Weierstrass elliptic function

A cubic of the form $C\_\{g\_2,g\_3\}^\backslash mathbb\{C\}=\backslash \{(x,y)\backslash in\backslash mathbb\{C\}^2:y^2=4x^3-g\_2x-g\_3\backslash \}$, where $g\_2,g\_3\backslash in\backslash mathbb\{C\}$ are complex numbers with $g\_2^3-27g\_3^2\backslash neq0$, cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.

For the quadric $K=\backslash left\backslash \{(x,y)\backslash in\backslash mathbb\{R\}^2:x^2+y^2=1\backslash right\backslash \}$; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:

$$\backslash psi:\backslash mathbb\{R\}/2\backslash pi\backslash mathbb\{Z\}\backslash to\; K,\; \backslash quad\; t\backslash mapsto(\backslash sin\; t,\backslash cos\; t).$$

Because of the periodicity of the sine and cosine $\backslash mathbb\{R\}/2\backslash pi\backslash mathbb\{Z\}$ is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of $C\_\{g\_2,g\_3\}^\backslash mathbb\{C\}$ by means of the doubly periodic $\backslash wp$-function (see in the section "Relation to elliptic curves"). This parameterization has the domain $\backslash mathbb\{C\}/\backslash Lambda$, which is topologically equivalent to a torus.{{citation|surname1=Rolf Busam| title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p. 259| isbn=978-3-540-32058-6|date=2006|language=German}}

There is another analogy to the trigonometric functions. Consider the integral function

$$a(x)=\backslash int\_0^x\backslash frac\{dy\}\{\backslash sqrt\{1-y^2\}\}\; .$$

It can be simplified by substituting $y=\backslash sin\; t$ and $s=\backslash arcsin\; x$:

$$a(x)=\backslash int\_0^s\; dt\; =\; s\; =\; \backslash arcsin\; x\; .$$

That means $a^\{-1\}(x)\; =\; \backslash sin\; x$. So the sine function is an inverse function of an integral function.{{citation| surname1=Jeremy Gray|title=Real and the complex: a history of analysis in the 19th century|publication-place=Cham|at=p. 71| isbn=978-3-319-23715-2|date=2015|language=German}}

Elliptic functions are the inverse functions of elliptic integrals. In particular, let:

$$u(z)=\backslash int\_z^\backslash infin\backslash frac\{ds\}\{\backslash sqrt\{4s^3-g\_2s-g\_3\}\}\; .$$

Then the extension of $u^\{-1\}$ to the complex plane equals the $\backslash wp$-function.{{citation|surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p. 294|isbn=978-3-540-32058-6|date=2006|language=German}} This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.{{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}}

File:Weierstrass elliptic function P.png

Let $\backslash omega\_1,\backslash omega\_2\backslash in\backslash mathbb\{C\}$ be two complex numbers that are linearly independent over $\backslash mathbb\{R\}$ and let $\backslash Lambda:=\backslash mathbb\{Z\}\backslash omega\_1+\backslash mathbb\{Z\}\backslash omega\_2:=\backslash \{m\backslash omega\_1+n\backslash omega\_2:\; m,n\backslash in\backslash mathbb\{Z\}\backslash \}$ be the period lattice generated by those numbers. Then the $\backslash wp$-function is defined as follows:

:$\backslash weierp(z,\backslash omega\_1,\backslash omega\_2):=\backslash wp(z)\; =\; \backslash frac\{1\}\{z^2\}\; +\; \backslash sum\_\{\backslash lambda\backslash in\backslash Lambda\backslash setminus\backslash \{0\backslash \}\}\backslash left(\backslash frac\; 1\; \{(z-\backslash lambda)^2\}\; -\; \backslash frac\; 1\; \{\backslash lambda^2\}\backslash right).$

This series converges locally uniformly absolutely in the complex torus $\backslash mathbb\{C\}\; /\; \backslash Lambda$.

