Lemniscate elliptic functions

In mathematics, and in particular the study of Weierstrass elliptic functions, the lemniscatic case occurs when the Weierstrass invariants satisfy g₂ = 1 and g₃ = 0. This page follows the terminology of Abramowitz and Stegun; see also the equianharmonic case.

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

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

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

${\displaystyle e_{1}={\tfrac {1}{2}},\qquad e_{2}=0,\qquad e_{3}=-{\tfrac {1}{2}}.}$

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

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

where

${\displaystyle r=\int _{0}^{s}{\frac {dt}{\sqrt {1-t^{4}}}}}$ 

and

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

${\displaystyle r=\int _{c}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}}$ 

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

${\displaystyle G={\frac {2}{\pi }}\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}=0.8346\ldots }$ 

Arclength of lemniscate

 

A lemniscate of Bernoulli and its two foci

The lemniscate of Bernoulli

${\displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}}$ 

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

${\displaystyle r=\int _{0}^{s}{\frac {dt}{\sqrt {1-t^{4}}}}}$ 

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 Ann, eds. (1983) [June 1964]. "Chapter 18". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 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, vol. 25, New York-London-Sydney: Wiley-Interscience A Division of John Wiley & Sons, ISBN 0-471-60844-0, MR 0257326

External links

• "Lemniscate functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]