![TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]](Files/WeierstrassE3.en/6.png "TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]"){kind=small}
WeierstrassE3
WeierstrassE3[{g2,g3}]
gives the value e3 of the Weierstrass elliptic function 
at the half-period ![TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]](Files/WeierstrassE3.en/2.png "TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3]"){kind=small}
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Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- WeierstrassE3 can be evaluated to arbitrary numerical precision.
Examples
open all close allBasic Examples (3)
WeierstrassE3 represents the value of WeierstrassP at its third half-period ω3:
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:
WeierstrassE3 is neither convex nor concave:
TraditionalForm formatting:
Applications (1)
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):
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