It is common to use $1$ and $\backslash tau$ in the upper half-plane $\backslash mathbb\{H\}:=\backslash \{z\backslash in\backslash mathbb\{C\}:\backslash operatorname\{Im\}(z)\; >\; 0\backslash \}$ as generators of the lattice. Dividing by $\backslash omega\_1$ maps the lattice $\backslash mathbb\{Z\}\backslash omega\_1+\backslash mathbb\{Z\}\backslash omega\_2$ isomorphically onto the lattice $\backslash mathbb\{Z\}+\backslash mathbb\{Z\}\backslash tau$ with $\backslash tau=\backslash tfrac\{\backslash omega\_2\}\{\backslash omega\_1\}$. Because $-\backslash tau$ can be substituted for $\backslash tau$, without loss of generality we can assume $\backslash tau\backslash in\backslash mathbb\{H\}$, and then define $\backslash wp(z,\backslash tau)\; :=\; \backslash wp(z,\; 1,\backslash tau)$. With that definition, we have $\backslash wp(z,\backslash omega\_1,\backslash omega\_2)\; =\; \backslash omega\_1^\{-2\}\backslash wp(z/\backslash omega\_1,\backslash omega\_2/\backslash omega\_1)$.

• $\backslash wp$ is a meromorphic function with a pole of order 2 at each period $\backslash lambda$ in $\backslash Lambda$.
• $\backslash wp$ is a homogeneous function in that:

::$\backslash wp(\backslash lambda\; z\; ,\; \backslash lambda\backslash omega\_\{1\},\; \backslash lambda\backslash omega\_\{2\})\; =\; \backslash lambda^\{-2\}\; \backslash wp\; (z,\; \backslash omega\_\{1\},\backslash omega\_\{2\}).$

• $\backslash wp$ is an even function. That means $\backslash wp(z)=\backslash wp(-z)$ for all $z\; \backslash in\; \backslash mathbb\{C\}\; \backslash setminus\; \backslash Lambda$, which can be seen in the following way:

::$\backslash begin\{align\}$

\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}

:The second last equality holds because $\backslash \{-\backslash lambda:\backslash lambda\; \backslash in\; \backslash Lambda\backslash \}=\backslash Lambda$. Since the sum converges absolutely this rearrangement does not change the limit.

• The derivative of $\backslash wp$ is given by:{{citation|surname1=Apostol, Tom M.|title=Modular functions and Dirichlet series in number theory|publisher=Springer-Verlag|publication-place=New York|at=p. 11|isbn=0-387-90185-X|date=1976| language=German}} $$\backslash wp\text{'}(z)=-2\backslash sum\_\{\backslash lambda\; \backslash in\; \backslash Lambda\}\backslash frac1\{(z-\backslash lambda)^3\}.$$
• $\backslash wp$ and $\backslash wp\text{'}$ are doubly periodic with the periods $\backslash omega\_1$ and $\backslash omega\_2$. This means: $$\backslash begin\{aligned\}$$

\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} It follows that $\backslash wp(z+\backslash lambda)=\backslash wp(z)$ and $\backslash wp\text{'}(z+\backslash lambda)=\backslash wp\text{'}(z)$ for all $\backslash lambda\; \backslash in\; \backslash Lambda$.

Let $r:=\backslash min\backslash \{\{|\backslash lambda\}|:0\backslash neq\backslash lambda\backslash in\backslash Lambda\backslash \}$. Then for

$$\backslash wp(z)=\backslash frac1\{z^2\}+\backslash sum\_\{n=1\}^\backslash infin\; (2n+1)G\_\{2n+2\}z^\{2n\}$$

where

$$G\_n=\backslash sum\_\{0\backslash neq\backslash lambda\backslash in\backslash Lambda\}\backslash lambda^\{-n\}$$ for $n\; \backslash geq\; 3$ are so called Eisenstein series.

Set $g\_2=60G\_4$ and $g\_3=140G\_6$. Then the $\backslash wp$-function satisfies the differential equation

$$\backslash wp\text{'}^2(z)\; =\; 4\backslash wp\; ^3(z)-g\_2\backslash wp(z)-g\_3.$$

This relation can be verified by forming a linear combination of powers of $\backslash wp$ and $\backslash wp\text{'}$ to eliminate the pole at $z=0$. This yields an entire elliptic function that has to be constant by Liouville's theorem.

Image:Gee three real.jpeg

Image:Gee three imag.jpeg

The coefficients of the above differential equation $g\_2$ and $g\_3$ are known as the invariants. Because they depend on the lattice $\backslash Lambda$ they can be viewed as functions in $\backslash omega\_1$ and $\backslash omega\_2$.

