Lemniscate elliptic functions: Difference between revisions

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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 {{math|''g''₂  {{= }} 1}} and {{math|''g''₃  {{= }} 0}}.

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

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

where ''{{math|Γ''}} is the [[gamma function]]. The second smallest half period is pure imaginary and equal to {{math|''iω''₁}}. In more algebraic terms, the [[period lattice]] is a real multiple of the [[Gaussian integer]]s.

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

The [[mathematical constant|constant]]s {{math|''e''_''1}}, {{math|''e''_''2}}, and {{math|''e''_''3}} are given by

:<math>e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12.

:<math>

e_1=\tfrac12,\qquad

e_2=0,\qquad

e_3=-\tfrac12.

</math>

The case {{math|''g''₂&nbsp; {{=&nbsp;}} ''a''}}, {{math|''g''₃&nbsp; {{=&nbsp;}} 0}} may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: {{math|''a''&nbsp; >&nbsp; 0}} and {{math|''a''&nbsp; <&nbsp; 0}}. The period paralleogram[[parallelogram]] is either a "[[square"]] or a "diamond"[[rhombus]].

==Lemniscate sine and cosine functions==

The [[lemniscate]] sine and cosine functions '''{{math|sl}}''' and '''{{math|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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and

:<math>\operatorname{cl}(r)=c</math>

where

:<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 {{math|2{{pi}}''G''}} and {{math|2{{pi}}''iG''}}, where [[Gauss's constant]] {{math|''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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[[Image:Lemniscate of Bernoulli.svg|thumb|400px|right|A lemniscate of Bernoulli and its two foci]]

The [[lemniscate of Bernoulli]]

:<math>\left(x^2+y^2\right)^2=x^2-y^2</math>

consists of the points such that the product of their distances from the two points {{math|({{fracsfrac|1|{{sqrt|2}}}},&nbsp; 0)}}, {{math|(−{{fracsfrac|1|{{sqrt|2}}}},&nbsp; 0)}} is the constant {{sfrac|1|2}}. The length {{math|''r''}} of the arc from the origin to a point at distance {{math|''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).