Weierstrass functions

In mathematics, the Weierstrass functions are functions that are auxilliary to the Weierstrass elliptic function.

Weierstrass sigma function

The Weierstrass sigma function is defined as the product

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

where $\Lambda ^{\ast }$ denotes $\Lambda -\{0\}$ and $\Lambda \subset \mathbb {C}$ is the two-dimensional lattice.

Weierstrass zeta function

The Weierstrass zeta function is defined by the sum

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

\item The \emph{Weierstrass eta function} is defined to be
$$\eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda), \text{for
any } z\in\Complex$$ (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.
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Weierstrass sigma function at PlanetMath.