Weierstrass functions

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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[edit]

The Weierstrass sigma-function associated to a two-dimensional lattice is defined to be the product

where denotes .

Weierstrass zeta-function[edit]

The Weierstrass zeta-function is defined by the sum

The Weierstrass zeta-function is the logarithmic derivative of the sigma-function. The zeta-function can be rewritten as:

where is the Eisenstein series of weight 2k + 2.

The derivative of the zeta-function is , where is the Weierstrass elliptic function

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

Weierstrass eta-function[edit]

The Weierstrass eta-function is defined to be

and any w in the lattice

This is well-defined, i.e. only depends on the lattice vector w. The Weierstrass eta-function should not be confused with the Dedekind eta-function.

Weierstrass p-function[edit]

The Weierstrass p-function is related to the zeta function by

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.

See also[edit]

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