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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

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

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), 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

\( \eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda), \mbox{ for any } z \in \Complex \)

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

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

The Weierstrass p-function is an even elliptic function of order N=2 with a double pole at each lattice and no others.
See also

Weierstrass function

This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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