The series expansion suggests that $g\_2$ and $g\_3$ are homogeneous functions of degree $-4$ and $-6$. That is{{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}}

$$g\_2(\backslash lambda\; \backslash omega\_1,\; \backslash lambda\; \backslash omega\_2)\; =\; \backslash lambda^\{-4\}\; g\_2(\backslash omega\_1,\; \backslash omega\_2)$$

$$g\_3(\backslash lambda\; \backslash omega\_1,\; \backslash lambda\; \backslash omega\_2)\; =\; \backslash lambda^\{-6\}\; g\_3(\backslash omega\_1,\; \backslash omega\_2)$$ for $\backslash lambda\; \backslash neq\; 0$.

If $\backslash omega\_1$ and $\backslash omega\_2$ are chosen in such a way that $\backslash operatorname\{Im\}\backslash left(\; \backslash tfrac\{\backslash omega\_2\}\{\backslash omega\_1\}\; \backslash right)>0$, $g\_2$ and $g\_3$ can be interpreted as functions on the upper half-plane $\backslash mathbb\{H\}:=\backslash \{z\backslash in\backslash mathbb\{C\}:\backslash operatorname\{Im\}(z)>0\backslash \}$.

Let $\backslash tau=\backslash tfrac\{\backslash omega\_2\}\{\backslash omega\_1\}$. One has:{{citation|title=Modular functions and Dirichlet series in number theory|date=1976|at=p. 14|publication-place=New York|publisher=Springer-Verlag|language=German|isbn=0-387-90185-X| surname1=Apostol, Tom M.}}

$$g\_2(1,\backslash tau)=\backslash omega\_1^4g\_2(\backslash omega\_1,\backslash omega\_2),$$

$$g\_3(1,\backslash tau)=\backslash omega\_1^6\; g\_3(\backslash omega\_1,\backslash omega\_2).$$

That means g₂ and g₃ are only scaled by doing this. Set

$$g\_2(\backslash tau):=g\_2(1,\backslash tau)$$ and $$g\_3(\backslash tau):=g\_3(1,\backslash tau).$$

As functions of $\backslash tau\backslash in\backslash mathbb\{H\}$, $g\_2$ and $g\_3$ are so called modular forms.

The Fourier series for $g\_2$ and $g\_3$ are given as follows:{{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}}

$$g\_2(\backslash tau)=\backslash frac43\backslash pi^4\; \backslash left[\; 1+\; 240\backslash sum\_\{k=1\}^\backslash infty\; \backslash sigma\_3(k)\; q^\{2k\}\; \backslash right]$$

$$g\_3(\backslash tau)=\backslash frac\{8\}\{27\}\backslash pi^6\; \backslash left[\; 1-\; 504\backslash sum\_\{k=1\}^\backslash infty\; \backslash sigma\_5(k)\; q^\{2k\}\; \backslash right]$$

where

$$\backslash sigma\_m(k):=\backslash sum\_\{d\backslash mid\{k\}\}d^m$$

is the divisor function and $q=e^\{\backslash pi\; i\backslash tau\}$ is the nome.

Image:Discriminant real part.jpeg

The modular discriminant $\backslash Delta$ is defined as the discriminant of the characteristic polynomial of the differential equation $$\backslash wp\text{'}^2(z)\; =\; 4\backslash wp\; ^3(z)-g\_2\backslash wp(z)-g\_3$$ as follows:

$$\backslash Delta=g\_2^3-27g\_3^2.$$

The discriminant is a modular form of weight $12$. That is, under the action of the modular group, it transforms as

$$\backslash Delta\; \backslash left(\; \backslash frac\; \{a\backslash tau+b\}\; \{c\backslash tau+d\}\backslash right)\; =\; \backslash left(c\backslash tau+d\backslash right)^\{12\}\; \backslash Delta(\backslash tau)$$

where $a,b,d,c\backslash in\backslash mathbb\{Z\}$ with $ad-bc\; =\; 1$.{{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}}

Note that $\backslash Delta=(2\backslash pi)^\{12\}\backslash eta^\{24\}$ where $\backslash eta$ is the Dedekind eta function.{{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}}

For the Fourier coefficients of $\backslash Delta$, see Ramanujan tau function.

