Equianharmonic

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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(\frac{1}{3})}{4\pi}

where Γ is the Gamma function. The half period is

\omega'=\omega_2\left(\frac{1}{2}+i\frac{\sqrt{3}}{2}\right).

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

The constants e1, e2 and e3 are given by

e_1=4^{-\frac{1}{3}}e^{\frac{2\pi i}{3}},\qquad e_2=4^{-\frac{1}{3}},\qquad e_3=4^{-\frac{1}{3}}e^{\frac{2\pi i}{3}}.