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.

Contents

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  1 Weierstrass sigma function
  


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  2 Weierstrass zeta function
  


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  3 Weierstrass eta function
  


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  4 Weierstrass p-function
  


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  5 See also
  

Weierstrass sigma function

The Weierstrass sigma function associated to a two-dimensional lattice Λ⊂Cdisplaystyle Lambda subset mathbb C is defined to be the product

σ(z;Λ)=z∏w∈Λ∗(1−zw)ez/w+12(z/w)2displaystyle sigma (z;Lambda )=zprod _win Lambda ^*left(1-frac zwright)e^z/w+frac 12(z/w)^2left(1-fraczwright) e^z/w+frac12(z/w)^2"/>

where Λ∗displaystyle Lambda ^* denotes Λ−0displaystyle Lambda -0.

Weierstrass zeta function

The Weierstrass zeta function is defined by the sum

ζ(z;Λ)=σ′(z;Λ)σ(z;Λ)=1z+∑w∈Λ∗(1z−w+1w+zw2).displaystyle zeta (z;Lambda )=frac sigma '(z;Lambda )sigma (z;Lambda )=frac 1z+sum _win Lambda ^*left(frac 1z-w+frac 1w+frac zw^2right).

The Weierstrass zeta function is the logarithmic derivative of the sigma-function. The zeta function can be rewritten as:

ζ(z;Λ)=1z−∑k=1∞G2k+2(Λ)z2k+1displaystyle zeta (z;Lambda )=frac 1z-sum _k=1^infty mathcal G_2k+2(Lambda )z^2k+1

where G2k+2displaystyle mathcal G_2k+2 is the Eisenstein series of weight 2k + 2.

The derivative of the zeta function is −℘(z)displaystyle -wp (z), where ℘(z)displaystyle wp (z) 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

η(w;Λ)=ζ(z+w;Λ)−ζ(z;Λ), for any z∈Cdisplaystyle eta (w;Lambda )=zeta (z+w;Lambda )-zeta (z;Lambda ),mbox for any zin mathbb C mbox for any z in Complex "/> and any w in the lattice Λdisplaystyle Lambda

This is well-defined, i.e. ζ(z+w;Λ)−ζ(z;Λ)displaystyle zeta (z+w;Lambda )-zeta (z;Lambda ) 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 p-function

The Weierstrass p-function is related to the zeta function by

℘(z;Λ)=−ζ′(z;Λ), for any z∈Cdisplaystyle wp (z;Lambda )=-zeta '(z;Lambda ),mbox for any zin mathbb C

The Weierstrass p-function is an even elliptic function of order N=2 with a double pole at each lattice point and no other poles.

See also

• Weierstrass function

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