WeierstrassE1

WeierstrassE1[{g2,g3}]

gives the value e1 of the Weierstrass elliptic function at the half-period TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW1].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • WeierstrassE1 can be evaluated to arbitrary numerical precision.

Examples

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Basic Examples  (3)

WeierstrassE1 represents the value of WeierstrassP at its first half-period ω1:

Evaluate numerically:

Plot the real and imaginary parts of e1:

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:

WeierstrassE1 has both singularities and discontinuities:

WeierstrassE1 is neither non-negative nor non-positive:

WeierstrassE1 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), WeierstrassE1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassE1.html.

Text

Wolfram Research (2017), WeierstrassE1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassE1.html.

CMS

Wolfram Language. 2017. "WeierstrassE1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeierstrassE1.html.

APA

Wolfram Language. (2017). WeierstrassE1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassE1.html

BibTeX

@misc{reference.wolfram_2024_weierstrasse1, author="Wolfram Research", title="{WeierstrassE1}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassE1.html}", note=[Accessed: 21-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_weierstrasse1, organization={Wolfram Research}, title={WeierstrassE1}, year={2017}, url={https://reference.wolfram.com/language/ref/WeierstrassE1.html}, note=[Accessed: 21-November-2024 ]}