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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>
where
:<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}</math>
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\
==See also==
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