Lemniscate elliptic functions: Difference between revisions

Revision as of 18:42, 28 September 2012

Revision as of 18:48, 28 September 2012

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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>sl(sr)=rs</math>

where

:<math> sr=\int_0^rs\frac{dt}{\sqrt{1-t^4}}</math>

and

:<math>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.