Lemniscate elliptic functions

In mathematics, and in particular the study of Weierstrass elliptic functions, the lemniscatic case occurs when the Weierstrass invariants satisfy $g_{2}=1$ and $g_{3}=0$. This page follows the terminology of Abramowitz and Stegun; see also the equianharmonic case.

In the lemniscatic case, the minimal half period $\omega _{1}$ is real and equal to

${\frac {\Gamma ^{2}({\frac {1}{4}})}{4{\sqrt {pi}}}}$

where $\Gamma$ is the Gamma function. The second smallest half period is pure imaginary and equal to $i\omega _{1}$.

The constants $e_{1}$, $e_{2}$ and $e_{3}$ are given by

$e_{1}={\frac {1}{2}},\qquad e_{2}=0,\qquad e_{3}=-{\frac {1}{2}}.$

See also pseudo-lemniscatic case.