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[[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).]]

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]] <!--
NOTE: THE RHS IN THE FOLLOWING IS x, NOT 1 --><math>x^2+y^2 = x,
</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>.

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>

The lemniscate functions and the hyperbolic lemniscate functions are [[#Relation to Weierstrass and Jacobi elliptic functions|related]] to the [[Weierstrass elliptic function]] <math>\wp (z;a,0)</math>.

==Lemniscate sine and cosine functions==
===Definitions===
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}z,\ \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{cl} z = -\bigl(1 + \operatorname{cl}^2 z\bigr)\operatorname{sl}z,\ \operatorname{sl} 0 = 0,\  \operatorname{cl} 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} z}^1\frac{\mathrm{d}t}{\sqrt{1-t^4}}.</math>

Beyond that square, the functions can be [[analytic continuation|analytically continued]] to the whole [[complex plane]] by a series of [[Schwarz reflection principle|reflections]].

By comparison, the circular sine and cosine can be defined as the solution to the initial value problem:

:<math>\frac{\mathrm{d}}{\mathrm{d}z} \sin z = \cos z,\ \frac{\mathrm{d}}{\mathrm{d}z} \cos z = -\sin z,\ \sin 0 = 0,\ \cos 0 = 1,</math>

or as inverses of a map from the [[upper half-plane]] to a half-infinite strip with real part between <math>-\tfrac12\pi, \tfrac12\pi</math> and positive imaginary part:
:<math> z = \int_0^{\sin z}\frac{\mathrm{d}t}{\sqrt{1-t^2}} = \int_{\cos z}^1\frac{\mathrm{d}t}{\sqrt{1-t^2}}.</math>

=== Relation to the lemniscate constant ===
{{main|Lemniscate constant}}

[[File:Lemniscate constant as an integral.png|thumb|upright=1.3|The lemniscate sine function and hyperbolic lemniscate sine functions are defined as inverses of elliptic integrals. The complete integrals are related to the lemniscate constant {{mvar|ϖ}}.]]
The lemniscate functions have minimal real period {{math|2''ϖ''}}, minimal [[Imaginary number|imaginary]] period {{math|2''ϖ''i}} and fundamental complex periods <math>(1+i)\varpi</math> and <math>(1-i)\varpi</math> for a constant {{math|''ϖ''}} called the ''[[lemniscate constant]]'',<ref>{{harvp|Schappacher|1997}}. OEIS sequence [https://oeis.org/A062539 A062539] lists the lemniscate constant's decimal digits.</ref>

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

<math display="block">
\frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots
</math>

An analogous formula for {{math|''ϖ''}} is:<ref>{{harvp|Levin|2006}}</ref>

<math display="block">
\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 +  \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 +  \frac12 \Bigg/ \!\sqrt{\frac12 +  \frac12 \bigg/ \!\sqrt\frac12}} \cdots
</math>

The [[Machin-like formula|Machin formula]] for {{math|''π''}} is <math display="inline">\tfrac14\pi = 4 \arctan \tfrac15 - \arctan \tfrac1{239},</math> and several similar formulas for {{math|''π''}} can be developed using trigonometric angle sum identities, e.g. Euler's formula <math display="inline">\tfrac14\pi = \arctan\tfrac12 + \arctan\tfrac13</math>. Analogous formulas can be developed for {{math|''ϖ''}}, including the following found by Gauss: <math>\tfrac12\varpi = 2 \operatorname{arcsl} \tfrac12 + \operatorname{arcsl} \tfrac7{23}.</math><ref>{{harvp|Todd|1975}}</ref>

The lemniscate and circle constants were found by Gauss to be related to each-other by the [[arithmetic-geometric mean]] {{math|''M''}}:<ref>{{harvp|Cox|1984}}</ref>

<math display="block">
\frac\pi\varpi = M{\left(1, \sqrt2\!~\right)}
</math>

== Argument identities ==

=== 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}z</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}(-z) &= \operatorname{cl} 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}z=-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}}\bigl(z \pm \tfrac12\varpi\bigr) &= \mp\operatorname{sl} z,&
{\operatorname{cl}}\bigl(z \pm \tfrac12i\varpi\bigr) &= \frac{\mp i}{\operatorname{sl} z} \\[6mu]
{\operatorname{sl}}\bigl(z \pm \tfrac12\varpi\bigr) &= \pm\operatorname{cl} z,& 
{\operatorname{sl}}\bigl(z \pm \tfrac12i\varpi\bigr) &= \frac{\pm i}{\operatorname{cl} z}
\end{aligned}</math>

Doubling these to translations by a [[unit (ring theory)|unit]]-Gaussian-integer multiple of <math>\varpi</math> (that is, <math>\pm \varpi</math> or <math>\pm i\varpi</math>), negates each function, an [[involution (mathematics)|involution]]:

:<math>\begin{aligned}
\operatorname{cl} (z + \varpi) &= \operatorname{cl} (z + i\varpi) = -\operatorname{cl} z \\[4mu]
\operatorname{sl} (z + \varpi) &= \operatorname{sl} (z + i\varpi) = -\operatorname{sl} z
\end{aligned}</math>

As a result, both functions are invariant under translation by an [[Gaussian integer#Examples|even-Gaussian-integer]] multiple of <math>\varpi</math>.<ref>The even Gaussian integers are the residue class of 0, modulo {{math|1 + ''i''}}, the black squares on a [[Checkerboard#Mathematical description|checkerboard]].</ref> That is, a displacement <math>(a + bi)\varpi,</math> with <math>a + b = 2k</math> for integers {{math|''a''}}, {{math|''b''}}, and  {{math|''k''}}.

:<math>\begin{aligned}
{\operatorname{cl}}\bigl(z + (1 + i)\varpi\bigr) &= {\operatorname{cl}} \bigl(z + (1 - i)\varpi\bigr) = \operatorname{cl} z \\[4mu]
{\operatorname{sl}}\bigl(z + (1 + i)\varpi\bigr) &= {\operatorname{sl}} \bigl(z + (1 - i)\varpi\bigr) = \operatorname{sl} z
\end{aligned}</math>

This makes them [[elliptic functions]] (doubly periodic [[meromorphic function]]s in the complex plane) with a [[square lattice|diagonal square]] [[period lattice]] of fundamental periods <math>(1 + i)\varpi</math> and <math>(1 - i)\varpi</math>.<ref>{{harvp|Prasolov|Solovyev|1997}}; {{harvp|Robinson|2019a}}</ref> Elliptic functions with a square period lattice are more symmetrical than arbitrary elliptic functions, following the symmetries of the square.

Reflections and quarter-turn rotations of lemniscate function arguments have simple expressions:

:<math>\begin{aligned}
\operatorname{cl} \bar{z} &= \overline{\operatorname{cl} z} \\[6mu]
\operatorname{sl} \bar{z} &= \overline{\operatorname{sl} z} \\[4mu]
\operatorname{cl} iz &= \frac{1}{\operatorname{cl} 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 {{math|''ϖ''}}, complex numbers of the form <math>a\varpi + b\varpi i</math> for integers {{math|''a''}} and {{math|''b''}}. It has simple [[zeros and poles|poles]] at Gaussian [[half-integer]] multiples of {{math|''ϖ''}}, 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}z = {\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}z=\operatorname{sl}w\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}z}{\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}z}=\sum_{(n,k)\in\mathbb{Z}^2}\frac{(-1)^{n+k}}{z+n\varpi+k\varpi i}</math>
:<math>\operatorname{sl}z=-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 ===
[[File:Algebraic curves (x² + y²) = a(1 - x²y²) for various values of a.png|thumb|upright=1.3|Curves {{math|1=''x''² ⊕ ''y''² = ''a''}} for various values of ''a''. Negative {{math|''a''}} in green, positive {{math|''a''}} in blue, {{math|1=''a'' = ±1}} in red, {{math|1=''a'' = ∞}} in black.]]

The lemniscate functions satisfy a [[Pythagorean trigonometric identity|Pythagorean]]-like identity:
:<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} 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>

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

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

Defining a [[List of trigonometric identities#Angle sum and difference identities|tangent-sum]] operator as <math>a \oplus b \mathrel{:=} \tan(\arctan a + \arctan b) = \frac{a+b}{1-ab},</math> gives:

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

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

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

=== Derivatives and integrals ===

The derivatives are as follows:

:<math>\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cl} z = \operatorname{cl'}z
&= -\bigl(1 + \operatorname{cl^2} z\bigr)\operatorname{sl}z=-\frac{2\operatorname{sl}z}{\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}z=\frac{2\operatorname{cl}z}{\operatorname{cl}^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}}\,z&=-2\,\tilde{\operatorname{sl}}\,z\,\operatorname{cl}z-\frac{\tilde{\operatorname{sl}}\,z}{\operatorname{cl}z}\\

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

\end{align}</math>

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}z = -2\operatorname{cl^3}z </math>

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

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

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

=== Argument sum and multiple identities ===

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

The derivative and Pythagorean-like identities can be used to rework the identity used by Fagano in terms of {{math|sl}} and {{math|cl}}. Defining a [[List of trigonometric identities#Angle sum and difference identities|tangent-sum]] operator <math>a \oplus b \mathrel{:=} \tan(\arctan a + \arctan b)</math> and tangent-difference operator <math>a \ominus b \mathrel{:=} a \oplus (-b),</math> the argument sum and difference identities can be expressed as:<ref>{{harvp|Euler|1761}} [https://archive.org/details/novicommentariia06impe/page/79/ §44 p. 79], §47 pp. 80–81</ref>

:<math>\begin{aligned}
\operatorname{cl}(u+v)
&= \operatorname{cl}u\,\operatorname{cl}v \ominus \operatorname{sl}u\, \operatorname{sl}v
= \frac{\operatorname{cl}u\, \operatorname{cl}v - \operatorname{sl}u\, \operatorname{sl}v}
{1 + \operatorname{sl}u\, \operatorname{cl}u\, \operatorname{sl}v\, \operatorname{cl}v} \\[2mu]
\operatorname{cl}(u-v)
&= \operatorname{cl}u\,\operatorname{cl}v \oplus \operatorname{sl}u\, \operatorname{sl}v \\[2mu]
\operatorname{sl}(u+v)
&= \operatorname{sl}u\,\operatorname{cl}v \oplus \operatorname{cl}u\,\operatorname{sl}v
= \frac{\operatorname{sl}u\, \operatorname{cl}v + \operatorname{cl}u\, \operatorname{sl}v}
{1 - \operatorname{sl}u\, \operatorname{cl}u\, \operatorname{sl}v\, \operatorname{cl}v} \\[2mu]
\operatorname{sl}(u-v)
&= \operatorname{sl}u\,\operatorname{cl}v \ominus \operatorname{cl}u\,\operatorname{sl}v
\end{aligned}</math>

These resemble their [[List of trigonometric identities#Angle sum and difference identities|trigonometric analogs]]:

:<math>\begin{aligned}
\cos(u \pm v) &= \cos u\,\cos v \mp \sin u\,\sin v \\[6mu]
\sin(u \pm v) &= \sin u\,\cos v \pm \cos u\,\sin v
\end{aligned}</math>

In particular, to compute the complex-valued functions in real components,

:<math>\begin{aligned}
\operatorname{cl}(x + iy)
&= \frac{\operatorname{cl}x - i \operatorname{sl}x\, \operatorname{sl}y\, \operatorname{cl}y}
{\operatorname{cl}y + i \operatorname{sl}x\, \operatorname{cl}x\, \operatorname{sl}y} \\[4mu]
&= \frac{\operatorname{cl}x\,\operatorname{cl}y\left(1 - \operatorname{sl}^2x\,\operatorname{sl}^2y\right)}{\operatorname{cl}^2y + \operatorname{sl}^2x\,\operatorname{cl}^2x\,\operatorname{sl}^2y}
- i \frac{\operatorname{sl}x\,\operatorname{sl}y\left(\operatorname{cl}^2x + \operatorname{cl}^2y\right)}{\operatorname{cl}^2y + \operatorname{sl}^2x\,\operatorname{cl}^2x\,\operatorname{sl}^2y}
\\[12mu]
\operatorname{sl}(x + iy)
&= \frac{\operatorname{sl}x + i \operatorname{cl}x\, \operatorname{sl}y\, \operatorname{cl}y}
{\operatorname{cl}y - i \operatorname{sl}x\, \operatorname{cl}x\, \operatorname{sl}y } \\[4mu]
&= \frac{\operatorname{sl}x\,\operatorname{cl}y\left(1 - \operatorname{cl}^2x\,\operatorname{sl}^2y\right)}{\operatorname{cl}^2y + \operatorname{sl}^2x\,\operatorname{cl}^2x\,\operatorname{sl}^2y}
+ i \frac{\operatorname{cl}x\,\operatorname{sl}y\left(\operatorname{sl}^2x + \operatorname{cl}^2y\right)}{\operatorname{cl}^2y + \operatorname{sl}^2x\,\operatorname{cl}^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>

