Equianharmonic

In the study of Weierstrass elliptic functions, the Equianharmonic case occurs when the invariants satisfy <math>g_2=0<math> and <math>g_3=1<math>; This page follows the terminology of Abramowitz and Stegun; see also the leminscatic case.

In the equianharmonic case, the minimal half period <math>\omega_2<math> is real and equal to <math>\Gamma^3(1/3)/(4\pi)<math> where <math>\Gamma<math> is the Gamma function. The half period <math>\omega'=\omega_2(1/2+i\sqrt{3}/2)<math>.

The constants <math>e_1<math>, <math>e_2<math> and <math>e_3<math> are given by

<math>

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