Lemniscate elliptic functions: Difference between revisions

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{{Short description|Mathematical functions}}

[[File:The lemniscate sine and lemniscate cosine functions of a real variable.png|thumb|upright=2.0|The lemniscate sine (red) and lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric sine {{math|''y'' {{=}} sin(''πx''/''ϖ'')}} (pale dashed red).]]

<span style="display: none;"><math>\def\sl{\operatorname{sl}} \def\cl{\operatorname{cl}}</math></span>In [[mathematics]], the '''lemniscate elliptic functions''' are [[elliptic function]]s related to the arc length of the [[lemniscate of Bernoulli]]. They were first studied by [[Giulio Carlo de' Toschi di Fagnano|Giulio Fagnano]] in 1718 and later by [[Leonhard Euler]] and [[Carl Friedrich Gauss]], among others.<ref>{{harvp|Fagnano|1718–1723}}; {{harvp|Euler|1761}}; {{harvp|Gauss|1917}}</ref>

The '''lemniscate sine''' and '''lemniscate cosine''' functions, usually written with the symbols {{math|sl}} and {{math|cl}} (sometimes the symbols {{math|sinlem}} and {{math|coslem}} or {{math|sin lemn}} and {{math|cos lemn}} are used instead),<ref>{{harvp|Gauss|1917}} p. 199 used the symbols {{math|sl}} and {{math|cl}} for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. {{harvp|Cox|1984}} p. 316, {{harvp|Eymard|Lafon|2004}} p. 204, and {{harvp|Lemmermeyer|2000}} p. 240. {{harvp|Ayoub|1984}} uses {{math|sinlem}} and {{math|coslem}}. {{harvp|Whittaker|Watson|1920}} use the symbols {{math|sin lemn}} and {{math|cos lemn}}. Some sources use the generic letters {{math|''s''}} and {{math|''c''}}. {{harvp|Prasolov|Solovyev|1997}} use the letter {{math|''φ''}} for the lemniscate sine and {{math|''φ′''}} for its derivative.</ref> are analogous to the [[trigonometric functions]] sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-[[diameter]] [[circle]] <!--

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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; -1),</math> <math>\operatorname{cl}sl z = \operatorname{cd}(z; -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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The lemniscate functions {{math|sl}} and {{math|cl}} can be defined as the solution to the [[initial value problem]]:<ref>{{harvp|Robinson|2019a}} starts from this definition and thence derives other properties of the lemniscate functions.</ref>

:<math>\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{sl} z = \bigl(1 + \operatorname{sl}^2 z\bigr)\operatorname{cl}zslz,\ \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{cl}sl z = -\bigl(1 + \operatorname{cl}sl^2 z\bigr)\operatorname{sl}zslz,\ \operatorname{sl} 0 = 0,\ \operatorname{cl}sl 0 = 1,</math>

or equivalently as the [[inverse function|inverses]] of an [[elliptic integral]], the [[Schwarz–Christoffel mapping|Schwarz–Christoffel map]] from the complex [[unit disk]] to a square with corners <math>\big\{\tfrac12\varpi, \tfrac12\varpi i, -\tfrac12\varpi, -\tfrac12\varpi i\big\}\colon</math><ref>This map was the first ever picture of a Schwarz–Christoffel mapping, in {{harvp|Schwarz|1869}} [https://archive.org/details/sim_journal-fuer-die-reine-und-angewandte-mathematik_1869_70/page/113 p. 113].</ref>

:<math> z = \int_0^{\operatorname{sl} z}\frac{\mathrm{d}t}{\sqrt{1-t^4}} = \int_{\operatorname{cl}sl z}^1\frac{\mathrm{d}t}{\sqrt{1-t^4}}.</math>

Beyond that square, the functions can be extended to the [[complex plane]] via [[analytic continuation]] by successive [[Schwarz reflection principle|reflections]].

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:<math>\varpi = 2\int_0^1\frac{\mathrm{d}t}{\sqrt{1-t^4}} = 2.62205\ldots</math>

The lemniscate functions satisfy the basic relation <math>\operatorname{cl}zslz = {\operatorname{sl}}\bigl(\tfrac12\varpi - z\bigr),</math> analogous to the relation <math>\cos z = {\sin}\bigl(\tfrac12\pi - z\bigr).</math>

The lemniscate constant {{tmath|\varpi}} is a close analog of the [[pi|circle constant {{tmath|\pi}}]], and many identities involving {{tmath|\pi}} have analogues involving {{tmath|\varpi}}, as identities involving the [[trigonometric functions]] have analogues involving the lemniscate functions. For example, [[Viète's formula]] for {{tmath|\pi}} can be written:

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=== Zeros, poles and symmetries ===

[[File:Lemniscate sine in the complex plane.svg|thumb|right|upright=1.3|<math>\operatorname{sl}</math> in the complex plane.<ref>Dark areas represent zeros, and bright areas represent poles. As the [[Argument (complex analysis)|argument]] of <math>\operatorname{sl}zslz</math> changes from <math>-\pi</math> (excluding <math>-\pi</math>) to <math>\pi</math>, the colors go through cyan, blue <math>(\operatorname{Arg}\approx -\pi/2)</math>, magneta, red <math>(\operatorname{Arg}\approx 0)</math>, orange, yellow <math>(\operatorname{Arg}\approx\pi/2)</math>, green, and back to cyan <math>(\operatorname{Arg}\approx\pi)</math>.</ref> In the picture, it can be seen that 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.]]

The lemniscate functions {{math|cl}} and {{math|sl}} are [[even and odd functions]], respectively,

:<math>\begin{aligned}

\operatorname{cl}sl(-z) &= \operatorname{cl}sl z \\[6mu]

\operatorname{sl}(-z) &= - \operatorname{sl} z

\end{aligned}</math>

At translations of <math>\tfrac12\varpi,</math> {{math|cl}} and {{math|sl}} are exchanged, and at translations of <math>\tfrac12i\varpi</math> they are additionally rotated and [[multiplicative inverse|reciprocated]]:<ref>Combining the first and fourth identity gives <math>\operatorname{sl}zslz=-i/\operatorname{sl}(z-(1+i)\varpi/2)</math>. This identity is (incorrectly) given in {{harvp|Eymard|Lafon|2004}} p. 226, without the minus sign at the front of the right-hand side.</ref>

:<math>\begin{aligned}

{\operatorname{cl}sl}\bigl(z \pm \tfrac12\varpi\bigr) &= \mp\operatorname{sl} z,&

{\operatorname{cl}sl}\bigl(z \pm \tfrac12i\varpi\bigr) &= \frac{\mp i}{\operatorname{sl} z} \\[6mu]

{\operatorname{sl}}\bigl(z \pm \tfrac12\varpi\bigr) &= \pm\operatorname{cl}sl z,&

{\operatorname{sl}}\bigl(z \pm \tfrac12i\varpi\bigr) &= \frac{\pm i}{\operatorname{cl}sl z}

\end{aligned}</math>

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:<math>\begin{aligned}

\operatorname{cl}sl (z + \varpi) &= \operatorname{cl}sl (z + i\varpi) = -\operatorname{cl}sl z \\[4mu]

\operatorname{sl} (z + \varpi) &= \operatorname{sl} (z + i\varpi) = -\operatorname{sl} z

\end{aligned}</math>

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:<math>\begin{aligned}

{\operatorname{cl}sl}\bigl(z + (1 + i)\varpi\bigr) &= {\operatorname{cl}sl} \bigl(z + (1 - i)\varpi\bigr) = \operatorname{cl}sl z \\[4mu]

{\operatorname{sl}}\bigl(z + (1 + i)\varpi\bigr) &= {\operatorname{sl}} \bigl(z + (1 - i)\varpi\bigr) = \operatorname{sl} z

\end{aligned}</math>

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:<math>\begin{aligned}

\operatorname{cl}sl \bar{z} &= \overline{\operatorname{cl}sl z} \\[6mu]

\operatorname{sl} \bar{z} &= \overline{\operatorname{sl} z} \\[4mu]

