WeierstrassE3
✖
WeierstrassE3
gives the value e3 of the Weierstrass elliptic function at the half-period
.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- WeierstrassE3 can be evaluated to arbitrary numerical precision.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
WeierstrassE3 represents the value of WeierstrassP at its third half-period ω3:

https://wolfram.com/xid/0g7lqxpg175-foj0mg


https://wolfram.com/xid/0g7lqxpg175-kl4dkv

Plot the real and imaginary parts of e3:

https://wolfram.com/xid/0g7lqxpg175-iaqoom

Scope (7)Survey of the scope of standard use cases
Evaluate to arbitrary precision:

https://wolfram.com/xid/0g7lqxpg175-lrxdjj

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

https://wolfram.com/xid/0g7lqxpg175-hfajh1

Evaluate symbolically for the equianharmonic case:

https://wolfram.com/xid/0g7lqxpg175-dmeqrb

Evaluate symbolically for the lemniscatic case:

https://wolfram.com/xid/0g7lqxpg175-bs95ea

WeierstrassE3 has both singularities and discontinuities:

https://wolfram.com/xid/0g7lqxpg175-mdtl3h


https://wolfram.com/xid/0g7lqxpg175-mn5jws

WeierstrassE3 is neither non-negative nor non-positive:

https://wolfram.com/xid/0g7lqxpg175-84dui


https://wolfram.com/xid/0g7lqxpg175-t326s

WeierstrassE3 is neither convex nor concave:

https://wolfram.com/xid/0g7lqxpg175-8kku21

TraditionalForm formatting:

https://wolfram.com/xid/0g7lqxpg175-fqv3hx

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:

https://wolfram.com/xid/0g7lqxpg175-go2o2z

https://wolfram.com/xid/0g7lqxpg175-c3nqna

Compute the modulus using an alternative formula:

https://wolfram.com/xid/0g7lqxpg175-6l5w

https://wolfram.com/xid/0g7lqxpg175-nzfkoa

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:

https://wolfram.com/xid/0g7lqxpg175-b2vjjn

https://wolfram.com/xid/0g7lqxpg175-jyps7u


https://wolfram.com/xid/0g7lqxpg175-g0g4n

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

https://wolfram.com/xid/0g7lqxpg175-bwnl1o

This identity holds for all arguments:

https://wolfram.com/xid/0g7lqxpg175-b0y60x

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

https://wolfram.com/xid/0g7lqxpg175-ce154e

https://wolfram.com/xid/0g7lqxpg175-1ekyk


https://wolfram.com/xid/0g7lqxpg175-jcbau


https://wolfram.com/xid/0g7lqxpg175-mycfzl


https://wolfram.com/xid/0g7lqxpg175-mjis1z

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
]}
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
]}