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
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.