Lemniscate elliptic functions: Difference between revisions

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In [[mathematics]], and in particular the study of [[Weierstrass elliptic function]]s, 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;<mathsub>\omega_11</mathsub> is real and equal to

:<math>\frac{\Gamma^2(\tfrac{1}{4})}{4\sqrt{\pi}}</math>

where <math>\&Gamma</math>; is the [[Gamma function]]. The second smallest half period is pure imaginary

and equal to ''i''&omega;<mathsub>i\omega_11</mathsub>. In more algebraic terms, the [[period lattice]] is a real multiple of the [[Gaussian integer]]s.

The [[mathematical constant|constant]]s ''e<mathsub>e_1''1</mathsub>, ''e<mathsub>e_2''2</mathsub>, and ''e<mathsub>e_3''3</mathsub> are given by

:<math>