# 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(27)

### 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 :

Plot the imaginary part of :

### Function Properties(10)

Real domain of WeierstrassZeta:

WeierstrassZeta is an odd function with respect to x:

WeierstrassZeta threads elementwise over lists in its first argument:

WeierstrassZeta is not an analytic function:

It has both singularities and discontinuities: is neither nondecreasing nor nonincreasing: is not injective: is surjective: is neither non-negative nor non-positive: is neither convex nor concave:

### 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: