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 $ω 2$ 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 $e 1$ , $e 2$ and $e 3$ are given by

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

This page was last modified 22:43, 17 January 2006 by Wikipedia user Charles Matthews. Based on work by Wikipedia user(s) Robinh, Michael Hardy, Oleg Alexandrov, Mathbot and Linas and Anonymous user(s) of Wikipedia.
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