WeierstrassZeta

WeierstrassZeta[u,{g2,g3}]

gives the Weierstrass zeta function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Related to WeierstrassP by the differential equation .
  • WeierstrassZeta is not periodic and is therefore not strictly an elliptic function.
  • For certain special arguments, WeierstrassZeta automatically evaluates to exact values.
  • WeierstrassZeta can be evaluated to arbitrary numerical precision.

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (21)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (3)

Value at zero:

WeierstrassZeta automatically evaluates to simpler functions for certain parameters:

Find a value of x for which WeierstrassZeta[x,1/2,1/2]=3:

Visualization  (2)

Plot the WeierstrassZeta function for various parameters:

Plot the real part of TemplateBox[{z, 2, 1}, WeierstrassZeta]:

Plot the imaginary part of TemplateBox[{z, 2, 1}, WeierstrassZeta]:

Function Properties  (4)

Real domain of WeierstrassZeta:

WeierstrassZeta is an odd function with respect to x:

WeierstrassZeta threads elementwise over lists in its first argument:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Applications  (1)

The 2D equations of motion of two point-like vertices having closed trajectories:

Solve the equations numerically:

Plot the vertex trajectories:

Properties & Relations  (6)

Derivatives of WeierstrassZeta:

Anti-derivative:

WeierstrassZeta is quasi-periodic with respect to translations by periods of the lattice:

WeierstrassZeta is an odd function:

WeierstrassZeta is quasi-periodic, with quasi-periods equal to periods of WeierstrassP:

Values of WeierstrassZeta at its half-quasi-periods:

Possible Issues  (1)

Machine-precision input may be insufficient to give the correct result:

Use arbitraryprecision arithmetic to obtain the correct result:

Neat Examples  (1)

Plot the quasidoubly periodic WeierstrassZeta over the complex plane:

Introduced in 1996
 (3.0)