Weierstrass functions

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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 functionEdit

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 functionEdit

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 functionEdit

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 ℘-functionEdit

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 alsoEdit

This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.