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.
Contents 
The Weierstrass sigmafunction associated to a twodimensional lattice is defined to be the product
where Λ ^{*} denotes Λ − {0}.
The Weierstrass zetafunction is defined by the sum
Note that the Weierstrass zetafunction is basically the logarithmic derivative of the sigmafunction. The zetafunction can be rewritten as:
where is the Eisenstein series of weight 2k + 2.
Also note that the derivative of the zetafunction is , where is the Weierstrass elliptic function
The Weierstrass zetafunction should not be confused with the Riemann zetafunction in number theory.
The Weierstrass etafunction is defined to be
It can be proved that this is welldefined, i.e. ζ(z + w;Λ) − ζ(z;Λ) only depends on w. The Weierstrass etafunction should not be confused with the Dedekind etafunction.
The Weierstrass pfunction is defined to be
The Weierstrass pfunction is an even elliptic function of order N=2 with a double pole at each lattice and no others.
This article incorporates material from Weierstrass sigma function on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
