WeierstrassE3

WeierstrassE3[{g2,g3}]

gives the value e3 of the Weierstrass elliptic function at the half-period TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • WeierstrassE3 can be evaluated to arbitrary numerical precision.

Examples

open allclose all

Basic Examples  (3)

WeierstrassE3 represents the value of WeierstrassP at its third half-period ω3:

Evaluate numerically:

Plot the real and imaginary parts of e3:

Scope  (7)

Evaluate to arbitrary precision:

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

Evaluate symbolically for the equianharmonic case:

Evaluate symbolically for the lemniscatic case:

WeierstrassE3 has both singularities and discontinuities:

WeierstrassE3 is neither non-negative nor non-positive:

It is inherently complex:

WeierstrassE3 is neither convex nor concave:

TraditionalForm formatting:

Applications  (1)

Find the elliptic modulus m corresponding to an elliptic curve specified by its Weierstrass invariants:

Compute the modulus using an alternative formula:

Properties & Relations  (3)

Values of WeierstrassP at its half-periods are the roots of the defining polynomial:

Values of WeierstrassP at its half-periods are not linearly independent:

This identity holds for all arguments:

The elementary symmetric polynomials evaluated at the values of WeierstrassP at half-periods yield WeierstrassInvariants (the Vieta relations):

Wolfram Research (2017), WeierstrassE3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassE3.html.

Text

Wolfram Research (2017), WeierstrassE3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassE3.html.

CMS

Wolfram Language. 2017. "WeierstrassE3." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeierstrassE3.html.

APA

Wolfram Language. (2017). WeierstrassE3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassE3.html

BibTeX

@misc{reference.wolfram_2025_weierstrasse3, author="Wolfram Research", title="{WeierstrassE3}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassE3.html}", note=[Accessed: 10-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_weierstrasse3, organization={Wolfram Research}, title={WeierstrassE3}, year={2017}, url={https://reference.wolfram.com/language/ref/WeierstrassE3.html}, note=[Accessed: 10-July-2025 ]}