# Lemniscatic elliptic function

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 *ω*_{1} is real and equal to

where Γ is the gamma function. The second smallest half period is pure imaginary and equal to *iω*_{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

The case *g*_{2} = *a*, *g*_{3} = 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.

## Lemniscate sine and cosine functions

The **lemniscate sine** (Latin: *sinus lemniscatus*) and **lemniscate cosine** (Latin: *cosinus lemniscatus*) functions **sinlemn** aka **sl** and **coslemn** aka **cl** are analogues of the usual sine and cosine functions, with a circle replaced by a lemniscate. They are defined by

where

and

where

They are doubly periodic (or elliptic) functions in the complex plane, with periods 2π*G* and 2π*iG*, where Gauss's constant *G* is given by

### Arclength of lemniscate

consists of the points such that the product of their distances from the two points (1/√2, 0), (−1/√2, 0) is the constant 1/2. The length *r* of the arc from the origin to a point at distance *s* from the origin is given by

In other words, the sine lemniscatic function gives the distance from the origin as a function of the arc length from the origin. Similarly the cosine lemniscate function gives the distance from the origin as a function of the arc length from (1, 0).

## Inverse functions

## See also

## References

- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 18".
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Applied Mathematics Series.**55**(Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 658. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. - Reinhardt, W. P.; Walker, P. L. (2010), "Lemniscate lattice", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
*NIST Handbook of Mathematical Functions*, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248 - Siegel, C. L. (1969). "Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory". Interscience Tracts in Pure and Applied Mathematics.
**25**. New York-London-Sydney: Wiley-Interscience A Division of John Wiley & Sons. ISBN 0-471-60844-0. MR 0257326. Cite journal requires`|journal=`

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## External links

- Hazewinkel, Michiel, ed. (2001) [1994], "Lemniscate functions",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4