PlanetMath: Weierstrass sigma function

Definition 1 Let $\Lambda\subset\mathbb{C}$ be a lattice. Let $\Lambda^{\ast}$ denote $\Lambda-\{ 0 \}$ .

The Weierstrass sigma function is defined as the product
$\displaystyle \sigma(z;\Lambda)=z\prod_{w\in\Lambda^{\ast}}\left(1-\frac{z}{w}\right)e^{z/w+\frac{1}{2}(z/w)^2}$
The Weierstrass zeta function is defined by the sum
$\displaystyle \zeta(z;\Lambda)=\frac{\sigma'(z;\Lambda)}{\sigma(z;\Lambda)}=\fr... ...+\sum_{w\in\Lambda^{\ast}}\left( \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right)$
Note that the Weierstrass zeta function is basically the derivative of the logarithm of the sigma function. The zeta function can be rewritten as:
$\displaystyle \zeta(z;\Lambda)=\frac{1}{z}-\sum_{k=1}^{\infty}\mathcal{G}_{2k+2}(\Lambda)z^{2k+1}$
where $\mathcal{G}_{2k+2}$ is the Eisenstein series of weight .
The Weierstrass eta function is defined to be
$\displaystyle \eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda),$ for any $\displaystyle z\in\mathbb{C}$
(It can be proved that this is well defined, i.e. $\zeta(z+w;\Lambda)-\zeta(z;\Lambda)$ only depends on ). The Weierstrass eta function must not be confused with the Dedekind eta function.