\operatorname{cl}sl iz &= \frac{1}{\operatorname{cl}sl z} \\[6mu]

\operatorname{sl} iz &= i \operatorname{sl} z

\end{aligned}</math>

The {{math|sl}} function has simple [[zeros and poles|zeros]] at Gaussian integer multiples of {{tmath|\varpi}}, complex numbers of the form <math>a\varpi + b\varpi i</math> for integers {{tmath|a}} and {{tmath|b}}. It has simple [[zeros and poles|poles]] at Gaussian [[half-integer]] multiples of {{tmath|\varpi}}, complex numbers of the form <math>\bigl(a + \tfrac12\bigr)\varpi + \bigl(b + \tfrac12\bigr)\varpi i</math>, with [[residue (complex analysis)|residue]]s <math>(-1)^{a-b+1}i</math>. The {{math|cl}} function is reflected and offset from the {{math|sl}} function, <math>\operatorname{cl}zslz = {\operatorname{sl}}\bigl(\tfrac12\varpi - z\bigr)</math>. It has zeros for arguments <math>\bigl(a + \tfrac12\bigr)\varpi + b\varpi i</math> and poles for arguments <math>a\varpi + \bigl(b + \tfrac12\bigr)\varpi i,</math> with residues <math>(-1)^{a-b}i.</math>

Also

:<math>\operatorname{sl}zslz=\operatorname{sl}wslw\leftrightarrow z=(-1)^{m+n}w+(m+ni)\varpi</math>

for some <math>m,n\in\mathbb{Z}</math> and

:<math>\operatorname{sl}((1\pm i)z)=(1\pm i)\frac{\operatorname{sl}zslz}{\operatorname{sl}'z}.</math>

The last formula is a special case of [[complex multiplication]]. Analogous formulas can be given for <math>\operatorname{sl}((n+mi)z)</math> where <math>n+mi</math> is any Gaussian integer – the function <math>\operatorname{sl}</math> has complex multiplication by <math>\mathbb{Z}[i]</math>.<ref name="harvp|Cox|2012">{{harvp|Cox|2012}}</ref>

There are also infinite series reflecting the distribution of the zeros and poles of {{math|sl}}:<ref>{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.12.6 §22.12.6], [https://dlmf.nist.gov/22.12.12 §22.12.12]</ref><ref>Analogously, <math>\frac{1}{\sin z}=\sum_{n\in\mathbb{Z}}\frac{(-1)^n}{z+n\pi}.</math></ref>

:<math>\frac{1}{\operatorname{sl}zslz}=\sum_{(n,k)\in\mathbb{Z}^2}\frac{(-1)^{n+k}}{z+n\varpi+k\varpi i}</math>

:<math>\operatorname{sl}zslz=-i\sum_{(n,k)\in\mathbb{Z}^2}\frac{(-1)^{n+k}}{z+(n+1/2)\varpi +(k+1/2)\varpi i}.</math>

=== Pythagorean-like identity ===

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:<math>\operatorname{cl^2} z + \operatorname{sl^2} z + \operatorname{cl^2} z \, \operatorname{sl^2} z = 1</math>

As a result, the parametric equation <math>(x, y) = (\operatorname{cl}sl t, \operatorname{sl} t)</math> parametrizes the [[quartic plane curve|quartic curve]] <math>x^2 + y^2 + x^2y^2 = 1.</math>

This identity can alternately be rewritten:<ref>{{harvp|Lindqvist|Peetre|2001}} generalizes the first of these forms.</ref>

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:<math>\operatorname{cl^2} z \oplus \operatorname{sl^2} z = 1.</math>

The functions <math>\tilde{\operatorname{cl}sl}</math> and <math>\tilde{\operatorname{sl}}</math> satisfy another Pythagorean-like identity:

:<math>\left(\int_0^x \tilde{\operatorname{cl}sl}\,t\,\mathrm dt\right)^2+\left(1-\int_0^x \tilde{\operatorname{sl}}\,t\,\mathrm dt\right)^2=1.</math>

=== Derivatives and integrals ===

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:<math>\begin{aligned}

\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cl}sl z = \operatorname{cl'}z

&= -\bigl(1 + \operatorname{cl^2} z\bigr)\operatorname{sl}zslz=-\frac{2\operatorname{sl}zslz}{\operatorname{sl}^2z+1} \\

\operatorname{cl'^2} z &= 1 - \operatorname{cl^4} z \\[5mu]

\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sl} z = \operatorname{sl'}z

&= \bigl(1 + \operatorname{sl^2} z\bigr)\operatorname{cl}zslz=\frac{2\operatorname{cl}zslz}{\operatorname{cl}sl^2z+1}\\

\operatorname{sl'^2} z &= 1 - \operatorname{sl^4} z\end{aligned}</math>

:<math>\begin{align}\frac{\mathrm d}{\mathrm dz}\,\tilde{\operatorname{cl}sl}\,z&=-2\,\tilde{\operatorname{sl}}\,z\,\operatorname{cl}zslz-\frac{\tilde{\operatorname{sl}}\,z}{\operatorname{cl}zslz}\\

\frac{\mathrm d}{\mathrm dz}\,\tilde{\operatorname{sl}}\,z&=2\,\tilde{\operatorname{cl}sl}\,z\,\operatorname{cl}zslz-\frac{\tilde{\operatorname{cl}sl}\,z}{\operatorname{cl}zslz}

\end{align}</math>

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The second derivatives of lemniscate sine and lemniscate cosine are their negative duplicated cubes:

:<math>\frac{\mathrm{d}^2}{\mathrm{d}z^2}\operatorname{cl}zslz = -2\operatorname{cl^3}z </math>

:<math>\frac{\mathrm{d}^2}{\mathrm{d}z^2}\operatorname{sl}zslz = -2\operatorname{sl^3}z </math>

The lemniscate functions can be integrated using the inverse tangent function:

:<math>\begin{align}\int\operatorname{cl}sl z \mathop{\mathrm{d}z}& = \arctan \operatorname{sl} z + C\\

\int\operatorname{sl} z \mathop{\mathrm{d}z}& = -\arctan \operatorname{cl}sl z + C\\

\int\tilde{\operatorname{cl}sl}\,z\,\mathrm dz&=\frac{\tilde{\operatorname{sl}}\,z}{\operatorname{cl}zslz}+C\\

\int\tilde{\operatorname{sl}}\,z\,\mathrm dz&=-\frac{\tilde{\operatorname{cl}sl}\,z}{\operatorname{cl}zslz}+C\end{align}</math>

=== Argument sum and multiple identities ===

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Like the trigonometric functions, the lemniscate functions satisfy argument sum and difference identities. The original identity used by Fagnano for bisection of the lemniscate was:<ref>{{harvp|Ayoub|1984}}; {{harvp|Prasolov|Solovyev|1997}}</ref>

: <math>\operatorname{sl}(u+v) = \frac{\operatorname{sl}uslu\,\operatorname{sl'}v + \operatorname{sl}vslv\,\operatorname{sl'}u}

{1 + \operatorname{sl^2}u\, \operatorname{sl^2}v}</math>

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:<math>\begin{aligned}

\operatorname{cl}sl(u+v)

&= \operatorname{cl}uslu\,\operatorname{cl}vslv \ominus \operatorname{sl}uslu\, \operatorname{sl}vslv

= \frac{\operatorname{cl}uslu\, \operatorname{cl}vslv - \operatorname{sl}uslu\, \operatorname{sl}vslv}

{1 + \operatorname{sl}uslu\, \operatorname{cl}uslu\, \operatorname{sl}vslv\, \operatorname{cl}vslv} \\[2mu]

\operatorname{cl}sl(u-v)

&= \operatorname{cl}uslu\,\operatorname{cl}vslv \oplus \operatorname{sl}uslu\, \operatorname{sl}vslv \\[2mu]

\operatorname{sl}(u+v)

&= \operatorname{sl}uslu\,\operatorname{cl}vslv \oplus \operatorname{cl}uslu\,\operatorname{sl}vslv

= \frac{\operatorname{sl}uslu\, \operatorname{cl}vslv + \operatorname{cl}uslu\, \operatorname{sl}vslv}

