Lemniscate elliptic functions: Difference between revisions

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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-1),</math> <math>\operatorname{cl} z = \operatorname{cd}(z; i-1)</math>.

Similarly, the '''hyperbolic lemniscate sine''' {{math|slh}} and '''hyperbolic lemniscate cosine''' {{math|clh}} have a square period lattice with fundamental periods <math>\bigl\{\sqrt2\varpi, \sqrt2\varpi i\bigr\}.</math>

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:<math>

\operatorname{cl}^2 \tfrac12x = \frac{1+\operatorname{cl}x \sqrt{1+\operatorname{sl}^2x}}{1 + \sqrt{1+\operatorname{sl}^2x}+1}

</math>

:<math>

\operatorname{sl}^2 \tfrac12x = \frac{1-\operatorname{cl}x\sqrt{1+\operatorname{sl}^2x}}{1 + \sqrt{1+\operatorname{sl}^2x}+1}

</math>

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&\quad {}= \frac 2 {2 + \sqrt{7} + \sqrt{21 + 8 \sqrt{7}} + \sqrt{2 {14 + 6 \sqrt{7} + \sqrt{455 + 172 \sqrt{7}}}}}

\\[18mu]

& {\operatorname{sl}}\bigl(\tfrac1{18}\varpi\bigr)\, {\operatorname{sl}}\bigl(\tfrac3{18}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac5{18}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac7{18}\varpi\bigr) \\[-3mu]

&\quad {}= \sqrt[8]{\frac{\lambda (9i)}{1-\lambda (9i)}}

= {\tan}\Bigglleft(\vphantom{\frac\Big|\Big|}\right. \frac\pi4 - {\arctan}\Bigglleft(\vphantom{\frac\Big|\Big|}\right.\frac{2\sqrt[3]{2\sqrt{3}-2}-2\sqrt[3]{2-\sqrt{3}}+\sqrt{3}-1}{\sqrt[4]{12}}\Biggrleft.\left.\vphantom{\frac\Big|\Big|}\right)\Biggrright)

\end{aligned}</math>