Lemniscate elliptic functions: Difference between revisions

Revision as of 11:30, 5 May 2014

Revision as of 18:36, 19 May 2016

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In [[mathematics]], a '''lemniscatic elliptic function''' is an [[elliptic function]] related to the arc length of a [[lemniscate of Bernoulli]] studied by [[Giulio Carlo de' Toschi di Fagnano]] in 1718. It has a square period lattice and is closely related to the [[Weierstrass elliptic function]] when the Weierstrass invariants satisfy ''g''₂ = 1 and ''g''₃ = 0.

In the lemniscatic case, the minimal half period ω''ω''₁ is real and equal to

:<math>\frac{\Gamma^2(\tfrac{1}{4})}{4\sqrt{\pi}}</math>

where &Gamma;''Γ'' is the [[Gammagamma function]]. The second smallest half period is pure imaginary

and equal to ''iiω''&omega;₁. In more algebraic terms, the [[period lattice]] is a real multiple of the [[Gaussian integer]]s.

The [[mathematical constant|constant]]s ''e_''1, ''e_''2, and ''e_''3 are given by

:<math>

e_1=\tfrac{1}{2}tfrac12,\qquad

e_2=0,\qquad

e_3=-\tfrac{1}{2}tfrac12.

</math>

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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>

where

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:<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 2π2{{pi}}''G'' and 2π2{{pi}}''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>

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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 ({{frac|1/|{{sqrt|2}}}},&radicnbsp;2, 0), (&minus;−{{frac|1/|{{sqrt|2}}}},&radicnbsp;2, 0) is the constant {{sfrac|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,&nbsp;0).

==See also==