In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.[1]
The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead),[2] are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle [3] the lemniscate sine relates the arc length to the chord length of a lemniscate
The lemniscate functions have periods related to a number 2.622057... called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic) 3.141592..., ratio of perimeter to diameter of a circle.
As complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods [4] and are a special case of two Jacobi elliptic functions on that lattice, .
Similarly, the hyperbolic lemniscate sineslh and hyperbolic lemniscate cosineclh have a square period lattice with fundamental periods
The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function .
^Fagnano (1718–1723); Euler (1761); Gauss (1917)
^Gauss (1917) p. 199 used the symbols sl and cl for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. Cox (1984) p. 316, Eymard & Lafon (2004) p. 204, and Lemmermeyer (2000) p. 240. Ayoub (1984) uses sinlem and coslem. Whittaker & Watson (1920) use the symbols sin lemn and cos lemn. Some sources use the generic letters s and c. Prasolov & Solovyev (1997) use the letter φ for the lemniscate sine and φ′ for its derivative.
^The circle is the unit-diameter circle centered at with polar equation the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle centered at the origin. Notice that the lemniscate is the degree-4 clover.
^The fundamental periods and are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.
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