Weierstrass sigma function

Definition 1.

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

1. The Weierstrass sigma function is defined as theproduct

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

2. The Weierstrass zeta function is defined by the sum

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

   Note that the Weierstrass zeta function is basically thederivative of the logarithm of the sigma function. The zetafunction can be rewritten as:

   $$\zeta (z;\mathrm{\Lambda })=\frac{1}{z}-\sum _{k=1}^{\mathrm{\infty }}{𝒢}_{2k+2}(\mathrm{\Lambda })z^{2k+1}$$

   where ${𝒢}_{2k+2}$ is the Eisenstein series of weight$2k+2$.

3. The Weierstrass eta function is defined to be

   $$\eta (w;\mathrm{\Lambda })=\zeta (z+w;\mathrm{\Lambda })-\zeta (z;\mathrm{\Lambda }),\mathit{\text{forany }}z\in ℂ$$

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