WeierstrassHalfPeriodW1

WeierstrassHalfPeriodW1[{g2,g3}]

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.

Examples

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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  (3)

Evaluate to arbitrary precision:

Precision of the output tracks the precision of the input:

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:

Introduced in 2017
 (11.2)