{1 - \operatorname{sl}uslu\, \operatorname{cl}uslu\, \operatorname{sl}vslv\, \operatorname{cl}vslv} \\[2mu]

\operatorname{sl}(u-v)

&= \operatorname{sl}uslu\,\operatorname{cl}vslv \ominus \operatorname{cl}uslu\,\operatorname{sl}vslv

\end{aligned}</math>

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:<math>\begin{aligned}

\operatorname{cl}sl(x + iy)

&= \frac{\operatorname{cl}xslx - i \operatorname{sl}xslx\, \operatorname{sl}ysly\, \operatorname{cl}ysly}

{\operatorname{cl}ysly + i \operatorname{sl}xslx\, \operatorname{cl}xslx\, \operatorname{sl}ysly} \\[4mu]

&= \frac{\operatorname{cl}xslx\,\operatorname{cl}ysly\left(1 - \operatorname{sl}^2x\,\operatorname{sl}^2y\right)}{\operatorname{cl}sl^2y + \operatorname{sl}^2x\,\operatorname{cl}sl^2x\,\operatorname{sl}^2y}

- i \frac{\operatorname{sl}xslx\,\operatorname{sl}ysly\left(\operatorname{cl}sl^2x + \operatorname{cl}sl^2y\right)}{\operatorname{cl}sl^2y + \operatorname{sl}^2x\,\operatorname{cl}sl^2x\,\operatorname{sl}^2y}

\\[12mu]

\operatorname{sl}(x + iy)

&= \frac{\operatorname{sl}xslx + i \operatorname{cl}xslx\, \operatorname{sl}ysly\, \operatorname{cl}ysly}

{\operatorname{cl}ysly - i \operatorname{sl}xslx\, \operatorname{cl}xslx\, \operatorname{sl}ysly } \\[4mu]

&= \frac{\operatorname{sl}xslx\,\operatorname{cl}ysly\left(1 - \operatorname{cl}sl^2x\,\operatorname{sl}^2y\right)}{\operatorname{cl}sl^2y + \operatorname{sl}^2x\,\operatorname{cl}sl^2x\,\operatorname{sl}^2y}

+ i \frac{\operatorname{cl}xslx\,\operatorname{sl}ysly\left(\operatorname{sl}^2x + \operatorname{cl}sl^2y\right)}{\operatorname{cl}sl^2y + \operatorname{sl}^2x\,\operatorname{cl}sl^2x\,\operatorname{sl}^2y}

\end{aligned}</math>

Gauss discovered that

:<math>\frac{\operatorname{sl}(u-v)}{\operatorname{sl}(u+v)}=\frac{\operatorname{sl}((1+i)u)-\operatorname{sl}((1+i)v)}{\operatorname{sl}((1+i)u)+\operatorname{sl}((1+i)v)}</math>

where <math>u,v\in\mathbb{C}</math> such that both sides are well-defined.

Also

:<math>\operatorname{sl}(u+v)\operatorname{sl}(u-v)=\frac{\operatorname{sl}^2u-\operatorname{sl}^2v}{1+\operatorname{sl}^2u\operatorname{sl}^2v}</math>

where <math>u,v\in\mathbb{C}</math> such that both sides are well-defined; this resembles the trigonometric analog

:<math>\sin (u+v)\sin (u-v)=\sin^2u-\sin^2v.</math>

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:<math>

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

</math>

:<math>

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

</math>

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:<math>

\operatorname{cl}sl 2x = \frac{-1+2\,\operatorname{cl}sl^2x + \operatorname{cl}sl^4x}{1+2\,\operatorname{cl}sl^2x - \operatorname{cl}sl^4x}

</math>

:<math>

\operatorname{sl} 2x = 2\,\operatorname{sl}xslx\,\operatorname{cl}xslx\frac{1+\operatorname{sl}^2x}{1+\operatorname{sl}^4x}

</math>

Line 255 ⟶ 254:

:<math>

\operatorname{cl}sl 3x = \frac{-3\,\operatorname{cl}xslx + 6\,\operatorname{cl}sl^5x + \operatorname{cl}sl^9x}{1+6\,\operatorname{cl}sl^4x - 3\,\operatorname{cl}sl^8x}

</math>

:<math>

\operatorname{sl} 3x = \frac{\color{red}{3}\,\color{black}{\operatorname{sl}xslx -\, }\color{green}{6}\,\color{black}{\operatorname{sl}^5x -\,}\color{blue}{1}\,\color{black}{ \operatorname{sl}^9x}}{\color{blue}{1}\,\color{black}{+\,}\,\color{green}{6}\,\color{black}{\operatorname{sl}^4x -\, }\color{red}{3}\,\color{black}{\operatorname{sl}^8x}}

</math>

Note the "reverse symmetry" of the coefficients of numerator and denominator of <math>\operatorname{sl}3xsl3x</math>. This phenomenon can be observed in multiplication formulas for <math>\operatorname{sl}\beta x</math> where <math>\beta=m+ni</math> whenever <math>m,n\in\mathbb{Z}</math> and <math>m+n</math> is odd.<ref name="harvp|Cox|2012" />

===Lemnatomic polynomials===

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Let <math>L</math> be the [[Lattice (group)|lattice]]

:<math>L=\mathbb{Z}(1+i)\varpi +\mathbb{Z}(1-i)\varpi.</math>

Furthermore, let <math>K=\mathbb{Q}(i)</math>, <math>\mathcal{O}=\mathbb{Z}[i]</math>, <math>z\in\mathbb{C}</math>, <math>\beta=m+in</math>, <math>\gamma=m'+in'</math> (where <math>m,n,m',n'\in\mathbb{Z}</math>), <math>m+n</math> be odd, <math>m'+n'</math> be odd, <math>\gamma\equiv 1\,\operatorname{mod}\, 2(1+i)</math> and <math>\operatorname{sl} \beta z=M_\beta (\operatorname{sl}zslz)</math>. Then

:<math>M_\beta (x)=i^\varepsilon x \frac{P_\beta (x^4)}{Q_\beta (x^4)}</math>

for some coprime polynomials <math>P_\beta (x), Q_\beta (x)\in \mathcal{O}[x]</math>

and some <math>\varepsilon\in \{0,1,2,3\}</math><ref>In fact, <math>i^\varepsilon=\operatorname{sl}\tfrac{\beta\varpi}{2}</math>.</ref> where

:<math>xP_\beta (x^4)=\prod_{\gamma |\beta}\Lambda_\gamma (x)</math>

and

:<math>\Lambda_\beta (x)=\prod_{[\alpha]\in (\mathcal{O}/\beta\mathcal{O})^\times}(x-\operatorname{sl}\alpha\delta_\beta)</math>

where <math>\delta_\beta</math> is any <math>\beta</math>-[[Torsion (algebra)|torsion]] generator (i.e. <math>\delta_\beta \in (1/\beta)L</math> and <math>[\delta_\beta]\in (1/\beta)L/L</math> generates <math>(1/\beta)L/L</math> as an <math>\mathcal{O}</math>-[[Module (mathematics)|module]]). Examples of <math>\beta</math>-torsion generators include <math>2\varpi/\beta</math> and <math>(1+i)\varpi/\beta</math>. The polynomial <math>\Lambda_\beta (x)\in\mathcal{O}[x]</math> is called the <math>\beta</math>-th '''lemnatomic polynomial'''. It is monic and is irreducible over <math>K</math>. The lemnatomic polynomials are the "lemniscate analogs" of the [[cyclotomic polynomials]],<ref name="CH">{{harvp|Cox|Hyde|2014}}</ref>

:<math>\Phi_k(x)=\prod_{[a]\in (\mathbb{Z}/k\mathbb{Z})^\times}(x-\zeta_k^a).</math>

