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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.
WeierstrassHalfPeriodW1 can be used with CenteredInterval objects. »

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 (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:

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:

WeierstrassHalfPeriodW1 can be used with CenteredInterval objects:

TraditionalForm formatting:

Applications (3)

Plot WeierstrassP over its real period:

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

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

Compare with the built‐in function value:

Compare with the expression in terms of WeierstrassP:

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:

WeierstrassHalfPeriodW1 gives a zero of WeierstrassPPrime in the lattice cell:

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