Bisection formulas:

:<math>
\operatorname{cl}^2 \tfrac12x = \frac{1+\operatorname{cl}x \sqrt{1+\operatorname{sl}^2x}}{\sqrt{1+\operatorname{sl}^2x}+1}
</math>

:<math>
\operatorname{sl}^2 \tfrac12x = \frac{1-\operatorname{cl}x\sqrt{1+\operatorname{sl}^2x}}{\sqrt{1+\operatorname{sl}^2x}+1}
</math>

Duplication formulas:<ref name="euler1761-46">{{harvp|Euler|1761}} [https://archive.org/details/novicommentariia06impe/page/80 §46 p. 80]</ref>

:<math>
\operatorname{cl} 2x = \frac{-1+2\,\operatorname{cl}^2x + \operatorname{cl}^4x}{1+2\,\operatorname{cl}^2x - \operatorname{cl}^4x}
</math>

:<math>
\operatorname{sl} 2x = 2\,\operatorname{sl}x\,\operatorname{cl}x\frac{1+\operatorname{sl}^2x}{1+\operatorname{sl}^4x}
</math>

Triplication formulas:<ref name="euler1761-46" />

:<math>
\operatorname{cl} 3x = \frac{-3\,\operatorname{cl}x + 6\,\operatorname{cl}^5x + \operatorname{cl}^9x}{1+6\,\operatorname{cl}^4x - 3\,\operatorname{cl}^8x}
</math>

:<math>
\operatorname{sl} 3x = \frac{\color{red}{3}\,\color{black}{\operatorname{sl}x -\, }\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}3x</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===

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}z)</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>
:<math>\omega_5=\sqrt[4]{-13+6\sqrt{5}+2\sqrt{85-38\sqrt{5}}}</math>
:<math>\tilde{\omega}_5=\sqrt[4]{-13-6\sqrt{5}+2\sqrt{85+38\sqrt{5}}}</math><ref name="ReferenceA">The fourth root with the least positive [[Argument (complex analysis)|principal argument]] is chosen.</ref>
(an equivalent expression is given in the table below). Another example is<ref name="CH" />
:<math>\Lambda_{-1+2i}(x)=x^4-1+2i</math>
which is the minimal polynomial of <math>\omega_{-1+2i}</math> (and also of <math>\tilde{\omega}_{-1+2i}</math>) in <math>K[x].</math>

If <math>p</math> is prime and <math>\beta</math> is positive and odd,<ref>The restriction to positive and odd <math>\beta</math> can be dropped in <math>\operatorname{deg}\Lambda_\beta=\left|(\mathcal{O}/\beta\mathcal{O})^\times\right|</math>.</ref> then<ref>{{harvp|Cox|2013}} p. 142, Example 7.29(c)</ref>
:<math>\operatorname{deg}\Lambda_{\beta}=\beta^2\prod_{p|\beta}\left(1-\frac{1}{p}\right)\left(1-\frac{(-1)^{(p-1)/2}}{p}\right)</math>
which can be compared to the cyclotomic analog
:<math>\operatorname{deg}\Phi_{k}=k\prod_{p|k}\left(1-\frac{1}{p}\right).</math>

===Specific values===

Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into {{math|''n''}} parts of equal length, using only basic arithmetic and square roots, if and only if {{math|''n''}} is of the form <math>n = 2^kp_1p_2\cdots p_m</math> where {{math|''k''}} is a non-negative [[integer]] and each {{math|''p''<sub>''i''</sub>}} (if any) is a distinct [[Fermat prime]].<ref>{{harvp|Rosen|1981}}</ref>

{| class="wikitable"
|- 
! <math>n</math> !! <math>\operatorname{cl}n\varpi</math> !! <math>\operatorname{sl}n\varpi</math> 
|-
| <math> 1</math>
| <math> -1</math>
| <math> 0</math>
|-
|<math> \tfrac{5}{6}</math>
|<math> -\sqrt[4]{2\sqrt{3}-3}</math>
|<math> \tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>
|-
|<math> \tfrac{3}{4}</math>
|<math> -\sqrt{\sqrt2-1}</math>
|<math> \sqrt{\sqrt2-1}</math>
|-
| <math> \tfrac{2}{3}</math>
| <math> -\tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>
| <math> \sqrt[4]{2\sqrt{3}-3}</math>
|-
| <math> \tfrac{1}{2}</math>
| <math> 0</math>
| <math> 1</math>
|-
| <math> \tfrac{1}{3}</math>
| <math> \tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>
| <math> \sqrt[4]{2\sqrt{3}-3}</math>
|-
| <math> \tfrac{1}{4}</math>
| <math> \sqrt{\sqrt2-1}</math>
| <math> \sqrt{\sqrt2-1}</math>
|-
|<math> \tfrac{1}{6}</math>
|<math> \sqrt[4]{2\sqrt{3}-3}</math>
|<math> \tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>
|}

== Relation to geometric shapes ==

===Arc length of Bernoulli's lemniscate===
[[File:The lemniscate sine and cosine related to the arclength of the lemniscate of Bernoulli.png|thumb|upright=1.8|The lemniscate sine and cosine relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.]]
[[File:The sine and cosine related to the arclength of the unit-diameter circle.png|thumb|upright=1.1|The trigonometric sine and cosine analogously relate the arc length of an arc of a unit-diameter circle to the distance of one endpoint from the origin.]]
<math>\mathcal{L}</math>, the [[lemniscate of Bernoulli]] with unit distance from its center to its furthest point (i.e. with unit "half-width"), is essential in the theory of the lemniscate elliptic functions. It can be [[Characterization (mathematics)|characterized]] in at least three ways:

'''Angular characterization:''' Given two points <math>A</math> and <math>B</math> which are unit distance apart, let <math>B'</math> be the [[Point reflection|reflection]] of <math>B</math> about <math>A</math>. Then <math>\mathcal{L}</math> is the [[Closure (topology)|closure]] of the locus of the points <math>P</math> such that <math>|APB-APB'|</math> is a [[right angle]].<ref>{{harvp|Eymard|Lafon|2004}} p. 200</ref>

'''Focal characterization:''' <math>\mathcal{L}</math> is the locus of points in the plane such that the product of their distances from the two focal points <math>F_1 = \bigl({-\tfrac1\sqrt2},0\bigr)</math> and <math>F_2 = \bigl(\tfrac1\sqrt2,0\bigr)</math> is the constant <math>\tfrac12</math>. 

'''Explicit coordinate characterization:''' <math>\mathcal{L}</math> is a [[quartic plane curve|quartic curve]] satisfying the [[polar coordinate system|polar]] equation <math>r^2 = \cos 2\theta</math> or the [[Cartesian coordinate system|Cartesian]] equation <math>\bigl(x^2+y^2\bigr){}^2=x^2-y^2.</math>

The [[perimeter]] of <math>\mathcal{L}</math> is <math>2\varpi</math>.<ref>And the area enclosed by <math>\mathcal{L}</math> is <math>1</math>, which stands in stark contrast to the unit circle ([[Squaring the circle|whose enclosed area is a non-constructible number]]).</ref>

The points on <math>\mathcal{L}</math> at distance <math>r</math> from the origin are the intersections of the circle <math>x^2+y^2=r^2</math> and the [[hyperbola]] <math>x^2-y^2=r^4</math>. The intersection in the positive quadrant has Cartesian coordinates:
:<math>\big(x(r), y(r)\big) = \biggl(\!\sqrt{\tfrac12r^2\bigl(1 + r^2\bigr)},\, \sqrt{\tfrac12r^2\bigl(1 - r^2\bigr)}\,\biggr).</math>

Using this [[Parametric equation|parametrization]] with <math>r \in [0, 1]</math> for a quarter of <math>\mathcal{L}</math>, the [[arc length]] from the origin to a point <math>\big(x(r), y(r)\big)</math> is:<ref>{{harvp|Euler|1761}}; {{harvp|Siegel|1969}}. {{harvp|Prasolov|Solovyev|1997}} use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same.</ref>
:<math>\begin{aligned}
&\int_0^r \sqrt{x'(t)^2 + y'(t)^2} \mathop{\mathrm{d}t} \\
& \quad {}= \int_0^r \sqrt{\frac{(1+2t^2)^2}{2(1+t^2)} + \frac{(1-2t^2)^2}{2(1-t^2)}} \mathop{\mathrm{d}t} \\[6mu]
& \quad {}= \int_0^r \frac{\mathrm{d}t}{\sqrt{1-t^4}} \\[6mu]
& \quad {}= \operatorname{arcsl} r.
\end{aligned}</math>

Likewise, the arc length from <math>(1,0)</math> to <math>\big(x(r), y(r)\big)</math> is:
:<math>\begin{aligned}
&\int_r^1 \sqrt{x'(t)^2 + y'(t)^2} \mathop{\mathrm{d}t} \\
& \quad {}= \int_r^1 \frac{\mathrm{d}t}{\sqrt{1-t^4}} \\[6mu]
& \quad {}= \operatorname{arccl} r = \tfrac12\varpi - \operatorname{arcsl} r.
\end{aligned}</math>

Or in the inverse direction, the lemniscate sine and cosine functions give the distance from the origin as functions of arc length from the origin and the point <math>(1,0)</math>, respectively.

Analogously, the circular sine and cosine functions relate the chord length to the arc length for the unit diameter circle with polar equation <math>r = \cos \theta</math> or Cartesian equation <math>x^2 + y^2 = x,</math> using the same argument above but with the parametrization:
:<math>\big(x(r), y(r)\big) = \biggl(r^2,\, \sqrt{r^2\bigl(1-r^2\bigr)}\,\biggr).</math>

Alternatively, just as the [[unit circle]] <math>x^2+y^2=1</math> is parametrized in terms of the arc length <math>s</math> from the point <math>(1,0)</math> by

:<math>(x(s),y(s))=(\cos s,\sin s),</math>

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

The notation <math>\tilde{\operatorname{cl}},\,\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>

:<math>\int_0^z \frac{\mathrm{d}t}{\sqrt{1 - t^4}} = 2 \int_0^u \frac{\mathrm{d}t}{\sqrt{1 - t^4}}, \quad \text{if }
z = \frac{2u\sqrt{1 - u^4}}{1 + u^4} \text{ and } 0\le u\le\sqrt{\sqrt{2}-1}.</math>

[[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 {{math|''n''}} sections of equal arc length using only [[straightedge and compass construction|straightedge and compass]] if and only if {{math|''n''}} is of the form <math>n = 2^kp_1p_2\cdots p_m</math> where {{math|''k''}} is a non-negative [[integer]] and each {{math|''p''<sub>''i''</sub>}} (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 {{math|''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 {{math|''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 ===
[[File:Rectangular elastica and lemniscatic sine.png|thumb|upright=1.3|The lemniscate sine relates the arc length to the x coordinate in the rectangular elastica.]]
The inverse lemniscate sine also describes the arc length {{math|''s''}} relative to the {{math|''x''}} coordinate of the rectangular [[Elastica theory|elastica]].<ref>{{harvp|Euler|1786}}; {{harvp|Sridharan|2004}}; {{harvp|Levien|2008}}</ref> This curve has {{math|''y''}} coordinate and arc length:
:<math>y = \int_x^1 \frac{t^2\mathop{\mathrm{d}t}}{\sqrt{1 - t^4}},\quad s = \operatorname{arcsl} x = \int_0^x \frac{\mathrm{d}t}{\sqrt{1 - t^4}}</math>

The rectangular elastica solves a problem posed by [[Jacob Bernoulli]], in 1691, to describe the shape of an idealized flexible rod fixed in a vertical orientation at the bottom end, and pulled down by a weight from the far end until it has been bent horizontal. Bernoulli's proposed solution established [[Euler–Bernoulli beam theory]], further developed by Euler in the 18th century.

