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


  • 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.


open allclose all

Basic Examples  (3)

Evaluate numerically:

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

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

Scope  (6)

Evaluate to arbitrary precision:

Precision of the output tracks the precision of the input:

WeierstrassHalfPeriodW1 has both singularities and discontinuities:

WeierstrassHalfPeriodW1 is neither non-negative nor non-positive:

However, it is positive in the first quadrant:

WeierstrassHalfPeriodW1 is neither convex nor concave:

TraditionalForm formatting:

Applications  (3)

Plot WeierstrassP over its real period:

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

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

Compare with the builtin function value:

Properties & Relations  (4)

WeierstrassHalfPeriods returns the pair and :

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

Weierstrass half-periods , and are not linearly independent:

This identity holds for all arguments:

WeierstrassHalfPeriodW1 gives a zero of WeierstrassPPrime in the lattice cell:

Wolfram Research (2017), WeierstrassHalfPeriodW1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html.


Wolfram Research (2017), WeierstrassHalfPeriodW1, Wolfram Language function, https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html.


Wolfram Language. 2017. "WeierstrassHalfPeriodW1." Wolfram Language & System Documentation Center. Wolfram Research. 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


@misc{reference.wolfram_2022_weierstrasshalfperiodw1, author="Wolfram Research", title="{WeierstrassHalfPeriodW1}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html}", note=[Accessed: 02-December-2022 ]}


@online{reference.wolfram_2022_weierstrasshalfperiodw1, organization={Wolfram Research}, title={WeierstrassHalfPeriodW1}, year={2017}, url={https://reference.wolfram.com/language/ref/WeierstrassHalfPeriodW1.html}, note=[Accessed: 02-December-2022 ]}