# Weierstrass elliptic function

In mathematics, the Weierstrass elliptic functions are elliptic functions 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 ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.
Symbol for Weierstrass $\wp$-function

## Motivation

A cubic of the form $C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}$, where $g_{2},g_{3}\in \mathbb {C}$ are complex numbers with $g_{2}^{3}-27g_{3}^{2}\neq 0$, cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.
For the quadric $K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}$; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: $\psi :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).$ Because of the periodicity of the sine and cosine $\mathbb {R} /2\pi \mathbb {Z}$ is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of $C_{g_{2},g_{3}}^{\mathbb {C} }$ by means of the doubly periodic $\wp$-function (see in the section "Relation to elliptic curves"). This parameterization has the domain $\mathbb {C} /\Lambda$, which is topologically equivalent to a torus.
There is another analogy to the trigonometric functions. Consider the integral function $a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.$ It can be simplified by substituting $y=\sin t$ and $s=\arcsin x$: $a(x)=\int _{0}^{s}dt=s=\arcsin x.$ That means $a^{-1}(x)=\sin x$. So the sine function is an inverse function of an integral function.
Elliptic functions are the inverse functions of elliptic integrals. In particular, let: $u(z)=\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.$ Then the extension of $u^{-1}$ to the complex plane equals the $\wp$-function. 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.

## Definition

Let $\omega _{1},\omega _{2}\in \mathbb {C}$ be two complex numbers that are linearly independent over $\mathbb {R}$ and let $\Lambda :=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}:=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}$ be the period lattice generated by those numbers. Then the $\wp$-function is defined as follows:
    $\wp (z,\omega _{1},\omega _{2}):=\wp (z)={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right).$
This series converges locally uniformly absolutely in the complex torus $\mathbb {C} /\Lambda$.
It is common to use $1$ and $\tau$ in the upper half-plane $\mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}$ as generators of the lattice. Dividing by ω$_1${\textstyle \omega _{1}} maps the lattice $\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}$ isomorphically onto the lattice $\mathbb {Z} +\mathbb {Z} \tau$ with τ=ω$_2$/ω$_1$ {\textstyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}}. Because $-\tau$ can be substituted for $\tau$, without loss of generality we can assume $\tau \in \mathbb {H}$, and then define $\wp (z,\tau ):=\wp (z,1,\tau )$. With that definition, we have $\wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp (z/\omega _{1},\omega _{2}/\omega _{1})$.

## Properties

* $\wp$ is a meromorphic function with a pole of order 2 at each period $\lambda$ in $\Lambda$.
* $\wp$ is a homogeneous function in that:
       $\wp (\lambda z,\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-2}\wp (z,\omega _{1},\omega _{2}).$
* $\wp$ is an even function. That means $\wp (z)=\wp (-z)$ for all $z\in \mathbb {C} \setminus \Lambda$, which can be seen in the following way:
       ${\begin{aligned}\wp (-z)&={\frac {1}{(-z)^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(-z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)=\wp (z).\end{aligned}}$
    The second last equality holds because $\{-\lambda :\lambda \in \Lambda \}=\Lambda$. Since the sum converges absolutely this rearrangement does not change the limit.
* The derivative of $\wp$ is given by: $\wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}.$
* $\wp$ and $\wp '$ are doubly periodic with the periods $\omega _{1}$ and $\omega _{2}$. This means: ${\begin{aligned}\wp (z+\omega _{1})&=\wp (z)=\wp (z+\omega _{2}),\ {\textrm {and}}\\[3mu]\wp '(z+\omega _{1})&=\wp '(z)=\wp '(z+\omega _{2}).\end{aligned}}$ It follows that $\wp (z+\lambda )=\wp (z)$ and $\wp '(z+\lambda )=\wp '(z)$ for all $\lambda \in \Lambda$.

## Laurent expansion

Let $r:=\min\{{|\lambda }|:0\neq \lambda \in \Lambda \}$. Then for $0<|z|<r$ the $\wp$-function has the following Laurent expansion $\wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}$ where $G_{n}=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-n}$ for $n\geq 3$ are so called Eisenstein series.