The <math>\beta</math>-th lemnatomic polynomial <math>\Lambda_\beta(x)</math> is the [[Minimal polynomial (field theory)|minimal polynomial]] of <math>\operatorname{sl}\delta_\beta</math> in <math>K[x]</math>. For convenience, let <math>\omega_{\beta}=\operatorname{sl}(2\varpi/\beta)</math> and <math>\tilde{\omega}_{\beta}=\operatorname{sl}((1+i)\varpi/\beta)</math>. So for example, the minimal polynomial of <math>\omega_5</math> (and also of <math>\tilde{\omega}_5</math>) in <math>K[x]</math> is

:<math>\Lambda_5(x)=x^{16}+52x^{12}-26x^8-12x^4+1,</math>

and<ref>{{harvp|Gómez-Molleda|Lario|2019}}</ref>

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{| class="wikitable"

|-

! <math>n</math> !! <math>\operatorname{cl}nsln\varpi</math> !! <math>\operatorname{sl}nsln\varpi</math>

|-

| <math> 1</math>

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<math>\mathcal{L}</math> is parametrized in terms of the arc length <math>s</math> from the point <math>(1,0)</math> by<ref>{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.18#E6 §22.18.E6]</ref>

:<math>(x(s),y(s))=\left(\frac{\operatorname{cl}ssls}{\sqrt{1+\operatorname{sl}^2 s}},\frac{\operatorname{sl}ssls\operatorname{cl}ssls}{\sqrt{1+\operatorname{sl}^2 s}}\right)=\left(\tilde{\operatorname{cl}sl}\,s,\tilde{\operatorname{sl}}\,s\right).</math>

The notation <math>\tilde{\operatorname{cl}sl},\,\tilde{\operatorname{sl}}</math> is used solely for the purposes of this article; in references, notation for general Jacobi elliptic functions is used instead.

The lemniscate integral and lemniscate functions satisfy an argument duplication identity discovered by Fagnano in 1718:<ref>{{harvp|Siegel|1969}}; {{harvp|Schappacher|1997}}</ref>

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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

:<math>\left(r_j\sqrt{\tfrac12\bigl(1+r_j^2\bigr)},\ (-1)^{\left\lfloor 4j/n\right\rfloor} \sqrt{\tfrac12r_j^2\bigl(1-r_j^2\bigr)}\right),\quad j\in\{1,2,\ldots ,n\}</math>

where <math>\lfloor\cdot\rfloor</math> is the [[floor function]]. See [[#Specific values|below]] for some specific values of <math>\operatorname{sl}\dfrac{2\varpi}{n}</math>.

=== Arc length of rectangular elastica ===

Line 407 ⟶ 406:

:<math>u=\int_0^{\varphi}r(\theta)\, \mathrm d\theta=\int_0^{\varphi}\frac{\mathrm d\theta}{\sqrt{1+\sin^2\theta}}.</math>

If <math>E</math> is the projection of <math>D</math> on the x-axis and if <math>F</math> is the projection of <math>C</math> on the x-axis, then the lemniscate elliptic functions are given by

:<math>\operatorname{cl}uslu=\overline{AF}, \quad \operatorname{sl}uslu=\overline{DE},</math>

:<math>\tilde{\operatorname{cl}sl}\, u=\overline{AF}\overline{AC}, \quad \tilde{\operatorname{sl}}\, u=\overline{AF}\overline{FC}.</math>

== Series Identities ==

Line 415 ⟶ 414:

The [[power series]] expansion of the lemniscate sine at the origin is<ref>{{cite web|url=https://oeis.org/A104203|website=The On-Line Encyclopedia of Integer Sequences|title=A104203}}</ref>

:<math>\operatorname{sl}zslz=\sum_{n=0}^\infty a_n z^n=z-12\frac{z^5}{5!}+3024\frac{z^9}{9!}-4390848\frac{z^{13}}{13!}+\cdots,\quad |z|< \tfrac{\varpi}{\sqrt{2}}</math>

where the coefficients <math>a_n</math> are determined as follows:

:<math>n\not\equiv 1\pmod 4\implies a_n=0,</math>

Line 424 ⟶ 423:

The expansion can be equivalently written as<ref>{{Cite book |last1=Lomont |first1=J.S.|last2=Brillhart|first2=John|title=Elliptic Polynomials|publisher=CRC Press |year=2001 |isbn=1-58488-210-7|pages=12, 44}}</ref>

:<math>\operatorname{sl}zslz=\sum_{n=0}^\infty p_{2n} \frac{z^{4n+1}}{(4n+1)!},\quad \left|z\right|<\frac{\varpi}{\sqrt{2}}</math>

where

:<math>p_{n+2}=-12\sum_{j=0}^n\binom{2n+2}{2j+2}p_{n-j}\sum_{k=0}^j \binom{2j+1}{2k+1}p_k p_{j-k},\quad p_0=1,\, p_1=0.</math>

The power series expansion of <math>\tilde{\operatorname{sl}}</math> at the origin is

:<math>\tilde{\operatorname{sl}}\,z=\sum_{n=0}^\infty \alpha_n z^n=z-9\frac{z^3}{3!}+153\frac{z^5}{5!}-4977\frac{z^7}{7!}+\cdots,\quad \left|z\right|<\frac{\varpi}{2}</math>

where <math>\alpha_n=0</math> if <math>n</math> is even and<ref name="OEIS_sl_tilde" />

:<math>\alpha_n=\sqrt{2}\frac{\pi}{\varpi}\frac{(-1)^{(n-1)/2}}{n!}\sum_{k=1}^{\infty}\frac{(2k\pi/\varpi)^{n+1}}{\cosh k\pi},\quad \left|\alpha_n\right|\sim 2^{n+5/2}\frac{n+1}{\varpi^{n+2}}</math>

Line 435 ⟶ 434:

The expansion can be equivalently written as<ref>{{Cite book |last1=Lomont |first1=J.S.|last2=Brillhart|first2=John|title=Elliptic Polynomials|publisher=CRC Press |year=2001 |isbn=1-58488-210-7}} p. 79, eq. 5.36</ref>

:<math>\tilde{\operatorname{sl}}\, z=\sum_{n=0}^\infty \frac{(-1)^n}{2^{n+1}} \left(\sum_{l=0}^n 2^l \binom{2n+2}{2l+1} s_l t_{n-l}\right)\frac{z^{2n+1}}{(2n+1)!} ,\quad \left|z\right|<\frac{\varpi}{2}</math>

where

:<math>s_{n+2}=3 s_{n+1} +24 \sum_{j=0}^n \binom{2n+2}{2j+2} s_{n-j} \sum_{k=0}^j \binom{2j+1}{2k+1} s_k s_{j-k},\quad s_0=1,\, s_1=3,</math>

Line 441 ⟶ 440:

For the lemniscate cosine,<ref>{{Cite book |last1=Lomont |first1=J.S.|last2=Brillhart|first2=John|title=Elliptic Polynomials|publisher=CRC Press |year=2001 |isbn=1-58488-210-7}} p. 79, eq. 5. 36 and p. 78, eq. 5.33</ref>

:<math>\operatorname{cl}sl{z}=1-\sum_{n=0}^\infty (-1)^n \left(\sum_{l=0}^n 2^l \binom{2n+2}{2l+1} q_l r_{n-l}\right) \frac{z^{2n+2}}{(2n+2)!}=1-2\frac{z^2}{2!}+12\frac{z^4}{4!}-216\frac{z^6}{6!}+\cdots ,\quad \left|z\right|<\frac{\varpi}{2},</math>

:<math>\tilde{\operatorname{cl}sl}\,z=\sum_{n=0}^\infty (-1)^n 2^n q_n \frac{z^{2n}}{(2n)!}=1-3\frac{z^2}{2!}+33\frac{z^4}{4!}-819\frac{z^6}{6!}+\cdots ,\quad\left|z\right|<\frac{\varpi}{2}</math>

where

:<math>r_{n+2}=3 \sum_{j=0}^n \binom{2n+2}{2j+2} r_{n-j} \sum_{k=0}^j \binom{2j+1}{2k+1} r_k r_{j-k},\quad r_0=1,\, r_1=0,</math>

Line 453 ⟶ 452:

