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 the product

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

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 the derivative of the logarithm of the sigma function. The zeta function 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

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

   (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 the Dedekind eta function.

Title	Weierstrass sigma function
Canonical name	WeierstrassSigmaFunction
Date of creation	2013-03-22 13:54:06
Last modified on	2013-03-22 13:54:06
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Definition
Classification	msc 33E05
Synonym	sigma function
Synonym	zeta function
Synonym	eta function
Related topic	EllipticFunction
Related topic	ModularDiscriminant
Defines	Weierstrass sigma function
Defines	Weierstrass zeta function
Defines	Weierstrass eta function