Lemniscate elliptic functions

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

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

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

where ${\displaystyle \Gamma }$ is the Gamma function. The second smallest half period is pure imaginary and equal to ${\displaystyle i\omega _{1}}$. In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.

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

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