\mathbb{H})</math>.<ref>{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=23.6.E7|title=Weierstrass Elliptic and Modular Functions}}</ref> This also provides a very rapid algorithm for computing <math>\wp (z,\tau)</math>.

== Relation to elliptic curves ==
{{see also|Elliptic curve#Elliptic curves over the complex numbers}}
Consider the embedding of the cubic curve in the [[complex projective plane]]
:<math>\bar C_{g_2,g_3}^\mathbb{C} = \{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x-g_3\}\cup\{O\}\subset \mathbb{C}^{2} \cup \mathbb{P}_1(\mathbb{C}) =  \mathbb{P}_2(\mathbb{C}).</math>

where <math>O</math> is a point lying on the [[line at infinity]] <math>\mathbb{P}_1(\mathbb{C})</math>. For this cubic there exists no rational parameterization, if <math>\Delta \neq 0</math>.<ref name=":5">{{citation|surname1=Hulek, Klaus.|title=Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen|edition=2., überarb. u. erw. Aufl. 2012|publisher=Vieweg+Teubner Verlag|publication-place=Wiesbaden|at=p.&nbsp;8|isbn=978-3-8348-2348-9|date=2012|language=German}}</ref> In this case it is also called an elliptic curve. Nevertheless there is a parameterization in [[homogeneous coordinates]] that uses the <math>\wp</math>-function and its derivative <math>\wp'</math>:<ref>{{citation|title=Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen|date=2012|at=p.&nbsp;12|edition=2., überarb. u. erw. Aufl. 2012|publication-place=Wiesbaden|publisher=Vieweg+Teubner Verlag|language=German|isbn=978-3-8348-2348-9|surname1=Hulek, Klaus.}}</ref>
:<math> \varphi(\wp,\wp'): \mathbb{C}/\Lambda\to\bar C_{g_2,g_3}^\mathbb{C}, \quad
z \mapsto \begin{cases}
\left[\wp(z):\wp'(z):1\right] & z \notin \Lambda \\
\left[0:1:0\right] \quad & z \in \Lambda 
\end{cases} </math>

Now the map <math>\varphi</math> is [[Bijection|bijective]] and parameterizes the elliptic curve <math>\bar C_{g_2,g_3}^\mathbb{C}</math>.

<math>\mathbb{C}/\Lambda </math> is an [[abelian group]] and a [[topological space]], equipped with the [[quotient topology]].

It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair <math>g_2,g_3\in\mathbb{C}</math> with <math>\Delta = g_2^3 - 27g_3^2 \neq 0 </math> there exists a lattice <math>\mathbb{Z}\omega_1+\mathbb{Z}\omega_2</math>, such that

<math>g_2=g_2(\omega_1,\omega_2) </math> and <math>g_3=g_3(\omega_1,\omega_2) </math>.<ref>{{citation|surname1=Hulek, Klaus.| title=Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen| edition=2., überarb. u. erw. Aufl. 2012|publisher=Vieweg+Teubner Verlag|publication-place=Wiesbaden|at=p.&nbsp;111| isbn=978-3-8348-2348-9| date=2012|language=German}}</ref>

The statement that elliptic curves over <math>\mathbb{Q}</math> can be parameterized over <math>\mathbb{Q}</math>, is known as the [[modularity theorem]]. This is an important theorem in [[number theory]]. It was part of [[Andrew Wiles|Andrew Wiles']] proof (1995) of [[Fermat's Last Theorem]].

==Addition theorems==
Let <math>z,w\in\mathbb{C}</math>, so that <math>z,w,z+w,z-w\notin\Lambda </math>. Then one has:<ref name=":3">{{citation| surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p.&nbsp;286|isbn=978-3-540-32058-6|date=2006|language=German}}</ref>
<math display="block">\wp(z+w)=\frac14 \left[\frac{\wp'(z)-\wp'(w)}{\wp(z)-\wp(w)}\right]^2-\wp(z)-\wp(w).</math>

As well as the duplication formula:<ref name=":3" />
<math display="block">\wp(2z)=\frac14\left[\frac{\wp''(z)}{\wp'(z)}\right]^2-2\wp(z).</math>

==== Proofs ====
1. These formulas can come with a geometric interpretation. If one looks at the elliptic curve <math>C_{g_2,g_3}^{\mathbb{C}} </math> a line <math>\lambda= \{(x,y)\in\mathbb{C}^2:y=mx+q\}</math> intersects it in three points:<math>C_{g_2,g_3}^{\mathbb{C}} \cap \lambda=\{P,Q,R\} </math>. Since these points belong to the elliptic curve they can be labeled as <math>P=(\wp(u),\wp'(u)) \quad Q=(\wp(v),\wp'(v)) \quad</math> <math> R=(\wp(u+v),\wp'(u+v))</math> with <math>(u,v)\notin \Lambda </math>. From the formula of a secant line we have <math>m=\frac{y_P-y_Q}{x_P-x_Q}=\frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)}</math> letting <math>C_{g_2,g_3}^{\mathbb{C}} = \lambda </math> we have the equation <math> (mx+q)^2=4x^3-g_2x-g_3</math> which becomes <math> 4x^3-m^2x^2-(2mq+g_2)x-g_3-q^2=0</math> using [[Vieta's formulas]] one obtains: 
<math> x_P+x_Q+x_R=\frac{m^2}4 </math> which provides the wanted formula 
<math>\wp(u+v)+\wp(u)+\wp(v)=\frac14 \left[ \frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)} \right]^2 </math>

