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 and.
In the lemniscatic case, the minimal half period is real and equal to
where is the gamma function. The second smallest half period is pure imaginary and equal to. In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.
The constants,, and are given by
The case, 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: and. The period parallelogram is either a square or a rhombus.

Lemniscate sine and cosine functions

The lemniscate sine and lemniscate cosine functions ' aka ' and ' aka ' 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 functions in the complex plane, with periods and, where Gauss's constant is given by

Arclength of lemniscate

The lemniscate of Bernoulli
consists of the points such that the product of their distances from the two points, is the constant. The length of the arc from the origin to a point at distance 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.

Inverse functions