WeierstrassHalfPeriodW2

WeierstrassHalfPeriodW2[{g₂,g₃}]

gives the half-period ω₂ 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.
WeierstrassHalfPeriodW2 can be evaluated to arbitrary precision.
WeierstrassHalfPeriodW2 can be used with CenteredInterval objects. »

Examples

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Basic Examples (3)

Evaluate numerically:

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

Compute the value of the Weierstrass function at the second 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:

WeierstrassHalfPeriodW2 has both singularities and discontinuities:

WeierstrassHalfPeriodW2 is neither non-negative nor non-positive:

WeierstrassHalfPeriodW2 is neither convex nor concave:

WeierstrassHalfPeriodW2 can be used with CenteredInterval objects:

TraditionalForm formatting:

Properties & Relations (4)

Up to a change in sign, the half-period is equal to the sum of the half-periods 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:

WeierstrassHalfPeriodW2 gives a zero of WeierstrassPPrime in the lattice cell:

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