# WeierstrassP

WeierstrassP[u,{g2,g3}]

gives the Weierstrass elliptic function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• gives the value of for which .
• For certain special arguments, WeierstrassP automatically evaluates to exact values.
• WeierstrassP can be evaluated to arbitrary numerical precision.
• WeierstrassP can be used with CenteredInterval objects. »

# 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(5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

WeierstrassP can be used with CenteredInterval objects:

### Specific Values(3)

Find the first positive minimum of WeierstrassP[x,1/2,1/2]:

WeierstrassP automatically evaluates to simpler functions for certain parameters:

Find a few singular points of WeierstrassP[x,{1/2,1/2}]:

### Visualization(2)

Plot the WeierstrassP function for various parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(10)

Real domain of WeierstrassP:

WeierstrassP is an even function with respect to x:

WeierstrassP threads elementwise over lists in its first argument: is not an analytic function of :

It has both singularities and discontinuities: is neither nondecreasing nor nonincreasing: is not injective: is not 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(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

## Applications(5)

Express roots of a cubic through WeierstrassP:

Uniformization of a generic elliptic curve :

The parametrized uniformization:

Check the correctness of the uniformization:

Special solution of the Kortewegde Vries equation:

The Kortewegde Vries equation:

A highprecision check of the solution:

Plot of the solution:

Define the Dixon elliptic functions:

These functions are cubic generalizations of Cos and Sin:

Real and imaginary periods of the Dixon elliptic functions:

Plot the Dixon elliptic functions on the real line:

Visualize the Dixon elliptic functions in the complex plane:

Series expansions of the Dixon elliptic functions:

Compute the invariants corresponding to the lemniscatic case of the Weierstrass elliptic function, in which the ratio of the periods is :

Parameterization of the ChenGackstatter minimal surface:

## Properties & Relations(5)

Derivatives:

Integrate expressions involving WeierstrassP:

WeierstrassP is closely related to the elliptic exponential function EllipticExp:

Compare numerical values:

WeierstrassP is periodic, with periods equal to twice the half-periods:

WeierstrassP values at its half-periods:

## Possible Issues(1)

Machine-precision input is insufficient to give a correct result:

Use arbitraryprecision arithmetic to obtain a correct result:

## Neat Examples(1)

Plot a doubly periodic function over the complex plane: