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.
The Weierstrass sigma-function associated to a two-dimensional lattice is defined to be the product
where denotes .
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.
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.
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.
This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.