Weierstrass sigma function
From Wikipedia, the free encyclopedia
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function
called 'pe'.
[edit] Weierstrass sigma-function
The Weierstrass sigma-function associated to a two-dimensional lattice 
is defined to be the product
where Λ * denotes Λ − {0}.
[edit] Weierstrass zeta-function
The Weierstrass zeta-function is defined by the sum
Note that the Weierstrass zeta-function is basically the logarithmic derivative of the sigma-function. The zeta-function can be rewritten as:
where 
is the Eisenstein series of weight 2k + 2.
Also note that the derivative of the zeta-function is 
.
The Weierstrass zeta-function should not be confused with the Riemann zeta-function in number theory.
[edit] Weierstrass eta-function
The Weierstrass eta-function is defined to be
It can be proved that this is well-defined, i.e. ζ(z + w;Λ) − ζ(z;Λ) only depends on w. The Weierstrass eta-function should not be confused with the Dedekind eta-function.
This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the GFDL.