===Elliptic characterization===
[[File:The lemniscate elliptic functions and an ellipse.jpg|thumb|The lemniscate elliptic functions and an ellipse]]
Let <math>C</math> be a point on the ellipse <math>x^2+2y^2=1</math> in the first quadrant and let <math>D</math> be the projection of <math>C</math> on the unit circle <math>x^2+y^2=1</math>. The distance <math>r</math> between the origin <math>A</math> and the point <math>C</math> is a function of <math>\varphi</math> (the angle <math>BAC</math> where <math>B=(1,0)</math>; equivalently the length of the circular arc <math>BD</math>). The parameter <math>u</math> is given by
:<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}u=\overline{AF}, \quad \operatorname{sl}u=\overline{DE},</math>
:<math>\tilde{\operatorname{cl}}\, u=\overline{AF}\overline{AC}, \quad \tilde{\operatorname{sl}}\, u=\overline{AF}\overline{FC}.</math>

== Series Identities ==

=== Power series ===

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}z=\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>
:<math>a_1=1,\, \forall n\in\mathbb{N}_0:\,a_{n+2}=-\frac{2}{(n+1)(n+2)}\sum_{i+j+k=n}a_ia_ja_k</math>

where <math>i+j+k=n</math> stands for all three-term [[Composition (combinatorics)|compositions]] of <math>n</math>. For example, to evaluate <math>a_{13}</math>, it can be seen that there are only six compositions of <math>13-2=11</math> that give a nonzero contribution to the sum: <math>11=9+1+1=1+9+1=1+1+9</math> and <math>11=5+5+1=5+1+5=1+5+5</math>, so
:<math>a_{13}=-\tfrac{2}{12\cdot 13}(a_9a_1a_1+a_1a_9a_1+a_1a_1a_9+a_5a_5a_1+a_5a_1a_5+a_1a_5a_5)=-\tfrac{11}{15600}.</math>

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}z=\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>
if <math>n</math> is odd.

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>
:<math>t_{n+2}=3 t_{n+1}+3 \sum_{j=0}^n \binom{2n+2}{2j+2} t_{n-j} \sum_{k=0}^j \binom{2j+1}{2k+1} t_k t_{j-k},\quad t_0=1,\, t_1=3.</math>

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}{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}}\,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>
:<math>q_{n+2}=\tfrac{3}{2} q_{n+1}+6 \sum_{j=0}^n \binom{2n+2}{2j+2} q_{n-j} \sum_{k=0}^j \binom{2j+1}{2k+1} q_k q_{j-k},\quad q_0=1, \,q_1=\tfrac{3}{2}.</math>

===Ramanujan's cos/cosh identity===
Ramanujan's famous cos/cosh identity states that if
:<math>R(s)=\frac{\pi}{\varpi\sqrt{2}}\sum_{n\in\mathbb{Z}}\frac{\cos (2n\pi s/\varpi)}{\cosh n\pi},</math>
then<ref name="OEIS_sl_tilde">{{cite web | url=https://oeis.org/A193543 | title=A193543 - Oeis }}</ref>
:<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}}\,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}t\, \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}t\operatorname{cl}t \, \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}x</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;
  | <math>a_0 \leftarrow 1;</math> <math>b_0 \leftarrow \tfrac{1}{\sqrt2};</math> <math>c_0 \leftarrow\sqrt{\tfrac12}</math>
  | '''for each''' <math>n\ge 1</math> '''do'''
  {{ubl |item_style=padding:0.2em 0 0 1.6em;
    | <math>a_n \leftarrow \tfrac12(a_{n-1}+b_{n-1});</math> <math>b_n \leftarrow \sqrt{a_{n-1}b_{n-1}};</math> <math>c_n \leftarrow \tfrac12(a_{n-1}-b_{n-1})</math>
    | '''if''' <math>c_n < \textrm{tolerance}</math> '''then'''
    {{ubl |item_style=padding:0.2em 0 0 1.6em;
      | <math>N \leftarrow n;</math> [[Control flow#Early exit from loops|'''break''']]
    }}
  }}
  | <math>\phi_N \leftarrow 2^N a_N \sqrt2x</math>
  | '''for each''' {{math|''n''}} from {{math|''N''}} to {{math|0}} '''do'''
  {{ubl |item_style=padding:0.2em 0 0 1.6em;
    | <math>\phi_{n-1} \leftarrow \tfrac12\left(\phi_n + {\arcsin}{\left(\frac{c_n}{a_n}\sin \phi_n\right)}\right)</math>
  }}
  | '''return''' <math>\frac{\sin \phi_0}{\sqrt{2-\sin^2\phi_0}}</math>
}}

This is effectively using the arithmetic-geometric mean and is based on [[Landen's transformation]]s.<ref>{{harvp|Carlson|2010}} [https://dlmf.nist.gov/19.8 §19.8]</ref>
| fontsize = 90%
| qalign = left
}}

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}\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}\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

:<math>\begin{aligned}
\theta_1(x,e^{-\pi})&=\sum_{n\in\mathbb{Z}}(-1)^{n+1}e^{-\pi (n+1/2+x/\pi)^2}=\sum_{n\in\mathbb{Z}} (-1)^n e^{-\pi (n+1/2)^2}\sin ((2n+1)x),\\
\theta_2(x,e^{-\pi})&=\sum_{n\in\mathbb{Z}}(-1)^n e^{-\pi (n+x/\pi)^2}=\sum_{n\in\mathbb{Z}} e^{-\pi (n+1/2)^2}\cos ((2n+1)x),\\
\theta_3(x,e^{-\pi})&=\sum_{n\in\mathbb{Z}}e^{-\pi (n+x/\pi)^2}=\sum_{n\in\mathbb{Z}} e^{-\pi n^2}\cos 2nx,\\
\theta_4(x,e^{-\pi})&=\sum_{n\in\mathbb{Z}}e^{-\pi (n+1/2+x/\pi)^2}=\sum_{n\in\mathbb{Z}} (-1)^n e^{-\pi n^2}\cos 2nx\end{aligned}</math>

are the [[Theta function|Jacobi theta functions]].<ref>{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.2.E7 §22.2.E7]</ref>

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}}</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}}\,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}}\,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>

:<math>\mathrm{sl}\Bigl(\frac\varpi\pi x\Bigr) = 2e^{-\pi/4}\sin x\prod_{n = 1}^{\infty} \frac{1-2e^{-2n\pi}\cos 2x+e^{-4n\pi}}{1+2e^{-(2n-1)\pi}\cos 2x+e^{-(4n-2)\pi}},\quad x\in\mathbb{C}</math>
:<math>\mathrm{cl}\Bigl(\frac\varpi\pi x\Bigr) = 2e^{-\pi/4}\cos x\prod_{n = 1}^{\infty} \frac{1+2e^{-2n\pi}\cos 2x+e^{-4n\pi}}{1-2e^{-(2n-1)\pi}\cos 2x+e^{-(4n-2)\pi}},\quad x\in\mathbb{C}</math>

A similar formula involving the <math>\operatorname{sn}</math> function can be given.<ref>{{harvp|Borwein|Borwein|1987}}</ref>

=== The lemniscate functions as a ratio of entire functions ===
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}z=\frac{M(z)}{N(z)}</math>

where

:<math>M(z)=z\prod_{\alpha}\left(1-\frac{z^4}{\alpha^4}\right),\quad N(z)=\prod_{\beta}\left(1-\frac{z^4}{\beta^4}\right).</math>

Here, <math>\alpha</math> and <math>\beta</math> denote, respectively, the zeros and poles of {{math|sl}} which are in the quadrant <math>\operatorname{Re}z>0,\operatorname{Im}z\ge 0</math>. A proof can be found in.<ref name="EL227"/><ref>{{Cite book |last=Cartan |first=H. |title=Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes|publisher=Hermann |year=1961|language=French |pages=160–164}}</ref> Importantly, the infinite products converge to the same value for all possible orders in which their terms can be multiplied, as a consequence of [[uniform convergence]].<ref>More precisely, suppose <math>\{a_n\}</math> is a sequence of bounded complex functions on a set <math>S</math>, such that <math display="inline">\sum\left|a_n(z)\right|</math> converges uniformly on <math>S</math>. If <math>\{n_1,n_2,n_3,\ldots\}</math> is any [[permutation]] of <math>\{1,2,3,\ldots\}</math>, then <math display="inline">\prod_{n=1}^\infty (1+a_n(z))=\prod_{k=1}^\infty (1+a_{n_k}(z))</math> for all <math>z\in S</math>. The theorem in question then follows from the fact that there exists a [[bijection]] between the natural numbers and <math>\alpha</math>'s (resp. <math>\beta</math>'s).</ref>
{{Collapse top|title=Proof of the infinite product for the lemniscate sine}}
'''Proof by logarithmic differentiation'''

It can be easily seen (using uniform and [[Absolute convergence|absolute]] convergence arguments to justify [[Interchange of limiting operations|interchanging of limiting operations]]) that
:<math>\frac{M'(z)}{M(z)}=-\sum_{n=0}^\infty 2^{4n}\mathrm{H}_{4n}\frac{z^{4n-1}}{(4n)!},\quad \left|z\right|<\varpi</math>
(where <math>\mathrm{H}_n</math> are the Hurwitz numbers defined in [[#Hurwitz numbers|Lemniscate elliptic functions § Hurwitz numbers]]) and
:<math>\frac{N'(z)}{N(z)}=(1+i)\frac{M'((1+i)z)}{M((1+i)z)}-\frac{M'(z)}{M(z)}.</math>
Therefore
:<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}z}=-\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}z}=-\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}z}=\frac{M'(z)}{M(z)}-\frac{N'(z)}{N(z)},\quad \left|z\right|<\frac{\varpi}{\sqrt{2}}.</math>
Therefore
:<math>\operatorname{sl}z=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}z}{z}=1</math>
gives <math>C=1</math> which completes the proof. <math>\blacksquare</math>

'''Proof by Liouville's theorem'''

Let
:<math>f(z)=\frac{M(z)}{N(z)}=\frac{(1+i)M(z)^2}{M((1+i)z)},</math>
with patches at removable singularities.
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}z}{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}z=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
[[File:The M function in the complex plane.png|thumb|The <math>M</math> function in the complex plane. The complex argument is represented by varying hue.]]
[[File:The N function in the complex plane.png|thumb|The <math>N</math> function in the complex plane. The complex argument is represented by varying hue.]]
:<math>N(z)=\frac{M((1+i)z)}{(1+i)M(z)},\quad z\notin \varpi\mathbb{Z}[i]</math>
and
:<math>N(2z)=M(z)^4+N(z)^4.</math>

