Lemniscate elliptic functions

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 $x^{2}+y^{2}=x,$ ^[3] the lemniscate sine relates the arc length to the chord length of a lemniscate ${\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}.$

The lemniscate functions have periods related to a number $\varpi =$ called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic) $\pi =$ , 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 $\{(1+i)\varpi ,(1-i)\varpi \},$ ^[4] and are a special case of two Jacobi elliptic functions on that lattice, $\operatorname {sl} z=\operatorname {sn} (z;i),$ $\operatorname {cl} z=\operatorname {cd} (z;i)$ .

Similarly, the hyperbolic lemniscate sine $slh$ and hyperbolic lemniscate cosine $clh$ have a square period lattice with fundamental periods ${\bigl \{}{\sqrt {2}}\varpi ,{\sqrt {2}}\varpi i{\bigr \}}.$

The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function $\wp (z;a,0)$ .

^ 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 $x^{2}+y^{2}=x$ is the unit-diameter circle centered at ${\textstyle {\bigl (}{\tfrac {1}{2}},0{\bigr )}}$ with polar equation $r=\cos \theta ,$ the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle $x^{2}+y^{2}=1$ centered at the origin. Notice that the lemniscate ${\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}$ is the degree-4 clover.
^ The fundamental periods $(1+i)\varpi$ and $(1-i)\varpi$ are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.

[1] Fagnano (1718–1723); Euler (1761); Gauss (1917)

[2] 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.

[3] The circle $x^{2}+y^{2}=x$ is the unit-diameter circle centered at ${\textstyle {\bigl (}{\tfrac {1}{2}},0{\bigr )}}$ with polar equation $r=\cos \theta ,$ the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle $x^{2}+y^{2}=1$ centered at the origin. Notice that the lemniscate ${\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}$ is the degree-4 clover.

[4] The fundamental periods $(1+i)\varpi$ and $(1-i)\varpi$ 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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