Lemniscatic elliptic function
- Lemniscatic elliptic function
In mathematics, and in particular the study of Weierstrass elliptic functions, the lemniscatic case occurs when the Weierstrass invariants satisfy and . This page follows the terminology of Abramowitz and Stegun; see also the equianharmonic case.
In the lemniscatic case, the minimal half period is real and equal to
:
where is the Gamma function. The second smallest half period is pure imaginary and equal to . In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.
The constants , and are given by
:
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