$e\_1$, $e\_2$ and $e\_3$ are usually used to denote the values of the $\backslash wp$-function at the half-periods.

$$e\_1\backslash equiv\backslash wp\backslash left(\backslash frac\{\backslash omega\_1\}\{2\}\backslash right)$$

$$e\_2\backslash equiv\backslash wp\backslash left(\backslash frac\{\backslash omega\_2\}\{2\}\backslash right)$$

$$e\_3\backslash equiv\backslash wp\backslash left(\backslash frac\{\backslash omega\_1+\backslash omega\_2\}\{2\}\backslash right)$$

They are pairwise distinct and only depend on the lattice $\backslash Lambda$ and not on its generators.{{citation| first=Rolf | last = Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin| at=p. 270|isbn=978-3-540-32058-6|date=2006|language=German}}

$e\_1$, $e\_2$ and $e\_3$ are the roots of the cubic polynomial $4\backslash wp(z)^3-g\_2\backslash wp(z)-g\_3$ and are related by the equation:

$$e\_1+e\_2+e\_3=0.$$

Because those roots are distinct the discriminant $\backslash Delta$ does not vanish on the upper half plane.{{citation| first=Tom M. |last = Apostol|title=Modular functions and Dirichlet series in number theory|publisher=Springer-Verlag|publication-place=New York|at=p. 13|isbn=0-387-90185-X|date=1976|language=German}} Now we can rewrite the differential equation:

$$\backslash wp\text{'}^2(z)=4(\backslash wp(z)-e\_1)(\backslash wp(z)-e\_2)(\backslash wp(z)-e\_3).$$

That means the half-periods are zeros of $\backslash wp\text{'}$.

The invariants $g\_2$ and $g\_3$ can be expressed in terms of these constants in the following way:{{citation|surname1=K. Chandrasekharan|title=Elliptic functions|publisher=Springer-Verlag|publication-place=Berlin|at=p. 33| isbn=0-387-15295-4|date=1985|language=German}}

$$g\_2\; =\; -4\; (e\_1\; e\_2\; +\; e\_1\; e\_3\; +\; e\_2\; e\_3)$$

$$g\_3\; =\; 4\; e\_1\; e\_2\; e\_3$$

$e\_1$, $e\_2$ and $e\_3$ are related to the modular lambda function:

$$\backslash lambda\; (\backslash tau)=\backslash frac\{e\_3-e\_2\}\{e\_1-e\_2\},\backslash quad\; \backslash tau=\backslash frac\{\backslash omega\_2\}\{\backslash omega\_1\}.$$

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

The basic relations are:{{cite book | author = Korn GA, Korn TM | year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw–Hill | location = New York | pages = 721 | lccn = 59014456}}

\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}

where $e\_1,e\_2$ and $e\_3$ are the three roots described above and where the modulus k of the Jacobi functions equals

$$k\; =\; \backslash sqrt\backslash frac\{e\_2\; -\; e\_3\}\{e\_1\; -\; e\_3\}$$

and their argument w equals

$$w\; =\; z\; \backslash sqrt\{e\_1\; -\; e\_3\}.$$

The function $\backslash wp\; (z,\backslash tau)=\backslash wp\; (z,1,\backslash omega\_2/\backslash omega\_1)$ can be represented by Jacobi's theta functions:

$$\backslash wp\; (z,\backslash tau)=\backslash left(\backslash pi\; \backslash theta\_2(0,q)\backslash theta\_3(0,q)\backslash frac\{\backslash theta\_4(\backslash pi\; z,q)\}\{\backslash theta\_1(\backslash pi\; z,q)\}\backslash right)^2-\backslash frac\{\backslash pi^2\}\{3\}\backslash left(\backslash theta\_2^4(0,q)+\backslash theta\_3^4(0,q)\backslash right)$$

where $q=e^\{\backslash pi\; i\backslash tau\}$ is the nome and $\backslash tau$ is the period ratio $(\backslash tau\backslash in\backslash mathbb\{H\})$.{{dlmf|first1=W. P.|last1=Reinhardt|first2=P. L.|last2=Walker|id=23.6.E7|title=Weierstrass Elliptic and Modular Functions}} This also provides a very rapid algorithm for computing $\backslash wp\; (z,\backslash tau)$.

