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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[[File:Lemniscate 15-gon.png|thumb|right|upright=1.5|A lemniscate divided into 15 sections of equal arclength (red curves). Because the prime factors of 15 (3 and 5) are both Fermat primes, this polygon (in black) is constructible using a straightedge and compass.]]

Later mathematicians generalized this result. Analogously to the [[constructible polygon]]s in the circle, the lemniscate can be divided into {{tmath|n}} sections of equal arc length using only [[straightedge and compass construction|straightedge and compass]] if and only if {{tmath|n}} is of the form <math>n = 2^kp_1p_2\cdots p_m</math> where {{tmath|k}} is a non-negative [[integer]] and each {{tmath|p_i}} (if any) is a distinct [[Fermat prime]].<ref>Such numbers are OEIS sequence [[oeis:A003401|A003401]].</ref> The "if" part of the theorem was proved by [[Niels Henrik Abel|Niels Abel]] in 1827–1828, and the "only if" part was proved by [[Michael Rosen (mathematician)|Michael Rosen]] in 1981.<ref>{{harvp|Abel|1827–1828}}; {{harvp|Rosen|1981}}; {{harvp|Prasolov|Solovyev|1997}}</ref> Equivalently, the lemniscate can be divided into {{tmath|n}} sections of equal arc length using only straightedge and compass if and only if <math>\varphi (n)</math> is a [[power of two]] (where <math>\varphi</math> is [[Euler's totient function]]). The lemniscate is ''not'' assumed to be already drawn, as that would go against the rules of straightedge and compass constructions; instead, it is assumed that we are given only two points by which the lemniscate is defined, such as its center and radial point (one of the two points on the lemniscate such that their distance from the center is maximal) or its two foci.

Let <math>r_j=\operatorname{sl}\dfrac{2j\varpi}{n}</math>. Then the {{tmath|n}}-division points for <math>\mathcal{L}</math> are the points

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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>

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:<math>\text{ctlh}(u) = \frac{\text{cd}(u;\tfrac{1}{2}\sqrt{2})}{\sqrt[4]{\text{cd}(u;\tfrac{1}{2}\sqrt{2})^4 + \text{sn}(u;\tfrac{1}{2}\sqrt{2})^4}} </math>

When <math>u</math> is real, the derivative and [[quarter period]] integral of <math> \operatorname{tlh} </math> and <math> \operatorname{ctlh} </math> can be expressed in this way:

:{|class = "wikitable"

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:<math>y(w) = \text{tlh}(w) </math>

The sketch also shows the fact that the derivation of the Areasinus hyperbolic lemniscatus function is equal to the reciprocal of the square root of the successor of the [[fourth power]] function.

==== First proof: comparison with the derivative of the arctangent ====