WeierstrassHalfPeriodW3

WeierstrassHalfPeriodW3[{g₂,g₃}]

gives the half-period ω₃ for the 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.
WeierstrassHalfPeriodW3 gives the second element ω₃ returned by WeierstrassHalfPeriods.
WeierstrassHalfPeriodW3 can be evaluated to arbitrary precision.
WeierstrassHalfPeriodW3 can be used with CenteredInterval objects. »

Examples

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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:

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