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'.

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_}\left(1-\frac\right) e^(z/w)^2}]

where [\Lambda^] denotes [\Lambda-\].

Weierstrass zeta-function

The Weierstrass zeta-function is defined by the sum

[\zeta(z;\Lambda)=\frac=\frac+\sum_}\left( \frac+\frac+\frac\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-\sum_^\mathcal_(\Lambda)z^]

where [\mathcal_] 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.

Weierstrass eta-function

The Weierstrass eta-function is defined to be

[\eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda), \mbox 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.

This article incorporates material from on PlanetMath, which is licensed under the [Text of the GNU Free Documentation LicenseGFDL].

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