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

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

where ${\displaystyle \Gamma }$ is the Gamma function. The half period is

${\displaystyle \omega _{1}={\tfrac {1}{2}}(-1+{\sqrt {3}}i)\omega _{2}.}$

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

The constants e₁, e₂ and e₃ are given by

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

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