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(1/3)}{4\pi}

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

\omega_1=\tfrac{1}{2}(-1%2B\sqrt3i)\omega_2.

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^{-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.