Thanks to a certain theorem<ref>More precisely, if for each <math>k</math>, <math display="inline">\lim_{n\to\infty} a_k(n)</math> exists and there is a convergent series <math display="inline">\sum_{k=1}^\infty M_k</math> of nonnegative real numbers such that <math>\left|a_k(n)\right|\le M_k</math> for all <math>n\in\mathbb{N}</math> and <math>1\le k\le n</math>, then
:<math>\lim_{n\to\infty}\sum_{k=1}^n a_k(n)=\sum_{k=1}^\infty \lim_{n\to\infty}a_k(n).</math></ref> on splitting limits, we are allowed to multiply out the infinite products and collect like powers of <math>z</math>. Doing so gives the following power series expansions that are convergent everywhere in the complex plane:<ref>Alternatively, it can be inferred that these expansions exist just from the analyticity of <math>M</math> and <math>N</math>. However, establishing the connection to "multiplying out and collecting like powers" reveals identities between sums of reciprocals and the coefficients of the power series, like <math display="inline">\sum_{\alpha}\frac{1}{\alpha^4}=-\,\text{the coefficient of}\,z^5</math> in the <math>M</math> series, and infinitely many others.</ref><ref>{{Cite book |last1=Gauss |first1=C. F. |url=https://gdz.sub.uni-goettingen.de/id/PPN235999628 |title=Werke (Band III) |publisher=Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen |year=1866 |language=Latin, German}} p. 405; there's an error on the page: the coefficient of <math>\varphi^{17}</math> should be <math>\tfrac{107}{7\,410\,154\,752\,000}</math>, not <math>\tfrac{107}{207\,484\,333\,056\,000}</math>.</ref><ref>If <math display="inline">M(z)=\sum_{n=0}^\infty a_nz^{n+1}</math>, then the coefficients <math>a_n</math> are given by the recurrence <math display="inline">a_{n+1}=-\frac{1}{n+1}\sum_{k=0}^n 2^{n-k+1} a_k \frac{\mathrm{H}_{n-k+1}}{(n-k+1)!}</math> with <math>a_0=1</math> where <math>\mathrm{H}_n</math> are the Hurwitz numbers defined in [[#Hurwitz numbers|Lemniscate elliptic functions § Hurwitz numbers]].</ref><ref>The power series expansions of <math>M</math> and <math>N</math> are useful for finding a <math>\beta</math>-division polynomial for the <math>\beta</math>-division of the lemniscate <math>\mathcal{L}</math> (where <math>\beta=m+ni</math> where <math>m,n\in\mathbb{Z}</math> such that <math>m+n</math> is odd). For example, suppose we want to find a <math>3</math>-division polynomial. Given that
:<math>M(3z)=d_9M(z)^9+d_5M(z)^5N(z)^4+d_1 M(z)N(z)^8</math>
for some constants <math>d_1,d_5,d_9</math>, from
:<math>3z-2\frac{(3z)^5}{5!}-36\frac{(3z)^9}{9!}+\operatorname{O}(z^{13})=d_9x^9+d_5x^5y^4+d_1xy^8,</math>
where
:<math>x=z-2\frac{z^5}{5!}-36\frac{z^9}{9!}+\operatorname{O}(z^{13}),\quad y=1+2\frac{z^4}{4!}-4\frac{z^8}{8!}+\operatorname{O}(z^{12}),</math>
we have
:<math>\{d_1,d_5,d_9\}=\{3,-6,-1\}.</math>
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
:<math>\begin{align}a_1(\beta)&=1,\\
a_5(\beta)&=\frac{\beta^4-\beta\overline{\beta}}{12},\\
a_9(\beta)&=\frac{-\beta^8-70\beta^5\overline{\beta}+336\beta^4+35\beta^2\overline{\beta}^2-300\beta\overline{\beta}}{10080}\end{align}</math>
and so on.</ref>

:<math>M(z)=z-2\frac{z^5}{5!}-36\frac{z^9}{9!}+552\frac{z^{13}}{13!}+\cdots,\quad z\in\mathbb{C}</math>
:<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}z=\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>

:<math>S(z)=1-\frac{z^2}{2!}-\frac{z^4}{4!}-3\frac{z^6}{6!}+17\frac{z^8}{8!}-9\frac{z^{10}}{10!}+111\frac{z^{12}}{12!}+\cdots,\quad z\in\mathbb{C}</math>
:<math>T(z)=1+\frac{z^2}{2!}-\frac{z^4}{4!}+3\frac{z^6}{6!}+17\frac{z^8}{8!}+9\frac{z^{10}}{10!}+111\frac{z^{12}}{12!}+\cdots,\quad z\in\mathbb{C}.</math>

Furthermore, the identities
:<math>M(2z)=2 M(z) N(z) S(z) T(z),</math>
:<math>S(2z)=S(z)^4-2M(z)^4,</math>
:<math>T(2z)=T(z)^4-2M(z)^4</math>
and the Pythagorean-like identities
:<math>M(z)^2+S(z)^2=N(z)^2,</math>
:<math>M(z)^2+N(z)^2=T(z)^2</math>
hold for all <math>z\in\mathbb{C}</math>.

The quasi-addition formulas
:<math>M(z+w)M(z-w)=M(z)^2N(w)^2-N(z)^2M(w)^2,</math>
:<math>N(z+w)N(z-w)=N(z)^2N(w)^2+M(z)^2M(w)^2</math>
(where <math>z,w\in\mathbb{C}</math>) imply further multiplication formulas for <math>M</math> and <math>N</math> by recursion.<ref>For example, by the quasi-addition formulas, the duplication formulas and the Pythagorean-like identities, we have
:<math>M(3z)=-M(z)^9-6M(z)^5N(z)^4+3M(z)N(z)^8,</math>
:<math>N(3z)=N(z)^9+6M(z)^4N(z)^5-3M(z)^8N(z),</math>
so
:<math>\operatorname{sl}3z=\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}3z=\frac{-\operatorname{sl}^9z-6\operatorname{sl}^5z+3\operatorname{sl}z}{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:
:<math>M(z)M''(z)=M'(z)^2-N(z)^2,</math>
:<math>N(z)N''(z)=N'(z)^2+M(z)^2</math>
where <math>z\in\mathbb{C}</math>. Both <math>M</math> and <math>N</math> satisfy the differential equation<ref>Gauss (1866), p. 408</ref>
:<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]].

An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see [[#Methods of computation|Lemniscate elliptic functions § Methods of computation]]); the relation between <math>M,N</math> and <math>\theta_1,\theta_3</math> is
:<math>M(z)=2^{-1/4}e^{\pi z^2/(2\varpi^2)}\sqrt{\frac{\pi}{\varpi}}\theta_1\left(\frac{\pi z}{\varpi},e^{-\pi}\right),</math>
:<math>N(z)=2^{-1/4}e^{\pi z^2/(2\varpi^2)}\sqrt{\frac{\pi}{\varpi}}\theta_3\left(\frac{\pi z}{\varpi},e^{-\pi}\right)</math>
where <math>z\in\mathbb{C}</math>.

== Relation to other functions ==

=== Relation to Weierstrass and Jacobi elliptic functions ===

The lemniscate functions are closely related to the [[Weierstrass elliptic function]] <math>\wp(z; 1, 0)</math> (the "lemniscatic case"), with invariants {{math|''g''<sub>2</sub> {{=}} 1}} and {{math|''g''<sub>3</sub> {{=}} 0}}. This lattice has fundamental periods <math>\omega_1 = \sqrt{2}\varpi,</math> and <math>\omega_2 = i\omega_1</math>. The associated constants of the Weierstrass function are <math>e_1=\tfrac12,\ e_2=0,\ e_3=-\tfrac12.</math>

The related case of a Weierstrass elliptic function with {{math|''g''<sub>2</sub> {{=}} ''a''}}, {{math|''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: {{math|''a'' > 0}} and {{math|''a'' < 0}}. The period [[parallelogram]] is either a [[square]] or a [[rhombus]]. The Weierstrass elliptic function <math>\wp (z;-1,0)</math> is called the "pseudolemniscatic case".<ref>{{harvp|Robinson|2019a}}</ref>

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 {{math|''g''<sub>2</sub>}} and {{math|''g''<sub>3</sub>}}. 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}z=-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 {{math|''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} 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} 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}}</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}}\,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 ===
{{Unreferenced section |date=August 2024}}

The lemniscate sine can be used for the computation of values of the [[modular lambda function]]:

:<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>

For example:

:<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)
\\[7mu]
&\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)
\end{aligned}</math>

==Inverse functions==

The inverse function of the lemniscate sine is the lemniscate arcsine, defined as<ref>{{harvp|Siegel|1969}}</ref>

:<math> \operatorname{arcsl} x = \int_0^x \frac{\mathrm dt}{\sqrt{1-t^4}}. </math>

It can also be represented by the [[hypergeometric function]]:

:<math>\operatorname{arcsl}x=x\,{}_2F_1\bigl(\tfrac12,\tfrac14;\tfrac54;x^4\bigr)</math>
which can be easily seen by using the [[binomial series]].

The inverse function of the lemniscate cosine is the lemniscate arccosine. This function is defined by following expression:

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

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

Furthermore there are the so called Hyperbolic lemniscate area functions:{{cn|date=September 2024}}

:<math> \operatorname{aslh}(x) = \int_{0}^{x} \frac{1}{\sqrt{y^4 + 1}} \mathrm{d}y = \tfrac{1}{2}F\left(2\arctan x; \tfrac{1}{\sqrt2}\right) </math>

:<math> \operatorname{aclh}(x) = \int_{x}^{\infty} \frac{1}{\sqrt{y^4 + 1}} \mathrm{d}y = \tfrac12 F\left(2\arccot x; \tfrac{1}{\sqrt2}\right) </math>

:<math> \operatorname{aclh}(x) = \frac{\varpi}{\sqrt{2}} - \operatorname{aslh}(x) </math>

:<math> \operatorname{aslh}(x) = \sqrt{2}\operatorname{arcsl}\left(x \Big/ \sqrt{\textstyle 1 + \sqrt{x^4 + 1}} \right) </math>

:<math> \operatorname{arcsl}(x) = \sqrt{2}\operatorname{aslh}\left(x \Big/ \sqrt{\textstyle 1 + \sqrt{1 - x^4}}\right) </math>

=== Expression using elliptic integrals ===

The lemniscate arcsine and the lemniscate arccosine can also be expressed by the Legendre-Form:

These functions can be displayed directly by using the incomplete [[elliptic integral]] of the first kind:{{cn|date=September 2024}}

:<math>\operatorname{arcsl} x = \frac{1}{\sqrt2}F\left({\arcsin}{\frac{\sqrt2x}{\sqrt{1+x^2}}};\frac{1}{\sqrt2}\right) </math>

:<math>\operatorname{arcsl} x = 2(\sqrt2-1)F\left({\arcsin}{\frac{(\sqrt2+1)x}{\sqrt{1+x^2}+1}};(\sqrt2-1)^2\right) </math>

The arc lengths of the lemniscate can also be expressed by only using the arc lengths of [[ellipse]]s (calculated by elliptic integrals of the second kind):{{cn|date=September 2024}}

:<math>\begin{aligned}
\operatorname{arcsl} x = {}&\frac{2+\sqrt2}{2}E\left({\arcsin}{\frac{(\sqrt2+1)x}{\sqrt{1+x^2}+1}};(\sqrt2-1)^2\right) \\[5mu]
&\ \ - E\left({\arcsin}{\frac{\sqrt2x}{\sqrt{1+x^2}}};\frac{1}{\sqrt2}\right) + \frac{x\sqrt{1-x^2}}{\sqrt2(1+x^2+\sqrt{1+x^2})}
\end{aligned}</math>

The lemniscate arccosine has this expression:{{cn|date=September 2024}}

:<math>\operatorname{arccl} x = \frac{1}{\sqrt2}F\left(\arccos x;\frac{1}{\sqrt2}\right) </math>

=== Use in integration ===

The lemniscate arcsine can be used to integrate many functions. Here is a list of important integrals (the [[Constant of integration|constants of integration]] are omitted):

:<math>\int\frac{1}{\sqrt{1-x^4}}\,\mathrm dx=\operatorname{arcsl} x</math>

:<math>\int\frac{1}{\sqrt{(x^2+1)(2x^2+1)}}\,\mathrm dx={\operatorname{arcsl}}{\frac{x}{\sqrt{x^2+1}}}</math>

:<math>\int\frac{1}{\sqrt{x^4+6x^2+1}}\,\mathrm dx={\operatorname{arcsl}}{\frac{\sqrt2x}{\sqrt{\sqrt{x^4+6x^2+1}+x^2+1}}}</math>

:<math>\int\frac{1}{\sqrt{x^4+1}}\,\mathrm dx={\sqrt2\operatorname{arcsl}}{\frac{x}{\sqrt{\sqrt{x^4+1}+1}}}</math>

:<math>\int\frac{1}{\sqrt[4]{(1-x^4)^3}}\,\mathrm dx={\sqrt2\operatorname{arcsl}}{\frac{x}{\sqrt{1+\sqrt{1-x^4}}}}</math>

:<math>\int\frac{1}{\sqrt[4]{(x^4+1)^3}}\,\mathrm dx={\operatorname{arcsl}}{\frac{x}{\sqrt[4]{x^4+1}}}</math>

:<math>\int\frac{1}{\sqrt[4]{(1-x^2)^3}}\,\mathrm dx={2\operatorname{arcsl}}{\frac{x}{1+\sqrt{1-x^2}}}</math>

:<math>\int\frac{1}{\sqrt[4]{(x^2+1)^3}}\,\mathrm dx={2\operatorname{arcsl}}{\frac{x}{\sqrt{x^2+1}+1}}</math>

:<math>\int\frac{1}{\sqrt[4]{(ax^2+bx+c)^3}}\,\mathrm dx={\frac{2\sqrt2}{\sqrt[4]{4a^2c-ab^2}}\operatorname{arcsl}}{\frac{2ax+b}{\sqrt{4a(ax^2+bx+c)}+\sqrt{4ac-b^2}}}</math>

:<math>\int\sqrt{\operatorname{sech} x}\,\mathrm dx={2\operatorname{arcsl}}\tanh \tfrac12x</math>

:<math>\int\sqrt{\sec x}\,\mathrm dx={2\operatorname{arcsl}}\tan \tfrac12x</math>

== Hyperbolic lemniscate functions ==

=== Fundamental information ===

[[File:The hyperbolic lemniscate sine and cosine functions of a real variable.png|thumb|upright=1.3|The hyperbolic lemniscate sine (red) and hyperbolic lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric tangent (pale dashed red).]]
[[File:Slh in the complex plane.png|thumb|The hyperbolic lemniscate sine in the complex plane. Dark areas represent zeros and bright areas represent poles. The complex argument is represented by varying hue.]]

