# Weierstrass functions

(Redirected from Weierstrass zeta function)

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 ${\displaystyle \Lambda \subset \mathbb {C} }$ is defined to be the product

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

where ${\displaystyle \Lambda ^{*}}$ denotes ${\displaystyle \Lambda -\{0\}}$.

## Weierstrass zeta-function

The Weierstrass zeta-function is defined by the sum

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

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

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

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

The derivative of the zeta-function is ${\displaystyle -\wp (z)}$, where ${\displaystyle \wp (z)}$ 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

${\displaystyle \eta (w;\Lambda )=\zeta (z+w;\Lambda )-\zeta (z;\Lambda ),{\mbox{ for any }}z\in \mathbb {C} }$ and any w in the lattice ${\displaystyle \Lambda }$

This is well-defined, i.e. ${\displaystyle \zeta (z+w;\Lambda )-\zeta (z;\Lambda )}$ only depends on the lattice vector w. The Weierstrass eta-function should not be confused with the Dedekind eta-function.

## Weierstrass p-function

The Weierstrass p-function is related to the zeta function by

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

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