:<math>R(s)^{-2}+R(is)^{-2}=2,\quad \left|\operatorname{Re}s\right|< \frac{\varpi}{2},\left|\operatorname{Im}s\right|< \frac{\varpi}{2}.</math>

There is a close relation between the lemniscate functions and <math>R(s)</math>. Indeed,<ref name="OEIS_sl_tilde" /><ref name="OEIS_cl_tilde">{{cite web | url=https://oeis.org/A289695 | title=A289695 - Oeis }}</ref>

:<math>\tilde{\operatorname{sl}}\,s=-\frac{\mathrm d}{\mathrm ds}R(s)\quad \left|\operatorname{Im}s\right|<\frac{\varpi}{2}</math>

:<math>\tilde{\operatorname{cl}sl}\,s=\frac{\mathrm d}{\mathrm ds}\sqrt{1-R(s)^2},\quad \left|\operatorname{Re}s-\frac{\varpi}{2}\right|<\frac{\varpi}{2},\,\left|\operatorname{Im}s\right|<\frac{\varpi}{2}</math>

and

:<math>R(s)=\frac{1}{\sqrt{1+\operatorname{sl}^2 s}},\quad \left|\operatorname{Im}s\right

|<\frac{\varpi}{2}.</math>

===Continued fractions===

For <math>z\in\mathbb{C}\setminus\{0\}</math>:<ref>{{Cite book |last1=Wall |first1=H. S. |title=Analytic Theory of Continued Fractions |publisher=Chelsea Publishing Company |year=1948 |pages=374–375}}</ref>

:<math>\int_0^\infty e^{-tz\sqrt{2}}\operatorname{cl}tslt\, \mathrm dt=\cfrac{1/\sqrt{2}}{z+\cfrac{a_1}{z+\cfrac{a_2}{z+\cfrac{a_3}{z+\ddots}}}},\quad a_n=\frac{n^2}{4}((-1)^{n+1}+3)</math>

:<math>\int_0^\infty e^{-tz\sqrt{2}}\operatorname{sl}tslt\operatorname{cl}tslt \, \mathrm dt=\cfrac{1/2}{z^2+b_1-\cfrac{a_1}{z^2+b_2-\cfrac{a_2}{z^2+b_3-\ddots}}},\quad a_n=n^2(4n^2-1),\, b_n=3(2n-1)^2</math>

=== Methods of computation ===

{{quote box

| quote = A fast algorithm, returning approximations to <math>\operatorname{sl} x</math> (which get closer to <math>\operatorname{sl}xslx</math> with increasing <math>N</math>), is the following:<ref>{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.20#ii §22.20(ii)]</ref>

{{ubl |item_style=padding:0.2em 0 0 1.6em;

Line 491 ⟶ 490:

}}

Several methods of computing <math>\operatorname{sl} x</math> involve first making the change of variables <math>\pi x = \varpi \tilde{x}</math> and then computing <math>\operatorname{sl}(\varpi \tilde{x} / \pi).</math>

A [[Hyperbolic function|hyperbolic]] series method:<ref>{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.12.12 §22.12.12]</ref><ref>In general, <math>\sinh(x-n\pi)</math> and <math>\sin (x-n\pi i)=-i\sinh (ix+n\pi)</math> are not equivalent, but the resulting infinite sum is the same.</ref>

:<math>\operatorname{sl}\left(\frac{\varpi}{\pi}x\right)=\frac{\pi}{\varpi}\sum_{n\in\mathbb{Z}} \frac{(-1)^n}{\cosh (x-(n+1/2)\pi)},\quad x\in\mathbb{C}</math>

:<math>\frac{1}{\operatorname{sl}(\varpi x/\pi)} = \frac\pi\varpi \sum_{n\in\mathbb{Z}}\frac{(-1)^n}{{\sinh} {\left(x-n\pi\right)}}=\frac\pi\varpi \sum_{n\in\mathbb{Z}}\frac{(-1)^n}{\sin (x-n\pi i)},\quad x\in\mathbb{C}</math>

[[Fourier series]] method:<ref>{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.11 §22.11]</ref>

:<math>\operatorname{sl}\Bigl(\frac{\varpi}{\pi}x\Bigr)=\frac{2\pi}{\varpi}\sum_{n=0}^\infty \frac{(-1)^n\sin ((2n+1)x)}{\cosh ((n+1/2)\pi)},\quad \left|\operatorname{Im}x\right|<\frac{\pi}{2}</math>

:<math>\operatorname{cl}sl\left(\frac{\varpi}{\pi}x\right)=\frac{2\pi}{\varpi}\sum_{n=0}^\infty \frac{\cos ((2n+1)x)}{\cosh ((n+1/2)\pi)},\quad\left|\operatorname{Im}x\right|<\frac{\pi}{2}</math>

:<math>\frac{1}{\operatorname{sl}(\varpi x/\pi)}=\frac{\pi}{\varpi}\left(\frac{1}{\sin x}-4\sum_{n=0}^\infty \frac{\sin ((2n+1)x)}{e^{(2n+1)\pi}+1}\right),\quad\left|\operatorname{Im}x\right|<\pi</math>

The lemniscate functions can be computed more rapidly by

:<math>\begin{align}\operatorname{sl}\Bigl(\frac\varpi\pi x\Bigr)& = \frac{{\theta_1}{\left(x, e^{-\pi}\right)}}{{\theta_3}{\left(x, e^{-\pi}\right)}},\quad x\in\mathbb{C}\\

\operatorname{cl}sl\Bigl(\frac\varpi\pi x\Bigr)&=\frac{{\theta_2}{\left(x, e^{-\pi}\right)}}{{\theta_4}{\left(x, e^{-\pi}\right)}},\quad x\in\mathbb{C}\end{align}</math>

where

Line 520 ⟶ 519:

Fourier series for the logarithm of the lemniscate sine:

:<math>\ln \operatorname{sl}\left(\frac\varpi\pi x\right)=\ln 2-\frac{\pi}{4}+\ln\sin x+2\sum_{n=1}^\infty \frac{(-1)^n \cos 2nx}{n(e^{n\pi}+(-1)^n)},\quad \left|\operatorname{Im}x\right|<\frac{\pi}{2}</math>

The following series identities were discovered by [[Srinivasa Ramanujan|Ramanujan]]:<ref>{{harvp|Berndt|1994}} p. 247, 248, 253</ref>

:<math>\frac{\varpi ^2}{\pi ^2\operatorname{sl}^2(\varpi x/\pi)}=\frac{1}{\sin ^2x}-\frac{1}{\pi}-8\sum_{n=1}^\infty \frac{n\cos 2nx}{e^{2n\pi}-1},\quad \left|\operatorname{Im}x\right|<\pi</math>

:<math>\arctan\operatorname{sl}\Bigl(\frac\varpi\pi x\Bigr)=2\sum_{n=0}^\infty \frac{\sin((2n+1)x)}{(2n+1)\cosh ((n+1/2)\pi)},\quad \left|\operatorname{Im}x\right|<\frac{\pi}{2}</math>

The functions <math>\tilde{\operatorname{sl}}</math> and <math>\tilde{\operatorname{cl}sl}</math> analogous to <math>\sin</math> and <math>\cos</math> on the unit circle have the following Fourier and hyperbolic series expansions:<ref name="OEIS_sl_tilde" /><ref name="OEIS_cl_tilde" /><ref>{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.11.E1 §22.11.E1]</ref>

:<math>\tilde{\operatorname{sl}}\,s=2\sqrt{2}\frac{\pi^2}{\varpi^2}\sum_{n=1}^\infty\frac{n\sin (2n\pi s/\varpi)}{\cosh n\pi},\quad \left|\operatorname{Im}s\right|<\frac{\varpi}{2}</math>

:<math>\tilde{\operatorname{cl}sl}\,s=\sqrt{2}\frac{\pi^2}{\varpi^2}\sum_{n=0}^\infty \frac{(2n+1)\cos ((2n+1)\pi s/\varpi)}{\sinh ((n+1/2)\pi)},\quad \left|\operatorname{Im}s\right|<\frac{\varpi}{2}</math>