For convenience, let <math>\sigma=\sqrt{2}\varpi</math>. <math>\sigma</math> is the "squircular" analog of <math>\pi</math> (see below). The decimal expansion of <math>\sigma</math> (i.e. <math>3.7081\ldots</math><ref>{{cite OEIS|A175576}}</ref>) appears in entry 34e of chapter 11 of Ramanujan's second notebook.<ref>{{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part II |publisher=Springer |year=1989 |isbn=978-1-4612-4530-8}} p. 96</ref>

The hyperbolic lemniscate sine ({{math|slh}}) and cosine ({{math|clh}}) can be defined as inverses of elliptic integrals as follows:

:<math>z \mathrel{\overset{*}{=}} \int_0^{\operatorname{slh} z} \frac{\mathrm{d}t}{\sqrt{1 + t^4}} = \int_{\operatorname{clh} z}^\infty \frac{\mathrm{d}t}{\sqrt{1 + t^4}} </math>

where in <math>(*)</math>, <math>z</math> is in the square with corners <math>\{\sigma/2, \sigma i/2,-\sigma/2,-\sigma i/2\}</math>. Beyond that square, the functions can be analytically continued to meromorphic functions in the whole complex plane.

The complete integral has the value:

:<math>\int_0^\infty \frac{\mathrm{d}t}{\sqrt{t^4 + 1}} = \tfrac14 \Beta\bigl(\tfrac14, \tfrac14\bigr) = \frac{\sigma}{2} = 1.85407\;46773\;01371\ldots</math>

Therefore, the two defined functions have following relation to each other:

:<math>\operatorname{slh} z = {\operatorname{clh}}{\Bigl(\frac{\sigma}{2} - z \Bigr)} </math>

The product of hyperbolic lemniscate sine and hyperbolic lemniscate cosine is equal to one:

:<math>\operatorname{slh}z\,\operatorname{clh}z = 1 </math>

The functions <math>\operatorname{slh}</math> and <math>\operatorname{clh}</math> have a square period lattice with fundamental periods <math>\{\sigma,\sigma i\}</math>.

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}^2 z)\operatorname{sl}z}{\sqrt2\operatorname{cl}z} </math>

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

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

:<math> \operatorname{slh}z = \frac{\operatorname{sn}(z;1/\sqrt2)}{\operatorname{cd}(z;1/\sqrt2)} </math>

:<math> \operatorname{clh}z = \frac{\operatorname{cd}(z;1/\sqrt2)}{\operatorname{sn}(z;1/\sqrt2)} </math>

The hyperbolic lemniscate sine has following imaginary relation to the lemniscate sine:

:<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>

This is analogous to the relationship between hyperbolic and trigonometric sine:

:<math>\sinh z
= -i \sin (iz)
= \frac{\sin\left(\sqrt[2]{-1}z\right) }{ \sqrt[2]{-1}}
</math>

=== Relation to quartic Fermat curve ===

==== Hyperbolic Lemniscate Tangent and Cotangent ====

This image shows the standardized superelliptic Fermat squircle curve of the fourth degree:

[[File:Superellipse chamfered square.svg|thumb|280x280px|Superellipse with the relation <math>x^4 + y^4 = 1</math>]]

In a quartic [[Fermat curve]] <math>x^4 + y^4 = 1</math> (sometimes called a [[squircle]]) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle <math>x^2 + y^2 = 1</math> (the quadratic Fermat curve). If the origin and a point on the curve are connected to each other by a line {{math|''L''}}, the hyperbolic lemniscate sine of twice the enclosed area between this line and the x-axis is the y-coordinate of the intersection of {{math|''L''}} with the line <math>x = 1</math>.<ref>{{harvp|Levin|2006}}; {{harvp|Robinson|2019b}}</ref> Just as <math>\pi</math> is the area enclosed by the circle <math>x^2+y^2=1</math>, the area enclosed by the squircle <math>x^4+y^4=1</math> is <math>\sigma</math>. Moreover,

:<math>M(1,1/\sqrt{2})=\frac{\pi}{\sigma}</math>

where <math>M</math> is the [[arithmetic–geometric mean]].

The hyperbolic lemniscate sine satisfies the argument addition identity:

:<math> \operatorname{slh}(a+b) = \frac{\operatorname{slh}a\operatorname{slh}'b + \operatorname{slh}b\operatorname{slh}'a}{1-\operatorname{slh}^2a\,\operatorname{slh}^2b} </math>

When <math>u</math> is real, the derivative and the original [[antiderivative]] of <math> \operatorname{slh} </math> and <math> \operatorname{clh} </math> can be expressed in this way:

:{|class = "wikitable"
|
<math> \frac{\mathrm{d}}{\mathrm{d}u}\operatorname{slh}(u) = \sqrt{1 + \operatorname{slh}(u)^4} </math>

<math> \frac{\mathrm{d}}{\mathrm{d}u}\operatorname{clh}(u) = -\sqrt{1 + \operatorname{clh}(u)^4} </math>

<math> \frac{\mathrm{d}}{\mathrm{d}u} \,\frac{1}{2} \operatorname{arsinh}\bigl[ \operatorname{slh}(u)^2 \bigr] = \operatorname{slh}(u) </math>

<math> \frac{\mathrm{d}}{\mathrm{d}u} -\,\frac{1}{2} \operatorname{arsinh}\bigl[ \operatorname{clh}(u)^2 \bigr] = \operatorname{clh}(u) </math>
|}

There are also the Hyperbolic lemniscate tangent and the Hyperbolic lemniscate coangent als further functions:

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

The functional designation sl stands for the lemniscatic sine and the designation cl stands for the lemniscatic cosine.
In addition, those relations to the [[Jacobi elliptic function]]s are valid:
:<math>\text{tlh}(u) = \frac{\text{sn}(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>
:<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"
|
<math> \frac{\mathrm{d}}{\mathrm{d}u}\operatorname{tlh}(u) = \operatorname{ctlh}(u)^3 </math>

<math> \frac{\mathrm{d}}{\mathrm{d}u}\operatorname{ctlh}(u) = -\operatorname{tlh}(u)^3 </math>

<math> \int_{0}^{\varpi/\sqrt{2}} \operatorname{tlh}(u) \,\mathrm{d}u = \frac{\varpi}{2} </math>

<math> \int_{0}^{\varpi/\sqrt{2}} \operatorname{ctlh}(u) \,\mathrm{d}u = \frac{\varpi}{2} </math>
|}

==== Derivation of the Hyperbolic Lemniscate functions ====

[[File:Quartic Fermat curve.png|thumb|upright=1.3|With respect to the quartic Fermat curve <math>x^4 + y^4 = 1</math>, the hyperbolic lemniscate sine is analogous to the trigonometric tangent function. Unlike <math>\operatorname{slh}</math> and <math>\operatorname{clh}</math>, the functions <math>\sin_4</math> and <math>\cos_4</math> cannot be analytically extended to meromorphic functions in the whole complex plane.<ref>{{harvp|Levin|2006}} p. 515</ref>]]

The horizontal and vertical coordinates of this superellipse are dependent on twice the enclosed area w = 2A, so the following conditions must be met:
:<math>x(w)^4 + y(w)^4 = 1 </math>
:<math>\frac{\mathrm{d}}{\mathrm{d}w} x(w) = -y(w)^3 </math>
:<math>\frac{\mathrm{d}}{\mathrm{d}w} y(w) = x(w)^3 </math>
:<math>x(w = 0) = 1 </math>
:<math>y(w = 0) = 0 </math>

The solutions to this system of equations are as follows:
:<math>x(w) = \operatorname{cl}(\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}(\tfrac{ 1}{2}\sqrt{2}w)^2]^{-1/2} </math>
:<math>y(w) = \operatorname{sl}(\tfrac{1}{2}\sqrt{2}w) [\operatorname{cl}(\tfrac{1}{2}\sqrt{2}w)^2+1]^{1/2} [\operatorname{sl}(\tfrac{1}{2}\sqrt{2}w)^2+\operatorname{cl}(\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}(\tfrac{1}{2}\sqrt{2}w)^2+1]^{1/2}}{\operatorname{cl}(\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>
:<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 ====
{{Unreferenced section |date=August 2024}}

There is a black diagonal on the sketch shown on the right. The length of the segment that runs perpendicularly from the intersection of this black diagonal with the red vertical axis to the point (1|0) should be called s. And the length of the section of the black diagonal from the coordinate origin point to the point of intersection of this diagonal with the cyan curved line of the superellipse has the following value depending on the slh value:

:<math>D(s) = \sqrt{\biggl(\frac{1}{\sqrt[4]{s^4 + 1}}\biggr)^2 + \biggl(\frac{s}{\sqrt[4]{s^4 + 1}}\biggr)^2} = \frac{\sqrt{s^2 + 1}}{\sqrt[4]{s^4 + 1}} </math>

This connection is described by the [[Pythagorean theorem]].

An analogous unit circle results in the arctangent of the circle trigonometric with the described area allocation.

The following derivation applies to this:
:<math>\frac{\mathrm{d}}{\mathrm{d}s} \arctan(s) = \frac{1}{s^2 + 1} </math>

To determine the derivation of the areasinus lemniscatus hyperbolicus, the comparison of the infinitesimally small triangular areas for the same diagonal in the superellipse and the unit circle is set up below. Because the summation of the infinitesimally small triangular areas describes the area dimensions. In the case of the superellipse in the picture, half of the area concerned is shown in green. Because of the quadratic ratio of the areas to the lengths of triangles with the same infinitesimally small angle at the origin of the coordinates, the following formula applies:
:<math>\frac{\mathrm{d}}{\mathrm{d}s} \text{aslh}(s) = \biggl[\frac{\mathrm{d}}{\mathrm{d}s} \arctan(s)\biggr] D(s)^2 = \frac{1}{s^2 + 1}D(s)^2 = \frac{1}{s^2 + 1}\biggl(\frac{\sqrt{s^2 + 1}}{\sqrt[4]{s^4 + 1}}\biggr)^2 = \frac{1}{\sqrt{s^4 + 1}} </math>

==== Second proof: integral formation and area subtraction ====
{{Unreferenced section |date=August 2024}}

In the picture shown, the area tangent lemniscatus hyperbolicus assigns the height of the intersection of the diagonal and the curved line to twice the green area. The green area itself is created as the difference integral of the superellipse function from zero to the relevant height value minus the area of the adjacent triangle:
:<math>\text{atlh}(v) = 2\biggl(\int_{0}^{v} \sqrt[4]{1 - w^4} \mathrm{d}w\biggr) - v\sqrt[4]{1 - v^4} </math>
:<math>\frac{\mathrm{d}}{\mathrm{d}v} \text{atlh}(v) = 2\sqrt[4]{1 - v^4} - \biggl(\frac{\mathrm{d}}{\mathrm{d}v} v\sqrt[4]{1 - v^4}\biggr) = \frac{1}{(1 - v^4)^{3/4}} </math>

The following transformation applies:
:<math>\text{aslh}(x) = \text{atlh}\biggl(\frac{x}{\sqrt[4]{x^4 + 1}}\biggr) </math>

And so, according to the [[chain rule]], this derivation holds:
:<math>\frac{\mathrm{d}}{\mathrm{d}x} \text{aslh}(x) = \frac{\mathrm{d}}{\mathrm{d}x} \text{atlh}\biggl(\frac{x}{\sqrt[4]{x^4 + 1}}\biggr) = \biggl(\frac{\mathrm{d}}{\mathrm{d}x} \frac {x}{\sqrt[4]{x^4 + 1}}\biggr) \biggl[1 - \biggl(\frac{x}{\sqrt[4]{x^4 + 1}}\biggr)^4\biggr]^{-3/4} = </math>
:<math>= \frac{1}{(x^4 + 1)^{5/4}} \biggl[1 - \biggl(\frac{x}{\sqrt[4]{x^4 + 1}}\biggr)^4\biggr]^{-3/4} = \frac{1}{(x^4 + 1)^{5/4}} \biggl(\frac{1}{x^4 + 1}\biggr )^{-3/4} = \frac{1}{\sqrt{x^4 + 1}} </math>

=== Specific values ===
{{Unreferenced section |date=August 2024}}

This list shows the values of the '''Hyperbolic Lemniscate Sine''' accurately. Recall that,

:<math>\int_0^\infty \frac{\operatorname{d}t}{\sqrt{t^4 + 1}} = \tfrac14 \Beta\bigl(\tfrac14, \tfrac14\bigr) = \frac{\varpi}{\sqrt2} = \frac{\sigma}{2} = 1.85407\ldots</math>

whereas <math>\tfrac12 \Beta\bigl(\tfrac12, \tfrac12\bigr) = \tfrac{\pi}2,</math> so the values below such as  <math>{\operatorname{slh}}\bigl(\tfrac{\varpi}{2\sqrt{2}}\bigr) = {\operatorname{slh}}\bigl(\tfrac{\sigma}{4}\bigr) = 1 </math> are analogous to the trigonometric <math> {\sin}\bigl(\tfrac{\pi}2\bigr) = 1</math>.

