Weierstrass functions

From Wikipedia, the free encyclopedia

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.

Contents 1 Weierstrass sigma-function 2 Weierstrass zeta-function 3 Weierstrass eta-function 4 Weierstrass p-function

[edit] Weierstrass sigma-function

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

where Λ ^* denotes Λ − {0}.

[edit] Weierstrass zeta-function

The Weierstrass zeta-function is defined by the sum

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

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

Also note that 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.

[edit] Weierstrass eta-function

The Weierstrass eta-function is defined to be

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.

[edit] Weierstrass p-function

The Weierstrass p-function is defined to be

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

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