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

:$frac\{Gamma^3(frac\{1\}\{3\})\}\{4pi\}$

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

:$omega\text{'}=omega\_2left(frac\{1\}\{2\}+ifrac\{sqrt\{3\{2\}\; ight).$

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

The constants $e\_1$, $e\_2$ and $e\_3$ are given by

:$e\_1=4^\{-frac\{1\}\{3e^\{frac\{2pi\; i\}\{3,qquade\_2=4^\{-frac\{1\}\{3,qquade\_3=4^\{-frac\{1\}\{3e^\{frac\{2pi\; i\}\{3.$