2. A second proof from Akhiezer's book<ref>Akhiezer's book Elements of the theory of elliptic functions https://www.ams.org/books/mmono/079/mmono079-endmatter.pdf</ref> is the following:
 
if <math> f </math> is arbitrary elliptic function then: 
<math>f(u)=c\prod_{i=1}^n \frac{\sigma(u-a_i)}{\sigma(u-b_i)} \quad c \in \mathbb{C}</math> 

where <math> \sigma </math> is one of the [[Weierstrass functions ]] and <math> a_i , b_i</math> are the respective zeros and poles in the period parallelogram. We then let a function 
<math>k(u,v)=\wp(u)-\wp(v)</math> From the previous lemma we have: 
<math>k(u,v)= 
\wp(u)-\wp(v)=c\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2}
 </math>

From some calculations one can find that
<math>c=\frac1{\sigma(v)^2} \implies\wp(u)-\wp(v)=\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2\sigma(v)^2}</math> 

By definition the Weierstrass Zeta function: <math> \frac{d}{dz}\ln \sigma(z)=\zeta(z)</math> therefore we logarithmicly differentiate both sides obtaining:
<math>\frac{\wp'(u)}{\wp(u)-\wp(v)}=\zeta(u+v)-2\zeta(u)-\zeta(u-v)</math> 
Once again by definition <math> \zeta'(z)=-\wp(z)</math> thus by differentiating once more on both sides and rearranging the terms we obtain 

<math>-\wp(u+v)=-\wp(u)+\frac12 \frac{ \wp''(v)[\wp(u)-\wp(v) ]-\wp'(u)[\wp'(u)-\wp'(v)] }{ [\wp(u)-\wp(v) ]^2 } </math>  

Knowing that <math>\wp'' </math> has the following differential equation <math>2\wp''=12\wp^2-g_2</math> and rearranging the terms one gets the  wanted formula  
<math display="block">\wp(u+v)=\frac14 \left[\frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)}\right]^2-\wp(u)-\wp(v).</math>

== Typography ==
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863.{{refn | group=footnote |
This symbol was also used in the version of Weierstrass's lectures published by Schwarz in the 1880s. The first edition of ''[[A Course of Modern Analysis]]'' by [[E. T. Whittaker]]<!-- The first edition was by Whittaker alone. --> in 1902 also used it.<ref>{{citation | title= The letter ℘ Name & origin? | author = teika kazura| publisher = [[MathOverflow]] | url= https://mathoverflow.net/q/278130 | date= 2017-08-17 | access-date=2018-08-30 }}</ref> }} It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅.

In computing, the letter ℘ is available as <code>\wp</code> in [[TeX]]. In [[Unicode]] the code point is {{unichar|2118|script capital p|html=}}, with the more correct alias {{smallcaps|1=weierstrass elliptic function}}.{{refn|group="footnote" |

The [[Unicode Consortium]] has acknowledged two problems with the letter's name: the letter is in fact lowercase, and it is not a "script" class letter, like {{unichar|1d4c5|mathematical script small p}}, but the letter for Weierstrass's elliptic function.

Unicode added the alias as a correction.<ref>{{cite web | work=Unicode Technical Note #27 | title=Known Anomalies in Unicode Character Names | url=http://unicode.org/notes/tn27/ | version=version 4 | publisher=Unicode, Inc. | date=2017-04-10 | access-date=2017-07-20 }}</ref><ref>{{cite web | url=https://www.unicode.org/Public/10.0.0/ucd/NameAliases.txt | title=NameAliases-10.0.0.txt | date=2017-05-06 | access-date=2017-07-20 | publisher=Unicode, Inc.}}</ref>

}} In [[HTML]], it can be escaped as <code>&amp;weierp;</code>.
{{charmap
|2118|name1=Script Capital P /<br />Weierstrass Elliptic Function
}}

== See also ==

* [[Weierstrass functions]]
* [[Jacobi elliptic functions]]
* [[Lemniscate elliptic functions]]

== Footnotes ==
{{Reflist|group=footnote}}

== References ==
{{Reflist}}

*{{AS ref|18|627}}
* [[Naum Akhiezer|N. I. Akhiezer]], ''Elements of the Theory of Elliptic Functions'', (1970) Moscow, translated into English as ''AMS Translations of Mathematical Monographs Volume 79'' (1990) AMS, Rhode Island {{isbn|0-8218-4532-2}}
* [[Tom M. Apostol]], ''Modular Functions and Dirichlet Series in Number Theory, Second Edition'' (1990), Springer, New York {{isbn|0-387-97127-0}} (See chapter 1.)
* K. Chandrasekharan, ''Elliptic functions'' (1980), Springer-Verlag {{isbn|0-387-15295-4}}
* [[Konrad Knopp]], ''Funktionentheorie II'' (1947), Dover Publications; Republished in English translation as ''Theory of Functions'' (1996), Dover Publications {{isbn|0-486-69219-1}}
* [[Serge Lang]], ''Elliptic Functions'' (1973), Addison-Wesley, {{isbn|0-201-04162-6}}
* [[E. T. Whittaker]] and [[G. N. Watson]],  ''[[A Course of Modern Analysis]]'', [[Cambridge University Press]], 1952, chapters 20 and 21

==External links==
{{commons category|Weierstrass's elliptic functions}}
* {{springer|title=Weierstrass elliptic functions|id=p/w097450}}
* [http://mathworld.wolfram.com/WeierstrassEllipticFunction.html Weierstrass's elliptic functions on Mathworld].
* Chapter 23, [http://dlmf.nist.gov/23 Weierstrass Elliptic and Modular Functions] in DLMF ([[Digital Library of Mathematical Functions]]) by W. P. Reinhardt and P. L. Walker.
* [https://github.com/daviddumas/weierstrass Weierstrass P function and its derivative implemented in C by David Dumas]
[[Category:Modular forms]]
[[Category:Algebraic curves]]
[[Category:Elliptic functions]]