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Weierstrass sigma function
Definition 1 Let $\Lambda\subset\Complex$ be a lattice. Let $\Lambda^{\ast}$ denote $\Lambda-\{ 0 \}$ .
- The Weierstrass sigma function is defined as the product $$\sigma(z;\Lambda)=z\prod_{w\in\Lambda^{\ast}}\left(1-\frac{z}{w}\right)e^{z/w+\frac{1}{2}(z/w)^2}$$
- 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 $2k+2$ .
- The 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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