Equianharmonic

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

${\frac {\Gamma ^{3}(1/3)}{4\pi }}$

where $\Gamma$ is the Gamma function. The half period is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \omega_1=\tfrac{1}{2}(-1+\sqrt3i)\omega_2.}

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

The constants e₁, e₂ and e₃ are given by

$e_{1}=4^{-1/3}e^{(2/3)\pi i},\qquad e_{2}=4^{-1/3},\qquad e_{3}=4^{-1/3}e^{-(2/3)\pi i}.$

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