Weierstrass functions

In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and $\wp$ functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant.

Weierstrass sigma function

Plot of #Weierstrass sigma function using Domain coloring.

The Weierstrass sigma function associated to a two-dimensional lattice $\Lambda\subset\Complex$ is defined to be the product

\sigma(z;\Lambda)=z\prod_{w\in\Lambda^{*}} \left(1-\frac{z}{w}\right) e^{z/w+\frac{1}{2}(z/w)^2}

where $\Lambda^{*}$ denotes $\Lambda-\{ 0 \}$ . See also fundamental pair of periods.

Weierstrass zeta function

Plot of #Weierstrass zeta function using Domain coloring

The Weierstrass zeta function is defined by the sum

\zeta(z;\Lambda)=\frac{\sigma'(z;\Lambda)}{\sigma(z;\Lambda)}=\frac{1}{z}+\sum_{w\in\Lambda^{*}}\left( \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right).

The Weierstrass zeta function is 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.

The derivative of the zeta function is $-\wp(z)$ , where $\wp (z)$ is the Weierstrass elliptic function

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

and any w in the lattice

\Lambda

This is well-defined, i.e. $\zeta(z+w;\Lambda)-\zeta(z;\Lambda)$ only depends on the lattice vector w. The Weierstrass eta function should not be confused with either the Dedekind eta function or the Dirichlet eta function.

Weierstrass ℘-function

Plot of #Weierstrass p-function using Domain coloring

The Weierstrass p-function is related to the zeta function by

\wp(z;\Lambda)= -\zeta'(z;\Lambda), \mbox{ for any } z \in \Complex

The Weierstrass ℘-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.