WeierstrassE3

WeierstrassE3[{g2,g3}]

gives the value e3 of the Weierstrass elliptic function at the half-period TemplateBox[{{g, _, 2}, {g, _, 3}}, WeierstrassHalfPeriodW3].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • WeierstrassE3 can be evaluated to arbitrary numerical precision.

Examples

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Basic Examples  (3)

WeierstrassE3 represents the value of WeierstrassP at its third half-period ω3:

Evaluate numerically:

Plot the real and imaginary parts of the e3:

Scope  (3)

Evaluate to arbitrary precision:

Precision of the output tracks the precision of the input:

TraditionalForm formatting:

Applications  (1)

Find the modulus corresponding to the elliptic curve, specified by Weierstrass invariants:

Compute the modulus alternatively:

Properties & Relations  (3)

Values of WeierstrassP at half-periods are the roots of the defining polynomial:

Values of WeierstrassP at half-periods are not linearly independent:

This identity holds for all arguments:

Symmetric polynomials evaluated at values of WeierstrassP at half-periods yield WeierstrassInvariants:

Introduced in 2017
 (11.2)