Weierstrass functions - Alchetron, The Free Social Encyclopedia

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.

Weierstrass sigma-function

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

σ ( z ; Λ ) = z ∏ w ∈ Λ ∗ ( 1 − z w ) e z / w + 1 2 ( z / w ) 2

where Λ ∗ denotes Λ − { 0 } .

Weierstrass zeta-function

The Weierstrass zeta-function is defined by the sum

ζ ( z ; Λ ) = σ ′ ( z ; Λ ) σ ( z ; Λ ) = 1 z + ∑ w ∈ Λ ∗ ( 1 z − w + 1 w + z w 2 ) .

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

ζ ( z ; Λ ) = 1 z − ∑ k = 1 ∞ G 2 k + 2 ( Λ ) z 2 k + 1

where G 2 k + 2 is the Eisenstein series of weight 2k + 2.

The derivative of the zeta-function is − ℘ ( z ) , where ℘ ( 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 ∈ C and any w in the lattice Λ

This is well-defined, i.e. ζ ( z + w ; Λ ) − ζ ( z ; Λ ) only depends on the lattice vector w. The Weierstrass eta-function should not be confused with the Dedekind eta-function.

Weierstrass p-function

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

℘ ( z ; Λ ) = − ζ ′ ( z ; Λ ) , for any z ∈ 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.

References

Weierstrass functions Wikipedia

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Contents

Weierstrass sigma-function

Weierstrass zeta-function

Weierstrass eta-function

Weierstrass p-function

References