WOLFRAM

gives the half-period ω1 for Weierstrass elliptic functions corresponding to the invariants {g2,g3}.

Details

Examples

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Basic Examples  (3)Summary of the most common use cases

Evaluate numerically:

Out[1]=1

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

Out[1]=1

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

Out[1]=1
Out[2]=2

Scope  (8)Survey of the scope of standard use cases

Evaluate to arbitrary precision:

Out[5]=5

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

Out[1]=1

Evaluate symbolically for the equianharmonic case:

Out[1]=1

Evaluate symbolically for the lemniscatic case:

Out[2]=2

WeierstrassHalfPeriodW1 has both singularities and discontinuities:

Out[1]=1
Out[2]=2

WeierstrassHalfPeriodW1 is neither non-negative nor non-positive:

Out[1]=1

However, it is positive in the first quadrant:

Out[2]=2

WeierstrassHalfPeriodW1 is neither convex nor concave:

Out[1]=1

WeierstrassHalfPeriodW1 can be used with CenteredInterval objects:

Out[1]=1

TraditionalForm formatting:

Applications  (3)Sample problems that can be solved with this function

Plot WeierstrassP over its real period:

Out[2]=2

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

Out[2]=2

Compute the first lattice root TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassE1]:

Out[2]=2

Compare with the builtin function value:

Out[3]=3

Compare with the expression in terms of WeierstrassP:

Out[4]=4

Properties & Relations  (4)Properties of the function, and connections to other functions

WeierstrassHalfPeriods returns the pair and :

Out[1]=1
Out[2]=2

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

Out[1]=1
Out[2]=2
Out[3]=3

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

Out[1]=1

This identity holds for all arguments:

Out[3]=3

WeierstrassHalfPeriodW1 gives a zero of WeierstrassPPrime in the lattice cell:

Out[1]=1
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).

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 ]}

@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 ]}

@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 ]}