weierstrass sigma function

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

\wp(z)

called 'pe'.

Weierstrass sigma function

The Weierstrass sigma function is defined as 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 \}

and

\Lambda\subset\Complex

is the two-dimensional lattice.

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 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.

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