Weierstrass sigma function

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

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.

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.

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.