## Differential equation

Set $g_{2}=60G_{4}$ and $g_{3}=140G_{6}$. Then the $\wp$-function satisfies the differential equation $\wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}.$ This relation can be verified by forming a linear combination of powers of $\wp$ and $\wp '$ to eliminate the pole at $z=0$. This yields an entire elliptic function that has to be constant by Liouville's theorem.

### Invariants

The coefficients of the above differential equation $g_{2}$ and $g_{3}$ are known as the invariants. Because they depend on the lattice $\Lambda$ they can be viewed as functions in $\omega _{1}$ and $\omega _{2}$.
The series expansion suggests that $g_{2}$ and $g_{3}$ are homogeneous functions of degree $-4$ and $-6$. That is $g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})$ $g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})$ for $\lambda \neq 0$.
If $\omega _{1}$ and $\omega _{2}$ are chosen in such a way that $\operatorname {Im} \left({\tfrac {\omega _{2}}{\omega _{1}}}\right)>0$, $g_{2}$ and $g_{3}$ can be interpreted as functions on the upper half-plane $\mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}$.
Let $\tau ={\tfrac {\omega _{2}}{\omega _{1}}}$. One has: $g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2}),$ $g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2}).$ That means g$_2$ and g$_3$ are only scaled by doing this. Set $g_{2}(\tau ):=g_{2}(1,\tau )$ and $g_{3}(\tau ):=g_{3}(1,\tau ).$ As functions of $\tau \in \mathbb {H}$, $g_{2}$ and $g_{3}$ are so called modular forms.
The Fourier series for $g_{2}$ and $g_{3}$ are given as follows: $g_{2}(\tau )={\frac {4}{3}}\pi ^{4}\left[1+240\sum _{k=1}^{\infty }\sigma _{3}(k)q^{2k}\right]$ $g_{3}(\tau )={\frac {8}{27}}\pi ^{6}\left[1-504\sum _{k=1}^{\infty }\sigma _{5}(k)q^{2k}\right]$ where $\sigma _{m}(k):=\sum _{d\mid {k}}d^{m}$ is the divisor function and $q=e^{\pi i\tau }$ is the nome.

### Modular discriminant

The modular discriminant $\Delta$ is defined as the discriminant of the characteristic polynomial of the differential equation $\wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}$ as follows: $\Delta =g_{2}^{3}-27g_{3}^{2}.$ The discriminant is a modular form of weight $12$. That is, under the action of the modular group, it transforms as $\Delta \left({\frac {a\tau +b}{c\tau +d}}\right)=\left(c\tau +d\right)^{12}\Delta (\tau )$ where $a,b,d,c\in \mathbb {Z}$ with $ad-bc=1$.
Note that $\Delta =(2\pi )^{12}\eta ^{24}$ where $\eta$ is the Dedekind eta function.
For the Fourier coefficients of $\Delta$, see Ramanujan tau function.

### The constants e$_1$, e$_2$ and e$_3$

$e_{1}$, $e_{2}$ and $e_{3}$ are usually used to denote the values of the $\wp$-function at the half-periods. $e_{1}\equiv \wp \left({\frac {\omega _{1}}{2}}\right)$ $e_{2}\equiv \wp \left({\frac {\omega _{2}}{2}}\right)$ $e_{3}\equiv \wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)$ They are pairwise distinct and only depend on the lattice $\Lambda$ and not on its generators.
$e_{1}$, $e_{2}$ and $e_{3}$ are the roots of the cubic polynomial $4\wp (z)^{3}-g_{2}\wp (z)-g_{3}$ and are related by the equation: $e_{1}+e_{2}+e_{3}=0.$ Because those roots are distinct the discriminant $\Delta$ does not vanish on the upper half plane. Now we can rewrite the differential equation: $\wp '^{2}(z)=4(\wp (z)-e_{1})(\wp (z)-e_{2})(\wp (z)-e_{3}).$ That means the half-periods are zeros of $\wp '$.
The invariants $g_{2}$ and $g_{3}$ can be expressed in terms of these constants in the following way: $g_{2}=-4(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})$ $g_{3}=4e_{1}e_{2}e_{3}$ $e_{1}$, $e_{2}$ and $e_{3}$ are related to the modular lambda function: $\lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.$

