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| Version | Summary | Created by | Modification | Content Size | Created at | Operation |
|---|---|---|---|---|---|---|
| 1 | Dean Liu | -- | 3339 | 2022-10-20 01:38:14 |
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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 g2 = 1 and g3 = 0. In the lemniscatic case, the minimal half period ω1 is real and equal to where Γ is the gamma function. The second smallest half period is pure imaginary and equal to iω1. In more algebraic terms, the period lattice is a real multiple of the Gaussian integers. The constants e1, e2, and e3 are given by The case g2 = a, g3 = 0 may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: a > 0 and a < 0. The period parallelogram is either a square or a rhombus.
References
- Ogawa, Takuma (2005). "SIMILARITIES BETWEEN THE TRIGONOMETRIC FUNCTION AND THE LEMNISCATE FUNCTION FROM ARITHEMETIC VIEW POINT". https://www.jstor.org/stable/43686710?seq=1.
- Carlson, B. C. (2010), "Jacobian Elliptic Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/22.20.E5
