WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

WeierstrassE3
✖
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)Summary of the most common use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-foj0mg
Out[1]=1

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-kl4dkv
Out[1]=1

Plot the real and imaginary parts of e3:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-iaqoom
Out[1]=1

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

Evaluate to arbitrary precision:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-lrxdjj
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-hfajh1
Out[1]=1

Evaluate symbolically for the equianharmonic case:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-dmeqrb
Out[1]=1

Evaluate symbolically for the lemniscatic case:

In[2]:=2
✖
https://wolfram.com/xid/0g7lqxpg175-bs95ea
Out[2]=2

WeierstrassE3 has both singularities and discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7lqxpg175-mn5jws
Out[2]=2

WeierstrassE3 is neither non-negative nor non-positive:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-84dui
Out[1]=1

It is inherently complex:

In[2]:=2
✖
https://wolfram.com/xid/0g7lqxpg175-t326s
Out[2]=2

WeierstrassE3 is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-8kku21
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
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:

In[8]:=8
✖
https://wolfram.com/xid/0g7lqxpg175-go2o2z
In[7]:=7
✖
https://wolfram.com/xid/0g7lqxpg175-c3nqna
Out[7]=7

Compute the modulus using an alternative formula:

In[5]:=5
✖
https://wolfram.com/xid/0g7lqxpg175-6l5w
In[6]:=6
✖
https://wolfram.com/xid/0g7lqxpg175-nzfkoa
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:

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-b2vjjn
In[2]:=2
✖
https://wolfram.com/xid/0g7lqxpg175-jyps7u
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0g7lqxpg175-g0g4n
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-bwnl1o
Out[1]=1

This identity holds for all arguments:

In[2]:=2
✖
https://wolfram.com/xid/0g7lqxpg175-b0y60x

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

In[1]:=1
✖
https://wolfram.com/xid/0g7lqxpg175-ce154e
In[2]:=2
✖
https://wolfram.com/xid/0g7lqxpg175-1ekyk
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0g7lqxpg175-jcbau
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0g7lqxpg175-mycfzl
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0g7lqxpg175-mjis1z
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 ]}