:<math>
\operatorname{slh}\,\left(\frac{\varpi}{2\sqrt{2}}\right) = 1
</math>
:<math>
\operatorname{slh}\,\left(\frac{\varpi}{3\sqrt{2}}\right) = \frac{1}{\sqrt[4]{3}}\sqrt[4]{2\sqrt{3}-3}
</math>
:<math>
\operatorname{slh}\,\left(\frac{2\varpi}{3\sqrt{2}}\right) = \sqrt[4]{2\sqrt{3}+3}
</math>
:<math>
\operatorname{slh}\,\left(\frac{\varpi}{4\sqrt{2}}\right) = \frac{1}{\sqrt[4]{2}}(\sqrt{\sqrt{2}+1}-1)
</math>
:<math>
\operatorname{slh}\,\left(\frac{3\varpi}{4\sqrt{2}}\right) = \frac{1}{\sqrt[4]{2}}(\sqrt{\sqrt{2}+1}+1)
</math>
:<math>
\operatorname{slh}\,\left(\frac{\varpi}{5\sqrt{2}}\right) = \frac{1}{\sqrt[4]{8}}\sqrt{\sqrt{5}-1}\sqrt{\sqrt[4]{20}-\sqrt{\sqrt{5}+1}} = 2\sqrt[4]{\sqrt{5} - 2}\sqrt{\sin(\tfrac{1}{20}\pi)\sin(\tfrac{3}{20}\pi)}
</math>
:<math>
\operatorname{slh}\,\left(\frac{2\varpi}{5\sqrt{2}}\right) = \frac{1}{2\sqrt[4]{2}}(\sqrt{5}+1)\sqrt{\sqrt[4]{20}-\sqrt{\sqrt{5}+1}} = 2\sqrt[4]{\sqrt{5} + 2}\sqrt{\sin(\tfrac{1}{20}\pi)\sin(\tfrac{3}{20}\pi)}
</math>
:<math>
\operatorname{slh}\,\left(\frac{3\varpi}{5\sqrt{2}}\right) = \frac{1}{\sqrt[4]{8}}\sqrt{\sqrt{5}-1}\sqrt{\sqrt[4]{20}+\sqrt{\sqrt{5}+1}} = 2\sqrt[4]{\sqrt{5} - 2}\sqrt{\cos(\tfrac{1}{20}\pi)\cos(\tfrac{3}{20}\pi)}
</math>
:<math>
\operatorname{slh}\,\left(\frac{4\varpi}{5\sqrt{2}}\right) = \frac{1}{2\sqrt[4]{2}}(\sqrt{5}+1)\sqrt{\sqrt[4]{20}+\sqrt{\sqrt{5}+1}} = 2\sqrt[4]{\sqrt{5} + 2}\sqrt{\cos(\tfrac{1}{20}\pi)\cos(\tfrac{3}{20}\pi)}
</math>
:<math>
\operatorname{slh}\,\left(\frac{\varpi}{6\sqrt{2}}\right) = \frac{1}{2}(\sqrt{2\sqrt{3}+3}+1)(1-\sqrt[4]{2\sqrt{3}-3})
</math>
:<math>
\operatorname{slh}\,\left(\frac{5\varpi}{6\sqrt{2}}\right) = \frac{1}{2}(\sqrt{2\sqrt{3}+3}+1)(1+\sqrt[4]{2\sqrt{3}-3})
</math>

That table shows the most important values of the '''Hyperbolic Lemniscate Tangent and Cotangent''' functions:
{| class="wikitable"
!<math>z</math>
!<math> \operatorname{clh} z</math>
!<math> \operatorname{slh} z</math>
!<math> \operatorname{ctlh} z = \cos_{4} z</math>
!<math> \operatorname{tlh} z = \sin_{4} z</math>
|-
|<math> 0</math>
|<math> \infty</math>
|<math> 0</math>
|<math> 1</math>
|<math> 0</math>
|-
|<math> {\tfrac14}\sigma</math>
|<math> 1</math>
|<math> 1</math>
|<math> 1\big/\sqrt[4]{2}</math>
|<math> 1\big/\sqrt[4]{2}</math>
|-
|<math> {\tfrac12}\sigma</math>
|<math> 0</math>
|<math> \infty</math>
|<math> 0</math>
|<math> 1</math>
|-
|<math> {\tfrac34}\sigma</math>
|<math> -1</math>
|<math> -1</math>
|<math> -1\big/\sqrt[4]{2}</math>
|<math> 1\big/\sqrt[4]{2}</math>
|-
|<math> \sigma</math>
|<math> \infty</math>
|<math> 0</math>
|<math> -1</math>
|<math> 0</math>
|}

=== Combination and halving theorems ===
{{Unreferenced section |date=August 2024}}

Given the ''hyperbolic lemniscate tangent'' (<math> \operatorname{tlh} </math>) and ''hyperbolic lemniscate cotangent'' (<math> \operatorname{ctlh} </math>). Recall the ''hyperbolic lemniscate area functions'' from the section on inverse functions,

:<math> \operatorname{aslh}(x) = \int_{0}^{x} \frac{1}{\sqrt{y^4 + 1}} \mathrm{d}y </math> 
:<math> \operatorname{aclh}(x) = \int_{x}^{\infty} \frac{1}{\sqrt{y^4 + 1}} \mathrm{d}y </math>  

Then the following identities can be established,

:<math>\text{tlh}\bigl[\text{aslh}(x)\bigr] = \text{ctlh}\bigl[\text{aclh}(x)\bigr] = \frac{x}{\sqrt[4]{x^4 + 1}} </math>
:<math>\text{ctlh}\bigl[\text{aslh}(x)\bigr] = \text{tlh}\bigl[\text{aclh}(x)\bigr] = \frac{1}{\sqrt[4]{x^4 + 1}} </math>

hence the 4th power of <math> \operatorname{tlh} </math> and <math> \operatorname{ctlh} </math> for these arguments is equal to one,

:<math>\text{tlh}\bigl[\text{aslh}(x)\bigr]^4 + \text{ctlh}\bigl[\text{aslh}(x)\bigr]^4=1 </math>
:<math>\text{tlh}\bigl[\text{aclh}(x)\bigr]^4 + \text{ctlh}\bigl[\text{aclh}(x)\bigr]^4=1 </math>

so a 4th power version of the [[Pythagorean theorem]]. The bisection theorem of the hyperbolic sinus lemniscatus reads as follows:
:<math>\text{slh}\bigl[\tfrac{1}{2}\text{aslh}(x)\bigr] = \frac{\sqrt{2}x}{\sqrt{x^2 + 1 + \sqrt{x^4 + 1}} + \sqrt{\sqrt{x^4 + 1} - x^2 + 1}} </math>

This formula can be revealed as a combination of the following two formulas:
:<math>\mathrm{aslh}(x) = \sqrt{2}\,\text{arcsl}\bigl[x(\sqrt{x^4 + 1} + 1)^{-1/2}\bigr]</math>
:<math>\text{arcsl}(x) = \sqrt{2}\,\text{aslh}\bigl(\frac{\sqrt{2}x}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\bigr) </math>

In addition, the following formulas are valid for all real values <math>x \in \R</math>:
:<math>\text{slh}\bigl[\tfrac{1}{2}\text{aclh}(x)\bigr] = \sqrt{\sqrt{x^4 + 1} + x^2 - \sqrt{2}x\sqrt{\sqrt{x^4 + 1} + x^2}} = \bigl(\sqrt{x^4 + 1} - x^2 + 1\bigr) ^{-1/2}\bigl(\sqrt{\sqrt{x^4 + 1} + 1} - x\bigr) </math>
:<math>\text{clh}\bigl[\tfrac{1}{2}\text{aclh}(x)\bigr] = \sqrt{\sqrt{x^4 + 1} + x^2 + \sqrt{2}x\sqrt{\sqrt{x^4 + 1} + x^2}} = \bigl(\sqrt{x^4 + 1} - x^2 + 1\bigr)^ {-1/2}\bigl(\sqrt{\sqrt{x^4 + 1} + 1} + x\bigr) </math>

These identities follow from the last-mentioned formula:
:<math>\text{tlh}[\tfrac{1}{2}\text{aclh}(x)]^2 = \tfrac{1}{2}\sqrt{2-2\sqrt{2}\,x\sqrt{\sqrt{x^4+1}-x^2}} = \bigl(2x^2 + 2 + 2\sqrt{x^4 + 1}\bigr)^{-1 /2}\bigl(\sqrt{\sqrt{x^4 + 1} + 1} - x\bigr) </math>
:<math>\text{ctlh}[\tfrac{1}{2}\text{aclh}(x)]^2 = \tfrac{1}{2}\sqrt{2+2\sqrt{2}\,x\sqrt{\sqrt{x^4+1}-x^2}} = \bigl(2x^2 + 2 + 2\sqrt{x^4 + 1}\bigr)^{-1 /2}\bigl(\sqrt{\sqrt{x^4 + 1} + 1} + x\bigr) </math>

Hence, their 4th powers again equal one,

:<math>\text{tlh}\bigl[\tfrac{1}{2}\text{aclh}(x)\bigr]^4 + \text{ctlh}\bigl[\tfrac{1}{2}\text{aclh}(x)\bigr]^4=1 </math>

The following formulas for the lemniscatic sine and lemniscatic cosine are closely related:
:<math>\text{sl}[\tfrac{1}{2}\sqrt{2}\,\text{aclh}(x)] = \text{cl}[\tfrac{1}{2}\sqrt{2}\,\text{aslh}(x)] = \sqrt{\sqrt{x^4 + 1} - x^2} </math>
:<math>\text{sl}[\tfrac{1}{2}\sqrt{2}\,\text{aslh}(x)] = \text{cl}[\tfrac{1}{2}\sqrt{2}\,\text{aclh}(x)] = x\bigl(\sqrt{x^4 + 1} + 1\bigr)^{-1/2} </math>

=== Coordinate Transformations ===
{{Unreferenced section |date=August 2024}}

Analogous to the determination of the improper integral in the [[normal distribution|Gaussian bell curve function]], the coordinate transformation of a [[cylinder (geometry)#general cylinder|general cylinder]] can be used to calculate the integral from 0 to the positive infinity in the function <math> f(x)= \exp(-x^4) </math> integrated in relation to x. In the following, the proofs of both integrals are given in a parallel way of displaying.