:<math>\tilde{\operatorname{sl}}\,s=\frac{\pi^2}{\varpi^2\sqrt{2}}\sum_{n\in\mathbb{Z}}\frac{\sinh (\pi (n+s/\varpi))}{\cosh^2 (\pi (n+s/\varpi))},\quad s\in\mathbb{C}</math>

:<math>\tilde{\operatorname{cl}sl}\,s=\frac{\pi^2}{\varpi^2\sqrt{2}}\sum_{n\in\mathbb{Z}}\frac{(-1)^n}{\cosh^2 (\pi (n+s/\varpi))},\quad s\in\mathbb{C}</math>

The following identities come from product representations of the theta functions:<ref>{{harvp|Whittaker|Watson|1927}}</ref>

Line 544 ⟶ 543:

Since the lemniscate sine is a meromorphic function in the whole complex plane, it can be written as a ratio of [[entire function]]s. Gauss showed that {{math|sl}} has the following product expansion, reflecting the distribution of its zeros and poles:<ref name="EL227">{{harvp|Eymard|Lafon|2004}} p. 227.</ref>

:<math>\operatorname{sl}zslz=\frac{M(z)}{N(z)}</math>

where

Line 561 ⟶ 560:

:<math>\frac{N'(z)}{N(z)}=\sum_{n=0}^\infty 2^{4n}(1-(-1)^n 2^{2n})\mathrm{H}_{4n}\frac{z^{4n-1}}{(4n)!},\quad \left|z\right|<\frac{\varpi}{\sqrt{2}}.</math>

It is known that

:<math>\frac{1}{\operatorname{sl}^2z}=\sum_{n=0}^\infty 2^{4n}(4n-1)\mathrm{H}_{4n}\frac{z^{4n-2}}{(4n)!},\quad \left|z\right|<\varpi.</math>

Then from

:<math>\frac{\mathrm d}{\mathrm dz}\frac{\operatorname{sl}'z}{\operatorname{sl}zslz}=-\frac{1}{\operatorname{sl}^2z}-\operatorname{sl}^2z</math>

and

:<math>\operatorname{sl}^2z=\frac{1}{\operatorname{sl}^2z}-\frac{(1+i)^2}{\operatorname{sl}^2((1+i)z)}</math>

we get

:<math>\frac{\operatorname{sl}'z}{\operatorname{sl}zslz}=-\sum_{n=0}^\infty 2^{4n}(2-(-1)^n 2^{2n})\mathrm{H}_{4n}\frac{z^{4n-1}}{(4n)!},\quad \left|z\right|<\frac{\varpi}{\sqrt{2}}.</math>

Hence

:<math>\frac{\operatorname{sl}'z}{\operatorname{sl}zslz}=\frac{M'(z)}{M(z)}-\frac{N'(z)}{N(z)},\quad \left|z\right|<\frac{\varpi}{\sqrt{2}}.</math>

Therefore

:<math>\operatorname{sl}zslz=C\frac{M(z)}{N(z)}</math>

for some constant <math>C</math> for <math>\left|z\right|<\varpi/\sqrt{2}</math> but this result holds for all <math>z\in\mathbb{C}</math> by analytic continuation. Using

:<math>\lim_{z\to 0}\frac{\operatorname{sl}zslz}{z}=1</math>

gives <math>C=1</math> which completes the proof. <math>\blacksquare</math>

Line 583 ⟶ 582:

The shifting formulas

:<math>M(z+2\varpi)=e^{2\frac{\pi}{\varpi}(z+\varpi)}M(z),\quad M(z+2\varpi i)=e^{-2\frac{\pi}{\varpi}(iz-\varpi)}M(z)</math>

imply that <math>f</math> is an elliptic function with periods <math>2\varpi</math> and <math>2\varpi i</math>, just as <math>\operatorname{sl}</math>.

It follows that the function <math>g</math> defined by

:<math>g(z)=\frac{\operatorname{sl}zslz}{f(z)},</math>

when patched, is an elliptic function without poles. By [[Liouville's theorem (complex analysis)|Liouville's theorem]], it is a constant. By using <math>\operatorname{sl}zslz=z+\operatorname{O}(z^5)</math>, <math>M(z)=z+\operatorname{O}(z^5)</math> and <math>N(z)=1+\operatorname{O}(z^4)</math>, this constant is <math>1</math>, which proves the theorem. <math>\blacksquare</math>

{{Collapse bottom}}

Gauss conjectured that <math>\ln N(\varpi)=\pi/2</math> (this later turned out to be true) and commented that this “is most remarkable and a proof of this property promises the most serious increase in analysis”.<ref>{{harvp|Bottazzini|Gray|2013}} p. 58</ref> Gauss expanded the products for <math>M</math> and <math>N</math> as infinite series (see below). He also discovered several identities involving the functions <math>M</math> and <math>N</math>, such as

Line 606 ⟶ 605:

Therefore, a <math>3</math>-division polynomial is

:<math>-X^9-6X^5+3X</math>

(meaning one of its roots is <math>\operatorname{sl}(2\varpi/3)</math>).

The equations arrived at by this process are the lemniscate analogs of

:<math>X^n=1</math>

(so that <math>e^{2\pi i/n}</math> is one of the solutions) which comes up when dividing the unit circle into <math>n</math> arcs of equal length. In the following note, the first few coefficients of the monic normalization of such <math>\beta</math>-division polynomials are described symbolically in terms of <math>\beta</math>.</ref><ref>By utilizing the power series expansion of the <math>N</math> function, it can be proved that a polynomial having <math>\operatorname{sl}(2\varpi/\beta)</math> as one of its roots (with <math>\beta</math> from the previous note) is

:<math>\sum_{n=0}^{(\beta\overline{\beta}-1)/4}a_{4n+1}(\beta)X^{\beta\overline{\beta}-4n}</math>

where

Line 620 ⟶ 619:

:<math>N(z)=1+2\frac{z^4}{4!}-4\frac{z^8}{8!}+408\frac{z^{12}}{12!}+\cdots,\quad z\in\mathbb{C}.</math>

This can be contrasted with the power series of <math>\operatorname{sl}</math> which has only finite radius of convergence (because it is not entire).

We define <math>S</math> and <math>T</math> by

:<math>S(z)=N\left(\frac{z}{1+i}\right)^2-iM\left(\frac{z}{1+i}\right)^2,\quad T(z)=S(iz).</math>

Then the lemniscate cosine can be written as

:<math>\operatorname{cl}zslz=\frac{S(z)}{T(z)}</math>

where<ref>{{cite book |last=Zhuravskiy |first=A. M. |title=Spravochnik po ellipticheskim funktsiyam |publisher=Izd. Akad. Nauk. U.S.S.R. |year=1941 |language=Russian}}</ref>

Line 647 ⟶ 646:

:<math>N(3z)=N(z)^9+6M(z)^4N(z)^5-3M(z)^8N(z),</math>

so

:<math>\operatorname{sl}3zsl3z=\frac{-M(z)^9-6M(z)^5N(z)^4+3M(z)N(z)^8}{N(z)^9+6M(z)^4N(z)^5-3M(z)^8N(z)}.</math>

On dividing the numerator and the denominator by <math>N(z)^9</math>, we obtain the triplication formula for <math>\operatorname{sl}</math>:

:<math>\operatorname{sl}3zsl3z=\frac{-\operatorname{sl}^9z-6\operatorname{sl}^5z+3\operatorname{sl}zslz}{1+6\operatorname{sl}^4z-3\operatorname{sl}^8z}.</math></ref>

Gauss' <math>M</math> and <math>N</math> satisfy the following system of differential equations:

Line 657 ⟶ 656:

:<math>X(z)X''''(z)=4X'(z)X'''(z)-3X''(z)^2+2X(z)^2,\quad z\in\mathbb{C}.</math>

The functions can be also expressed by integrals involving elliptic functions:

:<math>M(z)=z\exp\left(-\int_0^z\int_0^w \left(\frac{1}{\operatorname{sl}^2v}-\frac{1}{v^2}\right)\, \mathrm dv\,\mathrm dw\right),</math>

:<math>N(z)=\exp\left(\int_0^z\int_0^w \operatorname{sl}^2v\,\mathrm dv\,\mathrm dw\right)</math>

where the contours do not cross the poles; while the innermost integrals are path-independent, the outermost ones are path-dependent; however, the path dependence cancels out with the non-injectivity of the complex [[exponential function]].

