WeierstrassHalfPeriodW3

WeierstrassHalfPeriodW3[{g2,g3}]

gives the half-period ω3 for the Weierstrass elliptic functions corresponding to the invariants {g2,g3}.

Details

Examples

open allclose all

Basic Examples  (3)

Evaluate numerically:

Plot the real and imaginary parts of the third half-period:

Compute the value of the Weierstrass function at the third half-period:

Scope  (8)

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:

WeierstrassHalfPeriodW3 has both singularities and discontinuities:

WeierstrassHalfPeriodW3 is neither non-negative nor non-positive:

WeierstrassHalfPeriodW3 is neither convex nor concave:

WeierstrassHalfPeriodW3 can be used with CenteredInterval objects:

TraditionalForm formatting:

Applications  (1)

Compute the elliptic modulus corresponding to the pair of Weierstrass invariants and :

Properties & Relations  (4)

WeierstrassHalfPeriods returns the pair and :

WeierstrassP is periodic, with periods equal to twice the half-periods:

The half-periods , and of Weierstrass elliptic functions are not linearly independent:

This identity holds for all arguments:

WeierstrassHalfPeriodW3 gives a zero of WeierstrassPPrime in the lattice cell:

Wolfram Research (2017), WeierstrassHalfPeriodW3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW3.html (updated 2023).

Text

Wolfram Research (2017), WeierstrassHalfPeriodW3, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW3.html (updated 2023).

CMS

Wolfram Language. 2017. "WeierstrassHalfPeriodW3." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW3.html.

APA

Wolfram Language. (2017). WeierstrassHalfPeriodW3. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW3.html

BibTeX

@misc{reference.wolfram_2023_weierstrasshalfperiodw3, author="Wolfram Research", title="{WeierstrassHalfPeriodW3}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW3.html}", note=[Accessed: 14-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_weierstrasshalfperiodw3, organization={Wolfram Research}, title={WeierstrassHalfPeriodW3}, year={2023}, url={https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW3.html}, note=[Accessed: 14-April-2024 ]}