Weierstrass sigma function

Definition 1.

Let $\Lambda\subset\mathbb{C}$ be a lattice. Let $\Lambda^{\ast}$ denote $\Lambda-\{0\}$.

1. 1.

The Weierstrass sigma function is defined as the product

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

The Weierstrass zeta function is defined by the sum

 $\zeta(z;\Lambda)=\frac{\sigma^{\prime}(z;\Lambda)}{\sigma(z;\Lambda)}=\frac{1}% {z}+\sum_{w\in\Lambda^{\ast}}\left(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right)$

Note that the Weierstrass zeta function is basically the derivative of the logarithm 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 $2k+2$.

3. 3.

The Weierstrass eta function is defined to be

 $\eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda),\text{for any }z\in\mathbb{C}$

(It can be proved that this is well defined, i.e. $\zeta(z+w;\Lambda)-\zeta(z;\Lambda)$ only depends on $w$). The Weierstrass eta function must not be confused with the Dedekind eta function.

 Title Weierstrass sigma function Canonical name WeierstrassSigmaFunction Date of creation 2013-03-22 13:54:06 Last modified on 2013-03-22 13:54:06 Owner alozano (2414) Last modified by alozano (2414) Numerical id 4 Author alozano (2414) Entry type Definition Classification msc 33E05 Synonym sigma function Synonym zeta function Synonym eta function Related topic EllipticFunction Related topic ModularDiscriminant Defines Weierstrass sigma function Defines Weierstrass zeta function Defines Weierstrass eta function