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

:\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.