Weierstrass sigma function

Definition 1.

Let ΛC be a lattice. Let Λ denote Λ-{0}.

  1. 1.

    The Weierstrass sigma functionDlmfDlmfDlmfMathworldPlanetmath is defined as the product

  2. 2.

    The Weierstrass zeta function is defined by the sum


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


    where 𝒢2k+2 is the Eisenstein seriesMathworldPlanetmath of weight 2k+2.

  3. 3.

    The Weierstrass eta function is defined to be

    η(w;Λ)=ζ(z+w;Λ)-ζ(z;Λ),for any z

    (It can be proved that this is well defined, i.e. ζ(z+w;Λ)-ζ(z;Λ) only depends on w). The Weierstrass eta function must not be confused with the Dedekind eta functionMathworldPlanetmath.

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