{{see also|Elliptic curve#Elliptic curves over the complex numbers}}

Consider the embedding of the cubic curve in the complex projective plane

:$\backslash bar\; C\_\{g\_2,g\_3\}^\backslash mathbb\{C\}\; =\; \backslash \{(x,y)\backslash in\backslash mathbb\{C\}^2:y^2=4x^3-g\_2x-g\_3\backslash \}\backslash cup\backslash \{O\backslash \}\backslash subset\; \backslash mathbb\{C\}^\{2\}\; \backslash cup\; \backslash mathbb\{P\}\_1(\backslash mathbb\{C\})\; =\; \backslash mathbb\{P\}\_2(\backslash mathbb\{C\}).$

where $O$ is a point lying on the line at infinity $\backslash mathbb\{P\}\_1(\backslash mathbb\{C\})$. For this cubic there exists no rational parameterization, if $\backslash Delta\; \backslash neq\; 0$.{{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. 8|isbn=978-3-8348-2348-9|date=2012|language=German}} In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the $\backslash wp$-function and its derivative $\backslash wp\text{'}$:{{citation|title=Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen|date=2012|at=p. 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.}}

:$\backslash varphi(\backslash wp,\backslash wp\text{'}):\; \backslash mathbb\{C\}/\backslash Lambda\backslash to\backslash bar\; C\_\{g\_2,g\_3\}^\backslash mathbb\{C\},\; \backslash quad$

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}

Now the map $\backslash varphi$ is bijective and parameterizes the elliptic curve $\backslash bar\; C\_\{g\_2,g\_3\}^\backslash mathbb\{C\}$.

$\backslash mathbb\{C\}/\backslash Lambda$ 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 $g\_2,g\_3\backslash in\backslash mathbb\{C\}$ with $\backslash Delta\; =\; g\_2^3\; -\; 27g\_3^2\; \backslash neq\; 0$ there exists a lattice $\backslash mathbb\{Z\}\backslash omega\_1+\backslash mathbb\{Z\}\backslash omega\_2$, such that

$g\_2=g\_2(\backslash omega\_1,\backslash omega\_2)$ and $g\_3=g\_3(\backslash omega\_1,\backslash omega\_2)$.{{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. 111| isbn=978-3-8348-2348-9| date=2012|language=German}}

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

Let $z,w\backslash in\backslash mathbb\{C\}$, so that $z,w,z+w,z-w\backslash notin\backslash Lambda$. Then one has:{{citation| surname1=Rolf Busam|title=Funktionentheorie 1|edition=4., korr. und erw. Aufl|publisher=Springer|publication-place=Berlin|at=p. 286|isbn=978-3-540-32058-6|date=2006|language=German}}

$$\backslash wp(z+w)=\backslash frac14\; \backslash left[\backslash frac\{\backslash wp\text{'}(z)-\backslash wp\text{'}(w)\}\{\backslash wp(z)-\backslash wp(w)\}\backslash right]^2-\backslash wp(z)-\backslash wp(w).$$

As well as the duplication formula:

$$\backslash wp(2z)=\backslash frac14\backslash left[\backslash frac\{\backslash wp\text{'}\text{'}(z)\}\{\backslash wp\text{'}(z)\}\backslash right]^2-2\backslash wp(z).$$

1. These formulas can come with a geometric interpretation. If one looks at the elliptic curve $C\_\{g\_2,g\_3\}^\{\backslash mathbb\{C\}\}$ a line $\backslash lambda=\; \backslash \{(x,y)\backslash in\backslash mathbb\{C\}^2:y=mx+q\backslash \}$ intersects it in three points:$C\_\{g\_2,g\_3\}^\{\backslash mathbb\{C\}\}\; \backslash cap\; \backslash lambda=\backslash \{P,Q,R\backslash \}$. Since these points belong to the elliptic curve they can be labeled as $P=(\backslash wp(u),\backslash wp\text{'}(u))\; \backslash quad\; Q=(\backslash wp(v),\backslash wp\text{'}(v))\; \backslash quad$ $R=(\backslash wp(u+v),\backslash wp\text{'}(u+v))$ with $(u,v)\backslash notin\; \backslash Lambda$. From the formula of a secant line we have $m=\backslash frac\{y\_P-y\_Q\}\{x\_P-x\_Q\}=\backslash frac\{\backslash wp\text{'}(u)-\backslash wp\text{'}(v)\}\{\backslash wp(u)-\backslash wp(v)\}$ letting $C\_\{g\_2,g\_3\}^\{\backslash mathbb\{C\}\}\; =\; \backslash lambda$ we have the equation $(mx+q)^2=4x^3-g\_2x-g\_3$ which becomes $4x^3-m^2x^2-(2mq+g\_2)x-g\_3-q^2=0$ using Vieta's formulas one obtains:

