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==Lemniscate sine and cosine functions==
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
:<math>\operatorname{sl}(r)=s</math>
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and
:<math>\operatorname{cl}(r)=c</math>
:<math> r=\int_c^1\frac{dt}{\sqrt{1-t^4}}.</math>
They are doubly periodic (or elliptic) functions in the complex plane, with periods 2π''G'' and 2π''iG'', where [[Gauss's constant]] ''G'' is given by
:<math>G=\frac{2}{\pi}\int_0^1\frac{dt}{\sqrt{1-t^4}}= 0.8346\ldots</math>
===Arclength of lemniscate===
[[Image:Lemniscate of Bernoulli.svg|thumb|400px|right|A lemniscate of Bernoulli and its two foci]]
The [[lemniscate of Bernoulli]]
:<math>(x^2+y^2)^2=x^2-y^2</math>
consists of the points such that the product of their distances from two 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
:<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}.</math>
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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