This is the [[polar coordinates|cylindrical coordinate transformation]] in the Gaussian bell curve function:
:<math>\biggl[\int_{0}^{\infty} \exp(-x^2) \,\mathrm{d}x\biggr]^2 = \int_{0}^{\infty} \int_{0}^{\infty} \exp(-y^2-z^2) \,\mathrm{d}y \,\mathrm{d}z = </math>
:<math>= \int_{0}^{\pi/2} \int_{0}^{\infty} \det\begin{bmatrix} \partial/\partial r\,\,r\cos(\phi) & \partial/\partial \phi\,\,r\cos(\phi) \\ \partial/\partial r\,\,r\sin(\phi) & \partial/\partial \phi\,\, r\sin(\phi) \end{bmatrix} \exp\bigl\{-\bigl[r\cos(\phi)\bigr]^2-\bigl[r\sin(\phi)\bigr]^2\bigr\} \,\mathrm{d}r \,\mathrm{d}\phi = </math>
:<math>= \int_{0}^{\pi/2} \int_{0}^{\infty} r\exp(-r^2) \,\mathrm{d}r \,\mathrm{d}\phi = \int_{0}^{\pi/2} \frac{1}{2} \,\mathrm{d}\phi = \frac{\pi }{4} </math>
And this is the analogous coordinate transformation for the lemniscatory case:
:<math>\biggl[\int_{0}^{\infty} \exp(-x^4) \,\mathrm{d}x\biggr]^2 = \int_{0}^{\infty} \int_{0}^{\infty} \exp(-y^4-z^4) \,\mathrm{d}y \,\mathrm{d}z = </math>
:<math>= \int_{0}^{\varpi/\sqrt{2}} \int_{0}^{\infty} \det\begin{bmatrix} \partial/\partial r\,\,r\,\text{ctlh}(\phi) & \partial/\partial \phi\,\,r\,\text{ctlh}(\phi) \\ \partial/\partial r\,\,r\, \text{tlh}(\phi) & \partial/\partial \phi\,\,r\,\text{tlh}(\phi) \end{bmatrix} \exp\bigl\{-\bigl[r\,\text{ctlh}(\phi)\bigr]^4-\bigl[r\,\text{tlh}(\phi)\bigr]^4\bigr\} \,\mathrm{d}r \,\mathrm{d }\phi = </math>
:<math>= \int_{0}^{\varpi/\sqrt{2}} \int_{0}^{\infty} r\exp(-r^4) \,\mathrm{d}r \,\mathrm{d}\phi = \int_{0}^{\varpi/\sqrt{2}} \frac{\sqrt{\pi}}{4} \,\mathrm{d}\phi = \frac{\varpi\sqrt{\pi}}{4\sqrt{2}} </math>

In the last line of this elliptically analogous equation chain there is again the original Gauss bell curve integrated with the square function as the inner substitution according to the [[Chain rule]] of infinitesimal analytics (analysis).

In both cases, the determinant of the [[Jacobian matrix and determinant|Jacobi matrix]] is multiplied to the original function in the integration domain.

The resulting new functions in the integration area are then integrated according to the new parameters.

== Number theory ==

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===
The [[Bernoulli number]]s <math>\mathrm{B}_n</math> can be defined by

:<math>
\mathrm{B}_n
= \lim_{z\to 0}\frac{\mathrm d^n}{\mathrm dz^n}\frac{z}{e^z-1},\quad n\ge 0
</math>

and appear in

:<math>
\sum_{k\in\mathbb{Z}\setminus\{0\}}\frac{1}{k^{2n}}
= (-1)^{n-1}\mathrm{B}_{2n}\frac{(2\pi)^{2n}}{(2n)!}=2\zeta (2n),\quad n\ge 1
</math>

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>

:<math>
\mathrm{H}_n
= -\lim_{z\to 0}\frac{\mathrm d^n}{\mathrm dz^n}z\zeta (z;1/4,0),\quad n\ge 0
</math>

where <math>\zeta (\cdot;1/4,0)</math> is the [[Weierstrass functions|Weierstrass zeta function]] with lattice invariants <math>1/4</math> and <math>0</math>. They appear in

:<math>
\sum_{z\in\mathbb{Z}[i]\setminus\{0\}}\frac{1}{z^{4n}}
= \mathrm{H}_{4n}\frac{(2\varpi)^{4n}}{(4n)!}
= G_{4n}(i),\quad n\ge 1
</math>

where <math>\mathbb{Z}[i]</math> are the [[Gaussian integers]] and <math>G_{4n}</math> are the [[Eisenstein series]] of weight <math>4n</math>, and in

:<math>\displaystyle \begin{array}{ll}
\displaystyle\sum_{n=1}^\infty\dfrac{n^k}{e^{2\pi n}-1} = \begin{cases}
\dfrac{1}{24}-\dfrac{1}{8\pi}  & {\text{if}}\ k=1 \\
\dfrac{\mathrm{B}_{k+1}}{2k+2} & {\text{if}}\ k\equiv1\, (\mathrm{mod}\, 4)\ {\text{and}}\ k\ge 5 \\ 
\dfrac{\mathrm{B}_{k+1}}{2k+2}+\dfrac{\mathrm{H}_{k+1}}{2k+2}\left(\dfrac{\varpi}{\pi}\right)^{k+1} & {\text{if}}\ k\equiv 3\,(\mathrm{mod}\,4)\ {\text{and}}\ k\ge 3. \\
\end{cases}
\end{array}</math>

The Hurwitz numbers can also be determined as follows: <math>\mathrm{H}_4=1/10</math>,

:<math>
\mathrm{H}_{4n} 
= \frac{3}{(2n-3)(16n^2-1)}\sum_{k=1}^{n-1}\binom{4n}{4k}(4k-1)(4(n-k)-1)\mathrm{H}_{4k}\mathrm{H}_{4(n-k)},\quad n\ge 2
</math>

and <math>\mathrm{H}_n=0</math> if <math>n</math> is not a multiple of <math>4</math>.<ref>The Bernoulli numbers can be determined by an analogous recurrence: <math>\mathrm{B}_{2n}=-\frac{1}{2n+1}\sum_{k=1}^{n-1}\binom{2n}{2k}\mathrm{B}_{2k}\mathrm{B}_{2(n-k)}</math> where <math>n\ge 2</math> and <math>\mathrm{B}_2=1/6</math>.</ref> This yields<ref name="Arakawa"/>
:<math>\mathrm{H}_8=\frac{3}{10},\,\mathrm{H}_{12}=\frac{567}{130},\,\mathrm{H}_{16}=\frac{43\,659}{170},\,\ldots</math>

Also<ref>{{cite journal |last1=Katz |first1=Nicholas M. |date=1975 |title=The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers  |journal=Mathematische Annalen |volume=216 |issue=1 |pages=1–4|doi=10.1007/BF02547966 }} See eq. (9)</ref>
:<math>\operatorname{denom}\mathrm{H}_{4n}=\prod_{(p-1)|4n}p</math>
where <math>p\in\mathbb{P}</math> such that <math>p\not\equiv 3\,(\text{mod}\,4),</math>
just as
:<math>\operatorname{denom}\mathrm{B}_{2n}=\prod_{(p-1)|2n}p</math>
where <math>p\in\mathbb{P}</math> (by the [[von Staudt–Clausen theorem]]).

In fact, the von Staudt–Clausen theorem determines the [[fractional part]] of the Bernoulli numbers:

:<math>
\mathrm{B}_{2n}+\sum_{(p-1)|2n}\frac{1}{p}\in\mathbb{Z},\quad n\ge 1
</math>

{{OEIS|A000146}} where <math>p</math> is any prime, and an analogous theorem holds for the Hurwitz numbers: suppose that <math>a\in\mathbb{Z}</math> is odd, <math>b\in\mathbb{Z}</math> is even, <math>p</math> is a prime such that <math>p\equiv 1\,(\mathrm{mod}\,4)</math>, <math>p=a^2+b^2</math> (see [[Fermat's theorem on sums of two squares]]) and <math>a\equiv b+1\,(\mathrm{mod}\,4)</math>. Then for any given <math>p</math>, <math>2a=\nu (p)</math> is uniquely determined; equivalently <math>\nu (p)=p-\mathcal{N}_p</math> where <math>\mathcal{N}_p</math> is the number of solutions of the congruence <math>X^3-X\equiv Y^2\, (\operatorname{mod}p)</math> in variables <math>X,Y</math> that are non-negative integers.<ref>For more on the <math>\nu</math> function, see [[Lemniscate constant]].</ref> The Hurwitz theorem then determines the fractional part of the Hurwitz numbers:<ref name="Arakawa"/>

:<math>
\mathrm{H}_{4n}-\frac{1}{2}-\sum_{(p-1)|4n}\frac{\nu (p)^{4n/(p-1)}}{p}
\mathrel{\overset{\text{def}}{=}} \mathrm{G}_n\in\mathbb{Z},\quad n\ge 1.
</math>

The sequence of the integers <math>\mathrm{G}_n</math> starts with <math>0,-1,5,253,\ldots .</math><ref name="Arakawa"/>

Let <math>n\ge 2</math>. If <math>4n+1</math> is a prime, then <math>\mathrm{G}_n\equiv 1\,(\mathrm{mod}\,4)</math>. If <math>4n+1</math> is not a prime, then <math>\mathrm{G}_n\equiv 3\,(\mathrm{mod}\,4)</math>.<ref>{{Cite book |last1=Hurwitz |first1=Adolf |title=Mathematische Werke: Band II |language=German|publisher=Springer Basel AG |year=1963}} p. 370</ref>

Some authors instead define the Hurwitz numbers as <math>\mathrm{H}_n'=\mathrm{H}_{4n}</math>.

====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}z}
  &= \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
\end{align}</math>

Analogously, in terms of the Bernoulli numbers:

:<math>
\frac{1}{\sinh^2 z}
= \frac{1}{z^2}-\sum_{n=1}^\infty \frac{2^{2n}\mathrm{B}_{2n}}{2n}\frac{z^{2n-2}}{(2n-2)!},\quad
\left|z\right|<\pi.
</math>

===A quartic analog of the Legendre symbol===
Let <math>p</math> be a prime such that <math>p\equiv 1\,(\text{mod}\,4)</math>. A '''quartic residue''' (mod <math>p</math>) is any number congruent to the fourth power of an integer. Define <math>\left(\tfrac{a}{p}\right)_4</math>
to be <math>1</math> if <math>a</math> is a quartic residue (mod <math>p</math>) and define it to be <math>-1</math> if <math>a</math> is not a quartic residue (mod <math>p</math>).

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>
where <math>\left(\tfrac{\cdot}{\cdot}\right)</math> is the [[Legendre symbol]].

== World map projections ==

[[File:Peirce Quincuncial Projection 1879.jpg|thumb|upright=1.3|"The World on a Quincuncial Projection", from {{harvp|Peirce|1879}}.]]
The [[Peirce quincuncial projection]], designed by [[Charles Sanders Peirce]] of the [[United States Coast and Geodetic Survey|US Coast Survey]] in the 1870s, is a world [[map projection]] based on the inverse lemniscate sine of [[stereographic projection|stereographically projected]] points (treated as complex numbers).<ref>{{harvp|Peirce|1879}}. {{harvp|Guyou|1887}} and {{harvp|Adams|1925}} introduced [[Map projection#Aspect of the projection|transverse and oblique aspects]] of the same projection, respectively. Also see {{harvp|Lee|1976}}. These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice.</ref>

When lines of constant real or imaginary part are projected onto the complex plane via the hyperbolic lemniscate sine, and thence stereographically projected onto the sphere (see [[Riemann sphere]]), the resulting curves are [[spherical conic]]s, the spherical analog of planar [[ellipse]]s and [[hyperbola]]s.<ref>{{harvp|Adams|1925}}</ref> Thus the lemniscate functions (and more generally, the [[Jacobi elliptic functions]]) provide a parametrization for spherical conics.

