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:<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===
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