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:$zeta(z;Lambda)=frac\{sigma\text{'}(z;Lambda)\}\{sigma(z;Lambda)\}=frac\{1\}\{z\}+sum\_\{winLambda^\{*left(\; frac\{1\}\{z-w\}+frac\{1\}\{w\}+frac\{z\}\{w^2\}\; ight).$

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.