Line 676 ⟶ 675:

The square of the lemniscate sine can be represented as

:<math>\operatorname{sl}^2 z=\frac{1}{\wp (z;4,0)}=\frac{i}{2\wp ((1-i)z;-1,0)}={-2\wp}{\left(\sqrt2z+(i-1)\frac{\varpi}{\sqrt2};1,0\right)}</math>

where the second and third argument of <math>\wp</math> denote the lattice invariants {{tmath|g_2}} and {{tmath|g_3}}. The lemniscate sine is a [[rational function]] in the Weierstrass elliptic function and its derivative:<ref>{{harvp|Eymard|Lafon|2004}} p. 234</ref>

:<math>\operatorname{sl}zslz=-2\frac{\wp (z;-1,0)}{\wp '(z;-1,0)}.</math>

The lemniscate functions can also be written in terms of [[Jacobi elliptic functions]]. The Jacobi elliptic functions <math>\operatorname{sn}</math> and <math>\operatorname{cd}</math> with positive real elliptic modulus have an "upright" rectangular lattice aligned with real and imaginary axes. Alternately, the functions <math>\operatorname{sn}</math> and <math>\operatorname{cd}</math> with modulus {{tmath|i}} (and <math>\operatorname{sd}</math> and <math>\operatorname{cn}</math> with modulus <math>1/\sqrt{2}</math>) have a square period lattice rotated 1/8 turn.<ref>{{Cite book |last1=Armitage |first1=J. V. |title=Elliptic Functions |last2=Eberlein |first2=W. F. |publisher=Cambridge University Press |year=2006 |isbn=978-0-521-78563-1 |page=49}}</ref><ref>The identity <math>\operatorname{cl}sl z = {\operatorname{cn}}\left(\sqrt2z;\tfrac{1}{\sqrt2}\right)</math> can be found in {{harvp|Greenhill|1892}} [[iarchive:applicationselli00greerich/page/n48|p. 33]].</ref>

:<math> \operatorname{sl} z = \operatorname{sn}(z;i)=\operatorname{sc}(z;\sqrt{2})={\tfrac1{\sqrt2}\operatorname{sd}}\left(\sqrt2z;\tfrac{1}{\sqrt2}\right) </math>

:<math> \operatorname{cl}sl z = \operatorname{cd}(z;i)= \operatorname{dn}(z;\sqrt{2})={\operatorname{cn}}\left(\sqrt2z;\tfrac{1}{\sqrt2}\right)</math>

where the second arguments denote the elliptic modulus <math>k</math>.

The functions <math>\tilde{\operatorname{sl}}</math> and <math>\tilde{\operatorname{cl}sl}</math> can also be expressed in terms of Jacobi elliptic functions:

:<math>\tilde{\operatorname{sl}}\,z=\operatorname{cd}(z;i)\operatorname{sd}(z;i)=\operatorname{dn}(z;\sqrt{2})\operatorname{sn}(z;\sqrt{2})=\tfrac{1}{\sqrt{2}}\operatorname{cn}\left(\sqrt{2}z;\tfrac{1}{\sqrt{2}}\right)\operatorname{sn}\left(\sqrt{2}z;\tfrac{1}{\sqrt{2}}\right),</math>

:<math>\tilde{\operatorname{cl}sl}\,z=\operatorname{cd}(z;i)\operatorname{nd}(z;i)=\operatorname{dn}(z;\sqrt{2})\operatorname{cn}(z;\sqrt{2})=\operatorname{cn}\left(\sqrt{2}z;\tfrac{1}{\sqrt{2}}\right)\operatorname{dn}\left(\sqrt{2}z;\tfrac{1}{\sqrt{2}}\right).</math>

=== Relation to the modular lambda function ===

Line 699 ⟶ 698:

:<math>

\prod_{k=1}^n \;{\operatorname{sl}}{\left(\frac{2k-1}{2n+1}\frac{\varpi}{2}\right)}

=\sqrt[8]{\frac{\lambda ((2n+1)i)}{1-\lambda ((2n+1)i)}}</math>

Line 705 ⟶ 704:

:<math>\begin{aligned}

&{\operatorname{sl}}\bigl(\tfrac1{14}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac3{14}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac5{14}\varpi\bigr) \\[7mu]

&\quad {}= \sqrt[8]{\frac{\lambda (7i)}{1-\lambda (7i)}}

= {\tan}\Bigl({\tfrac{1}{2}\arccsc}\Bigl(\tfrac{1}{2}\sqrt{8\sqrt{7}+21}+\tfrac{1}{2}\sqrt{7}+1\Bigr)\Bigr)

Line 711 ⟶ 710:

&\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}\Biggl( \frac\pi4 - {\arctan}\Biggl(\frac{2\sqrt[3]{2\sqrt{3}-2}-2\sqrt[3]{2-\sqrt{3}}+\sqrt{3}-1}{\sqrt[4]{12}}\Biggr)\Biggr)

Line 731 ⟶ 730:

:<math> \operatorname{arccl} x = \int_{x}^{1} \frac{\mathrm dt}{\sqrt{1-t^4}} = \tfrac12\varpi - \operatorname{arcsl}x</math>

For {{tmath|x}} in the interval <math>-1 \leq x \leq 1</math>, <math>\operatorname{sl}\operatorname{arcsl} x = x</math> and <math>\operatorname{cl}sl\operatorname{arccl} x = x</math>

For the halving of the lemniscate arc length these formulas are valid:{{cn|date=September 2024}}

:<math>\begin{aligned}

{\operatorname{sl}}\bigl(\tfrac12\operatorname{arcsl} x\bigr) &= {\sin}\bigl(\tfrac12\arcsin x\bigr) \,{\operatorname{sech}}\bigl(\tfrac12\operatorname{arsinh} x\bigr) \\

{\operatorname{sl}}\bigl(\tfrac12\operatorname{arcsl} x\bigr)^2 &= {\tan}\bigl(\tfrac14\arcsin x^2\bigr)

\end{aligned}</math>

Line 830 ⟶ 829:

The hyperbolic lemniscate functions can be expressed in terms of lemniscate sine and lemniscate cosine:

:<math>\operatorname{slh}\bigl(\sqrt2 z\bigr) = \frac{(1+\operatorname{cl}sl^2 z)\operatorname{sl}zslz}{\sqrt2\operatorname{cl}zslz} </math>

:<math>\operatorname{clh}\bigl(\sqrt2 z\bigr) = \frac{(1 + \operatorname{sl}^2 z)\operatorname{cl}zslz}{\sqrt2\operatorname{sl}zslz} </math>

But there is also a relation to the [[Jacobi elliptic functions]] with the elliptic modulus one by square root of two:

Line 843 ⟶ 842:

:<math>\operatorname{slh}z

= \frac{1-i}{\sqrt2} \operatorname{sl}\left(\frac{1+i}{\sqrt2}z\right)

= \frac{\operatorname{sl}\left(\sqrt[4]{-1}z\right) }{ \sqrt[4]{-1} }

</math>

Line 888 ⟶ 887:

The functions tlh and ctlh fulfill the identities described in the differential equation mentioned:

:<math>\text{tlh}(\sqrt{2}\,u) = \sin_{4}(\sqrt{2}\,u) = \operatorname{sl}(u)\sqrt{\frac{\operatorname{cl}sl^2 u+1}{\operatorname{sl}^2 u+\operatorname{cl}sl^2 u}} </math>

:<math>\text{ctlh}(\sqrt{2}\,u) = \cos_{4}(\sqrt{2}\,u) = \operatorname{cl}sl(u)\sqrt{\frac{\operatorname{sl}^2 u+1}{\operatorname{sl}^2 u+\operatorname{cl}sl^2 u}} </math>

The functional designation sl stands for the lemniscatic sine and the designation cl stands for the lemniscatic cosine.

