# Weierstrass eta function: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

(Redirected to Weierstrass functions article)

In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass.

## Weierstrass sigma-function

The Weierstrass sigma-function associated to a two-dimensional lattice $\Lambda\subset\Complex$ is defined to be the product

$\sigma(z;\Lambda)=z\prod_{w\in\Lambda^{*}} \left(1-\frac{z}{w}\right) e^{z/w+\frac{1}{2}(z/w)^2}$

where Λ * denotes Λ − {0}.

## Weierstrass zeta-function

The Weierstrass zeta-function is defined by the sum

$\zeta(z;\Lambda)=\frac{\sigma'(z;\Lambda)}{\sigma(z;\Lambda)}=\frac{1}{z}+\sum_{w\in\Lambda^{*}}\left( \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right).$

Note that the Weierstrass zeta-function is basically the logarithmic derivative of the sigma-function. The zeta-function can be rewritten as:

$\zeta(z;\Lambda)=\frac{1}{z}-\sum_{k=1}^{\infty}\mathcal{G}_{2k+2}(\Lambda)z^{2k+1}$

where $\mathcal{G}_{2k+2}$ is the Eisenstein series of weight 2k + 2.

Also note that the derivative of the zeta-function is $-\wp(z)$, where $\wp(z)$ is the Weierstrass elliptic function

The Weierstrass zeta-function should not be confused with the Riemann zeta-function in number theory.

## Weierstrass eta-function

The Weierstrass eta-function is defined to be

$\eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda), \mbox{ for any } z \in \Complex$

It can be proved that this is well-defined, i.e. ζ(z + w;Λ) − ζ(z;Λ) only depends on w. The Weierstrass eta-function should not be confused with the Dedekind eta-function.

## Weierstrass p-function

The Weierstrass p-function is defined to be

$\wp(z;\Lambda)= -\zeta'(z;\Lambda), \mbox{ for any } z \in \Complex$

The Weierstrass p-function is an even elliptic function of order N=2 with a double pole at each lattice and no others.