# Weierstrass functions

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.

The Weierstrass sigma function associated to a two-dimensional lattice ${\displaystyle \Lambda \subset \mathbb {C} }$ is defined to be the product

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

The Weierstrass zeta function is defined by the sum

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

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

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