---
tag: 361NAK68QK
kind: object
name: Weierstrass elliptic function
msc: ["33E05", 30D30]
instance_of: [FF9ZPCZ59K]
uses: [FF9ZPCZ59K, 0NSD1TPHR0, 0WPQYW8MTE]
sources:
  - url: https://en.wikipedia.org/wiki/Weierstrass_elliptic_function
same_as:
  wikidata: Q2343600
  wikipedia: Weierstrass elliptic function
---

## Description

Let $\omega_1, \omega_2 \in \C$ be linearly independent over $\R$, and let $\Lambda = \Z\omega_1 + \Z\omega_2$ be the lattice they span.
The *Weierstrass elliptic function* of $\Lambda$ is
$$\wp(z) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus \set{0}} \left( \frac{1}{(z - \omega)^2} - \frac{1}{\omega^2} \right), \qquad z \in \C \setminus \Lambda.$$
The series converges absolutely and uniformly on compact subsets of $\C \setminus \Lambda$, because $\sum_{\omega \ne 0} \abs{\omega}^{-3} < \infty$.

1. $\wp$ is a [[meromorphic function|0NSD1TPHR0]] on $\C$ whose [[poles|0WPQYW8MTE]] are exactly the points of $\Lambda$, each a pole of order $2$.
2. $\wp$ is even, and $\wp(z + \omega) = \wp(z)$ for all $\omega \in \Lambda$, so $\wp$ is an [[elliptic function|FF9ZPCZ59K]] for $\Lambda$.
3. $\wp$ satisfies the differential equation $(\wp')^2 = 4\wp^3 - g_2 \wp - g_3$, where $g_2 = 60 \sum_{\omega \ne 0} \omega^{-4}$ and $g_3 = 140 \sum_{\omega \ne 0} \omega^{-6}$.

## Remarks

Periodicity is checked on the derivative first: $\wp'(z) = -2 \sum_{\omega \in \Lambda} (z - \omega)^{-3}$ is visibly $\Lambda$-periodic.
So $\wp(z + \omega_1) - \wp(z)$ is a constant $c$; putting $z = -\omega_1/2$ and using that $\wp$ is even gives $c = 0$.
The differential equation follows because $(\wp')^2 - 4\wp^3 + g_2\wp + g_3$ is elliptic without poles, hence constant, and its Laurent expansion at $0$ shows the constant is $0$.
Every elliptic function for $\Lambda$ is a rational function of $\wp$ and $\wp'$.
The map $z \mapsto (\wp(z), \wp'(z))$ identifies the torus $\C / \Lambda$ minus one point with the affine cubic curve $y^2 = 4x^3 - g_2 x - g_3$.

## Explanations

### L1 human

The Weierstrass function is the simplest nonconstant function on the plane that repeats itself along two independent directions.
It is built by placing a double pole at each point of a lattice and adding up, with correction terms that make the sum converge.
Its square-derivative identity turns the torus into a cubic curve, which is the link between complex analysis and elliptic curves.