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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''<sub>2</sub> = 1 and ''g''<sub>3</sub> = 0.
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| In the lemniscatic case, the minimal half period ω<sub>1</sub> is real and equal to
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| :<math>\frac{\Gamma^2(\tfrac{1}{4})}{4\sqrt{\pi}}</math>
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| where Γ is the [[Gamma function]]. The second smallest half period is pure imaginary | |
| and equal to ''i''ω<sub>1</sub>. In more algebraic terms, the [[period lattice]] is a real multiple of the [[Gaussian integer]]s.
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| The [[mathematical constant|constant]]s ''e<sub>''1</sub>, ''e<sub>''2</sub>, and ''e<sub>''3</sub> are given by
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| :<math>
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| e_1=\tfrac{1}{2},\qquad
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| e_2=0,\qquad
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| e_3=-\tfrac{1}{2}.
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| </math>
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| The case ''g''<sub>2</sub> = ''a'', ''g''<sub>3</sub> = 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: ''a'' > 0 and ''a'' < 0. The period paralleogram is either a "square" or a "diamond".
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| ==Lemniscate sine and cosine functions==
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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
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| :<math>\operatorname{sl}(r)=s</math>
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| where
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| :<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}</math>
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| and
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| :<math>\operatorname{cl}(r)=c</math>
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| :<math> r=\int_c^1\frac{dt}{\sqrt{1-t^4}}.</math>
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| 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
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| :<math>G=\frac{2}{\pi}\int_0^1\frac{dt}{\sqrt{1-t^4}}= 0.8346\ldots.</math> | |
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| ===Arclength of lemniscate===
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| [[Image:Lemniscate of Bernoulli.svg|thumb|400px|right|A lemniscate of Bernoulli and its two foci]]
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| The [[lemniscate of Bernoulli]]
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| :<math>(x^2+y^2)^2=x^2-y^2</math>
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| consists of the points such that the product of their distances from two the two points (1/√2, 0), (−1/√2, 0) is the constant 1/2. The length ''r'' of the arc from the origin to a point at distance ''s'' from the origin is given by
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| :<math> r=\int_0^s\frac{dt}{\sqrt{1-t^4}}.</math>
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| 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 (1,0).
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| ==See also==
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| *[[Gauss's constant]]
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| ==References==
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| *{{AS ref|18|658}}
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| *{{dlmf|id=23.5.iii|title=Lemniscate lattice|first1=W.P.|last1=Reinhardt|first2=P.L.|last2=Walker}}
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| *{{citation|MR=0257326|last= Siegel|first= C. L.|title= Topics in complex function theory. Vol. I: Elliptic functions and uniformization theory|series= Interscience Tracts in Pure and Applied Mathematics|volume=25 |publisher=Wiley-Interscience A Division of John Wiley & Sons, |place=New York-London-Sydney|year= 1969 | ISBN= 0-471-60844-0 }}
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| ==External links==
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| * {{springer|title=Lemniscate functions|id=p/l058120}}
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| [[Category:Modular forms]]
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| [[Category:Elliptic curves]]
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| [[Category:Elliptic functions]]
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