Equianharmonic

In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy $$g_2=0 and $$g_3=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 $$omega_2 is real and equal to

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

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

:$$omega'=omega_2left(frac{1}{2}+ifrac{sqrt{3{2} ight).

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

The constants $$e_1, $$e_2 and $$e_3 are given by

:$$e_1=4^{-frac{1}{3e^{frac{2pi i}{3,qquade_2=4^{-frac{1}{3,qquade_3=4^{-frac{1}{3e^{frac{2pi i}{3.