Weierstrass functions

In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function

:$$wp(z)called 'pe'.

Weierstrass sigma-function

The Weierstrass sigma-function associated to a two-dimensional lattice $$LambdasubsetComplex is defined to be the product:$$sigma(z;Lambda)=zprod_{winLambda^{*left(1-frac{z}{w} ight) e^{z/w+frac{1}{2}(z/w)^2}where $$Lambda^{*} denotes $$Lambda-{ 0 }.

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_{winLambda^{*left( frac{1}{z-w}+frac{1}{w}+frac{z}{w^2} ight).

Note that the Weierstrass zeta-function is basically the logarithmic derivative 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.

Also note that the derivative of the zeta-function is $$-wp(z).

The Weierstrass zeta-function should not be confused with the Riemann zeta-function in number theory.

Weierstrass eta-function

The Weierstrass eta-function is defined to be:$$eta(w;Lambda)=zeta(z+w;Lambda)-zeta(z;Lambda), mbox{ 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 should not be confused with the Dedekind eta-function.