Lemniscatic elliptic function

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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 $ω 1$ is real and equal to

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

where $Γ$ is the Gamma function. The second smallest half period is pure imaginary and equal to $i ω 1$ . In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.

The constants $e 1$ , $e 2$ and $e 3$ are given by

$e_1=\frac{1}{2},\qquad e_2=0,\qquad e_3=-\frac{1}{2}.$