Английская Википедия:Equianharmonic

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In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g₂ = 0 and g₃ = 1. This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. (These are special examples of complex multiplication.)

In the equianharmonic case, the minimal half period ω₂ is real and equal to

<math>\frac{\Gamma^3(1/3)}{4\pi}</math>

where <math>\Gamma</math> is the Gamma function. The half period is

<math>\omega_1=\tfrac{1}{2}(-1+\sqrt3i)\omega_2.</math>

Here the period lattice is a real multiple of the Eisenstein integers.

The constants e₁, e₂ and e₃ are given by

<math>

e_1=4^{-1/3}e^{(2/3)\pi i},\qquad e_2=4^{-1/3},\qquad e_3=4^{-1/3}e^{-(2/3)\pi i}. </math>

The case g₂ = 0, g₃ = a may be handled by a scaling transformation.