---
tag: 3D15ZW4YZ6
kind: definition
name: Weierstrass zeta and sigma functions
msc: ["33E05", 30D30]
defines: QATW7046PD
uses: [WN9V7WSJKF, JYV1Y8XX16, 0G4F0XB9X2, 361NAK68QK]
sources:
  - url: https://en.wikipedia.org/wiki/Weierstrass_sigma,_zeta,_and_eta_functions
---

## Definition

Let $\omega_1, \omega_2 \in \C$ be linearly independent over $\R$, and let $\Lambda = \Z\omega_1 + \Z\omega_2$ be the [[lattice|WN9V7WSJKF]] they span.
The *Weierstrass sigma function* of $\Lambda$ is the [[infinite product|JYV1Y8XX16]]
$$\sigma(z) = z \prod_{\omega \in \Lambda \setminus \set{0}} \left(1 - \frac{z}{\omega}\right) \exp\!\left(\frac{z}{\omega} + \frac{z^2}{2\omega^2}\right), \qquad z \in \C.$$
The *Weierstrass zeta function* of $\Lambda$ is the meromorphic function
$$\zeta(z) = \frac{\sigma'(z)}{\sigma(z)} = \frac{1}{z} + \sum_{\omega \in \Lambda \setminus \set{0}} \left(\frac{1}{z - \omega} + \frac{1}{\omega} + \frac{z}{\omega^2}\right), \qquad z \in \C \setminus \Lambda.$$

## Remarks

The product converges absolutely and locally uniformly, because each factor is $1 + O(\abs{z/\omega}^3)$ and $\sum_{\omega \ne 0}\abs{\omega}^{-3} < \infty$.
So $\sigma$ is an [[entire function|0G4F0XB9X2]], odd, with simple zeros exactly at the points of $\Lambda$.
Also $\zeta'(z) = -\wp(z)$, where $\wp$ is the [[Weierstrass elliptic function|361NAK68QK]] of $\Lambda$.
Since $\wp$ is periodic, $\eta_j = \zeta(z + \omega_j) - \zeta(z)$ is a constant, the *quasi-period*, for $j = 1, 2$.
Then $\sigma(z + \omega_j) = -\exp\big(\eta_j(z + \omega_j/2)\big)\,\sigma(z)$.
If $\operatorname{Im}(\omega_2/\omega_1) > 0$, the Legendre relation $\eta_1\omega_2 - \eta_2\omega_1 = 2\pi i$ holds; it follows by integrating $\zeta$ around a fundamental parallelogram, where $\zeta$ has one simple pole of residue $1$.
Example: replace $\Lambda$ by the one-dimensional group $\Z\pi$ in the same formulas.
Since $\sum_{n \ne 0} 1/(n\pi)^2 = 1/3$, they give $\zeta(z) = \cot z + z/3$ and $\sigma(z) = e^{z^2/6}\sin z$.
This is the limit of the lattice case when $\omega_1 = \pi$ is fixed and $\omega_2 \to i\infty$.
Non-example: $\zeta$ is not an elliptic function.
A nonconstant elliptic function cannot have only one simple pole in a period parallelogram, and indeed the Legendre relation shows that $\eta_1, \eta_2$ are not both $0$.