Weierstrass sigma function

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In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function

\wp(z)

called 'pe'.

[edit] 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 Λ * denotes Λ − {0}.

[edit] 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).

The Weierstrass zeta-function should not be confused with the Riemann zeta-function in number theory.

[edit] 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. ζ(z + w;Λ) − ζ(z;Λ) only depends on w. The Weierstrass eta-function should not be confused with the Dedekind eta-function.


This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the GFDL.