WeierstrassHalfPeriodW1
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WeierstrassHalfPeriodW1
gives the half-period ω1 for Weierstrass elliptic functions corresponding to the invariants {g2,g3}.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The half-periods define the fundamental period parallelogram for the Weierstrass elliptic functions.
- WeierstrassHalfPeriodW1 gives the first element ω1 returned by WeierstrassHalfPeriods.
- WeierstrassHalfPeriodW1 can be evaluated to arbitrary numerical precision.
- WeierstrassHalfPeriodW1 can be used with CenteredInterval objects. »
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-c6cxri
Plot the real and imaginary parts of the first half-period:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-fj6pog
Compute the value of the Weierstrass function at the first half-period:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-crxsod
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-jo6462
Scope (8)Survey of the scope of standard use cases
Evaluate to arbitrary precision:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-dpe4v9
The precision of the output tracks the precision of the input:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-dib651
Evaluate symbolically for the equianharmonic case:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-dmeqrb
Evaluate symbolically for the lemniscatic case:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-bs95ea
WeierstrassHalfPeriodW1 has both singularities and discontinuities:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-mdtl3h
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-mn5jws
WeierstrassHalfPeriodW1 is neither non-negative nor non-positive:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-m7ugtz
However, it is positive in the first quadrant:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-cbx5bn
WeierstrassHalfPeriodW1 is neither convex nor concave:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-8kku21
WeierstrassHalfPeriodW1 can be used with CenteredInterval objects:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-pgrx3
TraditionalForm formatting:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-01sfk
Applications (3)Sample problems that can be solved with this function
Plot WeierstrassP over its real period:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-e3naj
Compute the elliptic modulus corresponding to the pair of Weierstrass invariants and :
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-ik8rex
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-bu0fnn
Compute the first lattice root :
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-cllwc0
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-bdfuqb
Compare with the built‐in function value:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-m7fqxk
Compare with the expression in terms of WeierstrassP:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-dx3lul
Properties & Relations (4)Properties of the function, and connections to other functions
WeierstrassHalfPeriods returns the pair and :
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-ft0ojm
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-be0bnx
WeierstrassP is periodic, with periods equal to twice the half-periods:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-bxyhom
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-bbltl8
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-gh43p9
The half-periods , and of Weierstrass elliptic functions are not linearly independent:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-gfxdp
This identity holds for all arguments:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-joi5r0
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-ql2xt
WeierstrassHalfPeriodW1 gives a zero of WeierstrassPPrime in the lattice cell:
https://wolfram.com/xid/0bsxzi8p7e46hry6n4-cqnw8o
Wolfram Research (2017), WeierstrassHalfPeriodW1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html (updated 2023).
Text
Wolfram Research (2017), WeierstrassHalfPeriodW1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html (updated 2023).
Wolfram Research (2017), WeierstrassHalfPeriodW1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html (updated 2023).
CMS
Wolfram Language. 2017. "WeierstrassHalfPeriodW1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html.
Wolfram Language. 2017. "WeierstrassHalfPeriodW1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html.
APA
Wolfram Language. (2017). WeierstrassHalfPeriodW1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html
Wolfram Language. (2017). WeierstrassHalfPeriodW1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html
BibTeX
@misc{reference.wolfram_2024_weierstrasshalfperiodw1, author="Wolfram Research", title="{WeierstrassHalfPeriodW1}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html}", note=[Accessed: 08-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_weierstrasshalfperiodw1, organization={Wolfram Research}, title={WeierstrassHalfPeriodW1}, year={2023}, url={https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html}, note=[Accessed: 08-January-2025
]}