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.

Weierstrass sigma-function

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 $\Lambda-\{ 0 \}$ .

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_{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 the Dedekind eta-function.

Weierstrass p-function

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

Weierstrass functions

Weierstrass sigma-function

Weierstrass zeta-function

Weierstrass eta-function

Weierstrass p-function

See also