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 $\Lambda \subset \mathbb {C}$ 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).

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.

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 \mathbb {C}

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.

Weierstrass p-function

The Weierstrass p-function is defined to be

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

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 sigma-function

Weierstrass zeta-function

Weierstrass eta-function

Weierstrass p-function

See also