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 g₂ = 1 and g₃ = 0.

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

$\frac{\Gamma^2(\tfrac{1}{4})}{4\sqrt{\pi}}$

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

$e_1=\tfrac{1}{2},\qquad e_2=0,\qquad e_3=-\tfrac{1}{2}.$

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 paralleogram is either a "square" or a "diamond".

Lemniscate sine and cosine functions

The lemniscate sine and cosine functions sl and cl are analogues of the usual sine and cosine functions, with a circle replaced by a lemniscate. They are defined by

$\operatorname{sl}(r)=s$

where

$r=\int_0^s\frac{dt}{\sqrt{1-t^4}}$

and

$\operatorname{cl}(r)=c$

$r=\int_c^1\frac{dt}{\sqrt{1-t^4}}.$

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

$G=\frac{2}{\pi}\int_0^1\frac{dt}{\sqrt{1-t^4}}= 0.8346\ldots.$

Arclength of lemniscate

A lemniscate of Bernoulli and its two foci

The lemniscate of Bernoulli

$(x^2+y^2)^2=x^2-y^2$

consists of the points such that the product of their distances from two 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

$r=\int_0^s\frac{dt}{\sqrt{1-t^4}}.$

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).

See also

• Gauss's constant

References

• Abramowitz, Milton; Stegun, Irene A., eds. (December 1972) [1964]. "Chapter 18". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 (10 ed.). New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 658. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642.
• Reinhardt, W.P.; Walker, P.L. (2010), "Lemniscate lattice", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, 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

External links

• Hazewinkel, Michiel, ed. (2001), "Lemniscate functions", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4