$x\_P+x\_Q+x\_R=\backslash frac\{m^2\}4$ which provides the wanted formula

$\backslash wp(u+v)+\backslash wp(u)+\backslash wp(v)=\backslash frac14\; \backslash left[\; \backslash frac\{\backslash wp\text{'}(u)-\backslash wp\text{'}(v)\}\{\backslash wp(u)-\backslash wp(v)\}\; \backslash right]^2$

2. A second proof from Akhiezer's bookAkhiezer's book Elements of the theory of elliptic functions https://www.ams.org/books/mmono/079/mmono079-endmatter.pdf is the following:

if $f$ is arbitrary elliptic function then:

$f(u)=c\backslash prod\_\{i=1\}^n\; \backslash frac\{\backslash sigma(u-a\_i)\}\{\backslash sigma(u-b\_i)\}\; \backslash quad\; c\; \backslash in\; \backslash mathbb\{C\}$

where $\backslash sigma$ is one of the Weierstrass functions and $a\_i\; ,\; b\_i$ are the respective zeros and poles in the period parallelogram. We then let a function

$k(u,v)=\backslash wp(u)-\backslash wp(v)$ From the previous lemma we have:

$k(u,v)=$

\wp(u)-\wp(v)=c\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2}

From some calculations one can find that

$c=\backslash frac1\{\backslash sigma(v)^2\}\; \backslash implies\backslash wp(u)-\backslash wp(v)=\backslash frac\{\backslash sigma(u+v)\backslash sigma(u-v)\}\{\backslash sigma(u)^2\backslash sigma(v)^2\}$

By definition the Weierstrass Zeta function: $\backslash frac\{d\}\{dz\}\backslash ln\; \backslash sigma(z)=\backslash zeta(z)$ therefore we logarithmicly differentiate both sides obtaining:

$\backslash frac\{\backslash wp\text{'}(u)\}\{\backslash wp(u)-\backslash wp(v)\}=\backslash zeta(u+v)-2\backslash zeta(u)-\backslash zeta(u-v)$

Once again by definition $\backslash zeta\text{'}(z)=-\backslash wp(z)$ thus by differentiating once more on both sides and rearranging the terms we obtain

$-\backslash wp(u+v)=-\backslash wp(u)+\backslash frac12\; \backslash frac\{\; \backslash wp\text{'}\text{'}(v)[\backslash wp(u)-\backslash wp(v)\; ]-\backslash wp\text{'}(u)[\backslash wp\text{'}(u)-\backslash wp\text{'}(v)]\; \}\{\; [\backslash wp(u)-\backslash wp(v)\; ]^2\; \}$

Knowing that $\backslash wp$ has the following differential equation $2\backslash wp$=12\wp^2-g_2 and rearranging the terms one gets the wanted formula

$$\backslash wp(u+v)=\backslash frac14\; \backslash left[\backslash frac\{\backslash wp\text{'}(u)-\backslash wp\text{'}(v)\}\{\backslash wp(u)-\backslash wp(v)\}\backslash right]^2-\backslash wp(u)-\backslash wp(v).$$

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 in 1902 also used it.{{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 }} }} It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅.

In computing, the letter ℘ is available as \wp 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.{{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 }}{{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.}}

}} In HTML, it can be escaped as ℘.

{{charmap

|2118|name1=Script Capital P /
Weierstrass Elliptic Function

}}

• Weierstrass functions
• Jacobi elliptic functions
• Lemniscate elliptic functions

{{Reflist|group=footnote}}

{{Reflist}}

• {{AS ref|18|627}}
• 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

{{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