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 relation between the sigma, zeta, and ${\displaystyle \wp }$ functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant.

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 ${\displaystyle \Lambda -\{0\}}$ and ${\displaystyle (\omega _{1},\omega _{2})}$ is a fundamental pair of periods.

Through careful manipulation of the Weierstrass factorization theorem as it relates also to the sine function, another potentially more manageable infinite product definition is

for any ${\displaystyle \{i,j\}\in \{1,2,3\}}$ with ${\displaystyle i\neq j}$ and where we have used the notation ${\displaystyle \eta _{i}=\zeta (\omega _{i}/2;\Lambda )}$ (see zeta function below). Also it is a "quasi-periodic" function, with the following property:

${\displaystyle \sigma (z+2\omega _{i})=-e^{2\eta _{i}(z+\omega _{i})}\sigma (z)}$

The sigma function can be used to represent an elliptic function: ${\displaystyle f(z+\omega _{i})=f(z)\quad i\in \{1,\ldots ,n\}}$ when knowing its zeros and poles that lie in the period parallelogram:

Where ${\displaystyle c}$ is a constant in ${\displaystyle \mathbb {C} }$ and ${\displaystyle a_{j}}$ are the zeros in the parallelogram and ${\displaystyle b_{j}}$ are the poles