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

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

where Γ is the gamma function. The second smallest half period is pure imaginary and equal to 1. 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 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

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

where

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

and

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

where

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

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

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

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

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

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).

Inverse functions

• 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= (help)