Lemniscatic elliptic function

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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

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

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[edit]

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

where

and

where

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

Arclength of lemniscate[edit]

A lemniscate of Bernoulli and its two foci

The lemniscate of Bernoulli

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).

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