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:

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 (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 Korteweg–de Vries equation:

The Korteweg–de Vries equation:

A high‐precision 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 Chen–Gackstatter 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 arbitrary‐precision arithmetic to obtain a correct result:

Neat Examples (1)

Plot a doubly periodic function over the complex plane:

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WeierstrassP

Details

Examples

Basic Examples (4)