Line 921 ⟶ 920:

The solutions to this system of equations are as follows:

:<math>x(w) = \operatorname{cl}sl(\tfrac{1}{2}\sqrt{2}w) [\operatorname{sl}(\tfrac{1}{2}\sqrt{2}w)^2+1]^{1/2} [\operatorname{sl}(\tfrac{1}{2}\sqrt{2}w)^2+\operatorname{cl}sl(\tfrac{ 1}{2}\sqrt{2}w)^2]^{-1/2} </math>

:<math>y(w) = \operatorname{sl}(\tfrac{1}{2}\sqrt{2}w) [\operatorname{cl}sl(\tfrac{1}{2}\sqrt{2}w)^2+1]^{1/2} [\operatorname{sl}(\tfrac{1}{2}\sqrt{2}w)^2+\operatorname{cl}sl(\tfrac{ 1}{2}\sqrt{2}w)^2]^{-1/2} </math>

The following therefore applies to the quotient:

:<math>\frac{y(w)}{x(w)} = \frac{\operatorname{sl}(\tfrac{1}{2}\sqrt{2}w) [\operatorname{cl}sl(\tfrac{1}{2}\sqrt{2}w)^2+1]^{1/2}}{\operatorname{cl}sl(\tfrac{1}{2}\sqrt{2}w) [ \operatorname{sl}(\tfrac{1}{2}\sqrt{2}w)^2+1]^{1/2}} = \operatorname{slh}(w) </math>

The functions x(w) and y(w) are called '''cotangent hyperbolic lemniscatus''' and '''hyperbolic tangent'''.

:<math>x(w) = \text{ctlh}(w) </math>

Line 1,110 ⟶ 1,109:

In [[algebraic number theory]], every finite [[abelian extension]] of the [[Gaussian rational]]s <math>\mathbb{Q}(i)</math> is a [[Field extension#Subfield|subfield]] of <math>\mathbb{Q}(i,\omega_n)</math> for some positive integer <math>n</math>.<ref name="CH"/><ref name="Cox508509">{{harvp|Cox|2012}} p. 508, 509</ref> This is analogous to the [[Kronecker–Weber theorem]] for the rational numbers <math>\mathbb{Q}</math> which is based on division of the circle – in particular, every finite abelian extension of <math>\mathbb{Q}</math> is a subfield of <math>\mathbb{Q}(\zeta_n)</math> for some positive integer <math>n</math>. Both are special cases of Kronecker's Jugendtraum, which became [[Hilbert's twelfth problem]].

The [[Field (mathematics)|field]] <math>\mathbb{Q}(i,\operatorname{sl}(\varpi /n))</math> (for positive odd <math>n</math>) is the extension of <math>\mathbb{Q}(i)</math> generated by the <math>x</math>- and <math>y</math>-coordinates of the <math>(1+i)n</math>-[[Torsion (algebra)|torsion points]] on the [[elliptic curve]] <math>y^2=4x^3+x</math>.<ref name="Cox508509"/>

===Hurwitz numbers===

Line 1,129 ⟶ 1,128:

where <math>\zeta</math> is the [[Riemann zeta function]].

The '''Hurwitz numbers''' <math>\mathrm{H}_n,</math> named after [[Adolf Hurwitz]], are the "lemniscate analogs" of the Bernoulli numbers. They can be defined by<ref name="Arakawa">{{Cite book |last1=Arakawa |first1=Tsuneo |last2=Ibukiyama| first2=Tomoyoshi |last3=Kaneko|first3=Masanobu|title=Bernoulli Numbers and Zeta Functions |publisher=Springer |year=2014 |isbn=978-4-431-54918-5}} p. 203—206</ref><ref>Equivalently, <math>\mathrm{H}_n=-\lim_{z\to 0}\frac{\mathrm d^n}{\mathrm dz^n} \left(\frac{(1+i)z/2}{\operatorname{sl}((1+i)z/2)}+\frac{z}{2}\mathcal{E}\left(\frac{z}{2};i\right)\right)</math>

where <math>n\ge 4</math> and <math>\mathcal{E}(\cdot;i)</math> is the [[Jacobi elliptic functions#Definition in terms of inverses of elliptic integrals|Jacobi epsilon function]] with modulus <math>i</math>.</ref>

Line 1,192 ⟶ 1,191:

====Appearances in Laurent series====

The Hurwitz numbers appear in several [[Laurent series]] expansions related to the lemniscate functions:<ref>Arakawa et al. (2014) define <math>\mathrm{H}_{4n}</math> by the expansion of <math>1/\operatorname{sl}^2.</math></ref>

:<math>\begin{align}

\operatorname{sl}^2z

&= \sum_{n=1}^\infty \frac{2^{4n}(1-(-1)^{n} 2^{2n})\mathrm{H}_{4n}}{4n}\frac{z^{4n-2}}{(4n-2)!},\quad

\left|z\right|<\frac{\varpi}{\sqrt{2}} \\

\frac{\operatorname{sl}'z}{\operatorname{sl}{z}}

&= \frac{1}{z}-\sum_{n=1}^\infty \frac{2^{4n}(2-(-1)^n 2^{2n})\mathrm{H}_{4n}}{4n}\frac{z^{4n-1}}{(4n-1)!},\quad

\left|z\right|<\frac{\varpi}{\sqrt{2}} \\

\frac{1}{\operatorname{sl}zslz}

&= \frac{1}{z}-\sum_{n=1}^\infty \frac{2^{2n} ((-1)^n 2-2^{2n})\mathrm{H}_{4n}}{4n}\frac{z^{4n-1}}{(4n-1)!},\quad

\left|z\right|<\varpi \\

\frac{1}{\operatorname{sl}^2z}

&= \frac{1}{z^2}+\sum_{n=1}^\infty \frac{2^{4n}\mathrm{H}_{4n}}{4n}\frac{z^{4n-2}}{(4n-2)!},\quad

\left|z\right|<\varpi

Line 1,222 ⟶ 1,221:

If <math>a</math> and <math>p</math> are coprime, then there exist numbers <math>p'\in\mathbb{Z}[i]</math> (see<ref>{{cite journal |last1=Eisenstein |first1=G.

|title=Beiträge zur Theorie der elliptischen Functionen |language=German|journal=Journal für die reine und angewandte Mathematik|date=1846 |volume=30| url=https://gdz.sub.uni-goettingen.de/id/PPN243919689_0030?tify=%7B%22pages%22%3A%5B202%5D%2C%22view%22%3A%22scan%22%7D}} Eisenstein uses <math>\varphi=\operatorname{sl}</math> and <math>\omega=2\varpi</math>.</ref> for these numbers) such that<ref>{{harvp|Ogawa|2005}}</ref>

:<math>\left(\frac{a}{p}\right)_4=\prod_{p'} \frac{\operatorname{sl}(2\varpi ap'/p)}{\operatorname{sl}(2\varpi p'/p)}.</math>

This theorem is analogous to

:<math>\left(\frac{a}{p}\right)=\prod_{n=1}^{\frac{p-1}{2}}\frac{\sin (2\pi a n/p)}{\sin (2\pi n/p)}</math>

Line 1,344 ⟶ 1,343:

== External links ==

* {{Cite episode |last1=Parker |first1=Matt |author-link1=Matt Parker |title=What is the area of a Squircle?|url=https://www.youtube.com/watch?v=gjtTcyWL0NA |series=Stand-up Maths |date=2021 |network=YouTube }}{{cbignore}}

* {{Cite episode |last=Ramalingam |first=Muthu Veerappan|title=Bernoulli Lemniscate and the Squircle <math>||</math> A remarkable Geometric fun fact!!? |url=https://www.youtube.com/watch?v=mAzIE5OkqWE |series=Act of Learning |date=2023 |network=YouTube}}{{cbignore}} Relation shown in the video amounts to <math>\operatorname{cl}sl(\sqrt{2}t)=\frac{\cos_4^2(t)-\sin_4^2(t)}{\cos_4^2(t)+\sin_4^2(t)}</math>