Lemniscate elliptic functions: Difference between revisions

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</math><ref>The circle <math>x^2+y^2 = x</math> is the unit-diameter circle centered at <math display=inline>\bigl(\tfrac12, 0\bigr)</math> with polar equation <math>r = \cos \theta,</math> the degree-2 ''clover'' under the definition from {{harvp|Cox|Shurman|2005}}. This is ''not'' the unit-''radius'' circle <math>x^2+y^2=1</math> centered at the origin. Notice that the lemniscate <math>\bigl(x^2+y^2\bigr){}^2=x^2-y^2</math> is the degree-4 clover.</ref> the lemniscate sine relates the arc length to the chord length of a lemniscate <math>\bigl(x^2+y^2\bigr){}^2=x^2-y^2.</math>

The lemniscate functions have periods related to a number {{math|<math>\&varpi; {{=</math>}} 2.622057...}} called the [[lemniscate constant]], the ratio of a lemniscate's perimeter to its diameter. This number is a [[Quartic plane curve|quartic]] analog of the ([[Conic section|quadratic]]) {{math|<math>\&pi; {{=</math>}} 3.141592...}}, [[pi|ratio of perimeter to diameter of a circle]].

As [[complex analysis|complex functions]], {{math|sl}} and {{math|cl}} have a [[square lattice|square]] [[period lattice]] (a multiple of the [[Gaussian integer]]s) with [[Fundamental pair of periods|fundamental periods]] <math>\{(1 + i)\varpi, (1 - i)\varpi\},</math><ref>The fundamental periods <math>(1+i)\varpi</math> and <math>(1-i)\varpi</math> are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.</ref> and are a special case of two [[Jacobi elliptic functions]] on that lattice, <math>\operatorname{sl} z = \operatorname{sn}(z; i),</math> <math>\operatorname{cl} z = \operatorname{cd}(z; i)</math>.