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 and is a fundamental pair of periods.
Through careful manipulation of the Weierstrass factorization theorem as it relates also to the sine function, another potentially more manageable infinite product definition is
for any with and where we have used the notation .
Also it is a "quasi-periodic" function, with the following property:
The sigma function can be used to represent an elliptic function: when knowing its zeros and poles that lie in the period parallelogram:
Where is a constant in and are the zeros in the parallelogram and are the poles

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
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.

Degenerate case

Consider the situation where one period is real, which we can scale to be and the other is taken to the limit of so that the functions are only singly-periodic. The corresponding invariants are of discriminant. Then we have and thus from the above infinite product definition the following equality:
A generalization for other sine-like functions on other doubly-periodic lattices is