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
\( {\displaystyle {\frac {\Gamma ^{3}(1/3)}{4\pi }}} \)
where \( \Gamma \) is the Gamma function. The half period is
\( {\displaystyle \omega _{1}={\tfrac {1}{2}}(-1+{\sqrt {3}}i)\omega _{2}.}} \)
Here the period lattice is a real multiple of the Eisenstein integers.
The constants e1, e2 and e3 are given by
\({\displaystyle e_{1}=4^{-1/3}e^{(2/3)\pi i},\qquad e_{2}=4^{-1/3},\qquad e_{3}=4^{-1/3}e^{-(2/3)\pi i}.}} \)
The case g2 = 0, g3 = a may be handled by a scaling transformation.
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