Weierstrass functions

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 relation between the sigma, zeta, and functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant.

Weierstrass sigma function

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

where denotes . See also fundamental pair of periods.

Weierstrass zeta function

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

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 either the Dedekind eta function or the Dirichlet eta function.

Weierstrass ℘-function

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

The Weierstrass ℘-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.

See also

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