Lemniscate elliptic functions: Difference between revisions

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In [[mathematics]], a '''lemniscatic elliptic function''' is an [[elliptic function]] related to the arc length of a [[lemniscate of Bernoulli]] studied by [[Giulio Carlo de' Toschi di Fagnano]] in 1718. It has a square period lattice and is closely related to the [[Weierstrass elliptic function]] when the Weierstrass invariants satisfy ''g''₂ = 1 and ''g''₃ = 0.

In the lemniscatic case, the minimal half period ω₁ is real and equal to

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

where &Gamma; is the [[Gamma function]]. The second smallest half period is pure imaginary

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*{{AS ref|18|658}}

*{{dlmf|id=23.5.iii|title=Lemniscate lattice|first1=W.P.|last1=Reinhardt|first2=P.L.|last2=Walker}}

*{{citation|MRmr=0257326|last= Siegel|first= C. L.|title= Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory|series= Interscience Tracts in Pure and Applied Mathematics|volume=25 |publisher=Wiley-Interscience A Division of John Wiley & Sons, |place=New York-London-Sydney|year= 1969 | ISBN= 0-471-60844-0 }}

==External links==