## 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: $\wp (z)=e_{3}+{\frac {e_{1}-e_{3}}{\operatorname {sn} ^{2}w}}=e_{2}+(e_{1}-e_{3}){\frac {\operatorname {dn} ^{2}w}{\operatorname {sn} ^{2}w}}=e_{1}+(e_{1}-e_{3}){\frac {\operatorname {cn} ^{2}w}{\operatorname {sn} ^{2}w}}$ where $e_{1},e_{2}$ and $e_{3}$ are the three roots described above and where the modulus k of the Jacobi functions equals $k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}$ and their argument w equals $w=z{\sqrt {e_{1}-e_{3}}}.$

## Relation to Jacobi's theta functions

The function $\wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})$ can be represented by Jacobi's theta functions: $\wp (z,\tau )=\left(\pi \theta _{2}(0,q)\theta _{3}(0,q){\frac {\theta _{4}(\pi z,q)}{\theta _{1}(\pi z,q)}}\right)^{2}-{\frac {\pi ^{2}}{3}}\left(\theta _{2}^{4}(0,q)+\theta _{3}^{4}(0,q)\right)$ where $q=e^{\pi i\tau }$ is the nome and $\tau$ is the period ratio $(\tau \in \mathbb {H} )$. This also provides a very rapid algorithm for computing $\wp (z,\tau )$.

## Relation to elliptic curves

Consider the embedding of the cubic curve in the complex projective plane
    ${\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}\cup \{O\}\subset \mathbb {C} ^{2}\cup \mathbb {P} _{1}(\mathbb {C} )=\mathbb {P} _{2}(\mathbb {C} ).$
where $O$ is a point lying on the line at infinity $\mathbb {P} _{1}(\mathbb {C} )$. For this cubic there exists no rational parameterization, if $\Delta \neq 0$. In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the $\wp$-function and its derivative $\wp '$:
    $\varphi (\wp ,\wp '):\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} },\quad z\mapsto {\begin{cases}\left[\wp (z):\wp '(z):1\right]&z\notin \Lambda \\\left[0:1:0\right]\quad &z\in \Lambda \end{cases}}$
Now the map $\varphi$ is bijective and parameterizes the elliptic curve ${\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$.
$\mathbb {C} /\Lambda$ 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 $g_{2},g_{3}\in \mathbb {C}$ with $\Delta =g_{2}^{3}-27g_{3}^{2}\neq 0$ there exists a lattice $\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}$, such that
$g_{2}=g_{2}(\omega _{1},\omega _{2})$ and $g_{3}=g_{3}(\omega _{1},\omega _{2})$.
The statement that elliptic curves over $\mathbb {Q}$ can be parameterized over $\mathbb {Q}$, 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.

## Addition theorems

Let $z,w\in \mathbb {C}$, so that $z,w,z+w,z-w\notin \Lambda$. Then one has: $\wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w).$
As well as the duplication formula: $\wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z).$
These formulas also have a geometric interpretation, if one looks at the elliptic curve ${\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ together with the mapping ${\varphi }:\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ as in the previous section.
The group structure of $(\mathbb {C} /\Lambda ,+)$ translates to the curve ${\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ and can be geometrically interpreted there:
The sum of three pairwise different points $a,b,c\in {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }$ is zero if and only if they lie on the same line in $\mathbb {P} _{\mathbb {C} }^{2}$.
This is equivalent to: $\det \left({\begin{array}{rrr}1&\wp (u+v)&-\wp '(u+v)\\1&\wp (v)&\wp '(v)\\1&\wp (u)&\wp '(u)\\\end{array}}\right)=0,$ where $\wp (u)=a$, $\wp (v)=b$ and $u,v\notin \Lambda$.

## 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. 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 U+2118 ℘ SCRIPT CAPITAL P (&weierp;, &wp;), with the more correct alias weierstrass elliptic function. In HTML, it can be escaped as &weierp;.
Character information :
| Preview | ℘ | ℘ |
| Unicode name | SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION | SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION |
| Encodings | decimal | hex |
| Unicode | 8472 | U+2118 |
| UTF-8 | 226 132 152 | E2 84 98 |
| Numeric character reference | &#8472; | &#x2118; |
| Named character reference | &weierp;, &wp; | &weierp;, &wp; |