WOLFRAM

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

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Basic Examples  (3)Summary of the most common use cases

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

Out[1]=1

Evaluate numerically:

Out[1]=1

Plot the real and imaginary parts of e3:

Out[1]=1

Scope  (7)Survey of the scope of standard use cases

Evaluate to arbitrary precision:

Out[1]=1

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

Out[1]=1

Evaluate symbolically for the equianharmonic case:

Out[1]=1

Evaluate symbolically for the lemniscatic case:

Out[2]=2

WeierstrassE3 has both singularities and discontinuities:

Out[1]=1
Out[2]=2

WeierstrassE3 is neither non-negative nor non-positive:

Out[1]=1

It is inherently complex:

Out[2]=2

WeierstrassE3 is neither convex nor concave:

Out[1]=1

TraditionalForm formatting:

Applications  (1)Sample problems that can be solved with this function

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

Out[7]=7

Compute the modulus using an alternative formula:

Out[6]=6

Properties & Relations  (3)Properties of the function, and connections to other functions

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

Out[2]=2
Out[3]=3

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

Out[1]=1

This identity holds for all arguments:

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

Out[2]=2
Out[3]=3
Out[4]=4
Out[5]=5
Wolfram Research (2017), WeierstrassE3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassE3.html.
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.

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.

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

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: 11-July-2025 ]}

@misc{reference.wolfram_2025_weierstrasse3, author="Wolfram Research", title="{WeierstrassE3}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassE3.html}", note=[Accessed: 11-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: 11-July-2025 ]}

@online{reference.wolfram_2025_weierstrasse3, organization={Wolfram Research}, title={WeierstrassE3}, year={2017}, url={https://reference.wolfram.com/language/ref/WeierstrassE3.html}, note=[Accessed: 11-July-2025 ]}