WeierstrassP

WeierstrassP[u,{g2,g3}]

gives the Weierstrass elliptic function TemplateBox[{u, {g, _, 2}, {g, _, 3}}, WeierstrassP].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{u, {g, _, 2}, {g, _, 3}}, WeierstrassP] 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  (29)

Numerical Evaluation  (7)

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:

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix WeierstrassP function using MatrixFunction:

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 TemplateBox[{z, 1, 2}, WeierstrassP]:

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

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:

TemplateBox[{x, g1, g2}, WeierstrassP] is not an analytic function of :

It has both singularities and discontinuities:

TemplateBox[{x, 1, 2}, WeierstrassP] is neither nondecreasing nor nonincreasing:

TemplateBox[{x, 1, 2}, WeierstrassP] is not injective:

TemplateBox[{x, 3, 1}, WeierstrassP] is not surjective:

TemplateBox[{x, 1, 2}, WeierstrassP] is neither non-negative nor non-positive:

TemplateBox[{x, 1, 2}, WeierstrassP] 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  (6)

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:

Plot an elliptic function over a period parallelogram:

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:

Wolfram Research (1988), WeierstrassP, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassP.html (updated 2023).

Text

Wolfram Research (1988), WeierstrassP, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassP.html (updated 2023).

CMS

Wolfram Language. 1988. "WeierstrassP." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassP.html.

APA

Wolfram Language. (1988). WeierstrassP. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassP.html

BibTeX

@misc{reference.wolfram_2024_weierstrassp, author="Wolfram Research", title="{WeierstrassP}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassP.html}", note=[Accessed: 03-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_weierstrassp, organization={Wolfram Research}, title={WeierstrassP}, year={2023}, url={https://reference.wolfram.com/language/ref/WeierstrassP.html}, note=[Accessed: 03-December-2024 ]}