A conformal map projection from the globe onto the 6 square faces of a [[cube]] can also be defined using the lemniscate functions.<ref>{{harvp|Adams|1925}}; {{harvp|Lee|1976}}.</ref> Because many [[partial differential equations]] can be effectively solved by conformal mapping, this map from sphere to cube is convenient for [[atmospheric model]]ing.<ref>{{harvp|Rančić|Purser|Mesinger|1996}}; {{harvp|McGregor|2005}}.</ref>

==See also==
* [[Elliptic function]]
** [[Abel elliptic functions]]
** [[Dixon elliptic functions]]
** [[Jacobi elliptic functions]]
** [[Weierstrass elliptic function]]
* [[Elliptic Gauss sum]]
* [[Lemniscate constant]]
* [[Peirce quincuncial projection]]
* [[Schwarz–Christoffel mapping]]

== Notes ==
{{Reflist|30em}}

== References ==
{{refbegin|30em}}
* {{wikicite
|ref={{harvid|Abel|1827–1828}}
|reference =
[[Niels Henrik Abel|Abel, Niels Henrik]] (1827–1828) "Recherches sur les fonctions elliptiques" [Research on elliptic functions] (in French). ''[[Crelle's Journal]]''.<br />[https://gdz.sub.uni-goettingen.de/id/PPN243919689_0002?tify={%22pages%22:%5B113%5D} Part 1]. 1827. '''2''' (2): 101–181. [[doi (identifier)|doi]]:[https://doi.org/10.1515%2Fcrll.1827.2.101 10.1515/crll.1827.2.101].<br />[https://gdz.sub.uni-goettingen.de/id/PPN243919689_0003?tify={%22pages%22:%5B166%5D} Part 2]. 1828. '''3''' (3): 160–190. [[doi (identifier)|doi]]:[https://doi.org/10.1515%2Fcrll.1828.3.160 10.1515/crll.1828.3.160].
}}
* {{cite book
  | last = Adams |first = Oscar S. | author-link = Oscar S. Adams
  | year = 1925
  | title = Elliptic Functions Applied to Conformal World Maps
  | id = Special Pub. No. 112
  | others = U.S. Coast and Geodetic Survey
  | publisher = US Government Printing Office
  | url = https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_112.pdf
  }}
* {{cite journal |last1=Ayoub |first1=Raymond |authorlink1=Raymond Ayoub |date=1984 |title=The Lemniscate and Fagnano's Contributions to Elliptic Integrals |journal=Archive for History of Exact Sciences |volume=29 |issue=2 |pages=131–149 |doi=10.1007/BF00348244 }}
* {{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part IV |publisher=Springer |year=1994 |edition=First |isbn=978-1-4612-6932-8 }}
* {{cite book |last1=Borwein |first1=Jonatham M. |authorlink1=Jonathan Borwein |last2=Borwein |first2=Peter B. |authorlink2=Peter Borwein |date= 1987 |chapter=2.7 The Landen Transformation |pages=60 |title=Pi and the AGM |publisher=Wiley-Interscience }}
* {{cite book |last1=Bottazzini |first1=Umberto |authorlink1=Umberto Bottazzini |last2=Gray |first2=Jeremy |authorlink2=Jeremy Gray |date=2013 |title=Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory |publisher=Springer |doi=10.1007/978-1-4614-5725-1 |isbn=978-1-4614-5724-4 }}
* {{cite book |last=Carlson |first=Billie C. |year=2010 |authorlink1=Bille C. Carlson |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor1-link=Frank W. J. Olver |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=19. Elliptic Integrals |chapter-url=https://dlmf.nist.gov/19 |title-link=Digital Library of Mathematical Functions }}
* {{cite journal |last=Cox |first=David Archibald |authorlink1=David A. Cox |date=1984 |title=The Arithmetic-Geometric Mean of Gauss |url=https://www.researchgate.net/publication/248675540 |journal=L'Enseignement Mathématique |volume=30 |issue=2 |pages=275–330 }}
* {{cite journal |last1=Cox |first1=David Archibald |last2=Shurman |first2=Jerry |date=2005 |title=Geometry and number theory on clovers |journal=The American Mathematical Monthly |volume=112 |issue=8 |pages=682–704 |doi=10.1080/00029890.2005.11920241 |url=https://people.reed.edu/~jerry/Clover/cloverexcerpt.pdf }}
* {{cite book |last1=Cox |first1=David Archibald |year=2012 |chapter=The Lemniscate |pages=463–514 |title=Galois Theory |publisher=Wiley |doi=10.1002/9781118218457.ch15 |isbn=978-1-118-07205-9 }}
* {{Cite book |last1=Cox |first1=David Archibald |title=Primes of the Form ''{{math|''x''<sup>2</sup> + ''ny''<sup>2</sup>}}'' |publisher=Wiley |year=2013 |edition=Second |url=https://dacox.people.amherst.edu/primes.html }}
* {{cite journal |last1=Cox |first1=David Archibald |first2=Trevor |last2=Hyde |year=2014 |title=The Galois theory of the lemniscate |journal=Journal of Number Theory |volume=135 |pages=43–59 |url=http://www-personal.umich.edu/~tghyde/Cox,%20Hyde%20--%20Galois%20theory%20on%20the%20lemniscate.pdf |doi=10.1016/j.jnt.2013.08.006 |arxiv=1208.2653 }}
* {{cite book |last1=Enneper |first1=Alfred |author-link1=Alfred Enneper |date=1890 |orig-date=1st ed. 1876 |chapter=Note III: Historische Notizen über geometrische Anwendungen elliptischer Integrale. |trans-chapter=Historical notes on geometric applications of elliptic integrals |pages=524–547 |title=Elliptische Functionen, Theorie und Geschichte |language=de |publisher=Nebert |chapter-url=https://archive.org/details/elliptischefunct00enneuoft/page/524/ }}
* {{cite journal |authorlink1=Leonhard Euler |last1=Euler |first1=Leonhard |year=1761 |title=Observationes de comparatione arcuum curvarum irrectificibilium |trans-title=Observations on the comparison of arcs of irrectifiable curves |journal=Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae |volume=6 |pages=58–84 |language=la |url=https://archive.org/details/novicommentariia06impe/page/58/ }} [[Leonhard Euler#Selected bibliography|E]][https://scholarlycommons.pacific.edu/euler-works/252/ 252]. ([https://archive.org/details/novicommentariia06impe/page/n617/ Figures])
* {{cite journal |last1=Euler |first1=Leonhard |year=1786 |title=De miris proprietatibus curvae elasticae sub aequatione <math display=inline>y = \int xx\mathop{\mathrm{d}x} \big/ \sqrt{1-x^4}</math> contentae |trans-title=On the amazing properties of elastic curves contained in equation <math display=inline>y = \int xx\mathop{\mathrm{d}x} \big/ \sqrt{1-x^4}</math> |journal=Acta Academiae Scientiarum Imperialis Petropolitanae |volume=1782 |issue=2 |pages=34–61 |language=la|url=https://archive.org/details/actaacademiae82impe/page/34/ }} E [https://scholarlycommons.pacific.edu/euler-works/605/ 605].
* {{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=The Number Pi |translator-last=Wilson|translator-first=Stephen|publisher=American Mathematical Society |year=2004 |isbn=0-8218-3246-8}}
* {{wikicite
|ref={{harvid|Fagnano|1718–1723}}
|reference=
[[Giulio Carlo de' Toschi di Fagnano|Fagnano, Giulio Carlo]] (1718–1723) "Metodo per misurare la lemniscata" [Method for measuring the lemniscate]. ''Giornale de' letterati d'Italia'' (in Italian).<br />[https://archive.org/details/giornaledeletter1718roya/page/258/ "Schediasma primo"] [Part 1]. 1718. '''29''': 258–269.<br />[https://archive.org/details/giornaledeletter1723roya/page/197 "Giunte al primo schediasma"] [Addendum to part 1]. 1723. '''34''': 197–207.<br />[https://archive.org/details/giornaledelette1718roya_0/page/87/ "Schediasma secondo"] [Part 2]. 1718. '''30''': 87–111.<br />Reprinted as {{cite book |last1=Fagnano |date=1850 |title=Opere Matematiche, vol. 2 |chapter=32–34. Metodo per misurare la lemniscata |pages=293–313 |publisher=Allerighi e Segati |chapter-url=https://archive.org/details/operematfagnano02itws0006/page/n322/ }} ([https://archive.org/details/operematfagnano02itws0006/page/n498/ Figures])
}}
* {{Cite book |last1=Gauss |first1=Carl Friedrich |authorlink1=Carl Friedrich Gauss |title=Werke (Band X, Abteilung I) |publisher=Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen |language=Latin, German|year=1917|url=https://gdz.sub.uni-goettingen.de/id/PPN236018647 }}
* {{cite journal |last1=Gómez-Molleda |first1=M. A. |last2=Lario |first2=Joan-C. |date=2019 |title=Ruler and Compass Constructions of the Equilateral Triangle and Pentagon in the Lemniscate Curve |doi=10.1007/s00283-019-09892-w |journal=The Mathematical Intelligencer |volume=41 |issue=4 |pages=17–21 }}
* {{cite book |last1=Greenhill |first1=Alfred George |authorlink1=Alfred George Greenhill|year=1892 |title=The Applications of Elliptic Functions| publisher= MacMillan |url=https://archive.org/details/applicationselli00greerich/page/n5/ }}
* {{cite journal| last1=Guyou |first1=Émile |date=1887 |title=Nouveau système de projection de la sphère: Généralisation de la projection de Mercator |trans-title=New system of projection of the sphere |journal=Annales Hydrographiques |series=Série 2 |volume=9 |pages=16–35 |language=fr |url=https://www.retronews.fr/journal/annales-hydrographiques/1-janvier-1887/1877/4868382/23 }}
* {{cite book |last1= Houzel |first1=Christian |author-link1=Christian Houzel |date=1978 |chapter=Fonctions elliptiques et intégrales abéliennes |trans-chapter=Elliptic functions and Abelian integrals |editor-last=Dieudonné |editor-first= Jean |editor-link1=Jean Dieudonné |title= Abrégé d'histoire des mathématiques, 1700–1900. II |publisher=Hermann |pages=1–113 |language=fr}}
* {{cite journal |last1=Hyde |first1=Trevor |year=2014 |title=A Wallis product on clovers |journal=The American Mathematical Monthly |volume=121 |issue=3 |pages=237–243 |url=https://math.uchicago.edu/~tghyde/Hyde%20--%20A%20Wallis%20product%20on%20clovers.pdf |doi=10.4169/amer.math.monthly.121.03.237 }}
* {{cite journal |last1=Kubota |first1=Tomio |authorlink1=Tomio Kubota |date=1964 |title=Some arithmetical applications of an elliptic function |journal=Crelle's Journal |volume='''214'''/'''215''' |pages=141–145 |doi=10.1515/crll.1964.214-215.141 }}
* {{cite journal |last1=Langer |first1=Joel C. |first2=David A. |last2=Singer |year=2010 |title=Reflections on the Lemniscate of Bernoulli: The Forty-Eight Faces of a Mathematical Gem |journal=Milan Journal of Mathematics |volume=78 |issue=2 |pages=643–682 |url=https://case.edu/artsci/math/langer/jlpreprints/fortyeight2010.pdf |doi=10.1007/s00032-010-0124-5 }}
* {{cite journal |last1=Langer |first1=Joel C. |first2=David A. |last2=Singer |year=2011 |title=The lemniscatic chessboard |journal=Forum Geometricorum |volume=11 |pages=183–199 |url=https://forumgeom.fau.edu/FG2011volume11/FG201119index.html }}
* {{cite book |last1=Lawden |first1=Derek Frank |authorlink1=Derek Frank Lawden |title=Elliptic Functions and Applications |series=Applied Mathematical Sciences |date=1989 |volume=80 |publisher=Springer-Verlag |doi=10.1007/978-1-4757-3980-0 |isbn=978-1-4419-3090-3 }}
* {{cite book
| last = Lee | first = L. P. | author-link = Laurence Patrick Lee
| year = 1976
| title = Conformal Projections Based on Elliptic Functions
| location = Toronto | publisher = B. V. Gutsell, York University
| series = ''Cartographica Monographs'' | volume = 16
| url = https://archive.org/details/conformalproject0000leel | url-access = limited
| isbn = 0-919870-16-3
}} Supplement No. 1 to [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer'' '''13'''].
* {{Cite book |last1=Lemmermeyer |first1=Franz |title=Reciprocity Laws: From Euler to Eisenstein |publisher=Springer |year=2000 |isbn=3-540-66957-4}}
* {{cite tech report |last1=Levien |first1=Raph |year=2008 |authorlink1=Raph Levien |title=The elastica: a mathematical history |number=UCB/EECS-2008-103 |institution=University of California at Berkeley |url=https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.pdf}}
* {{cite journal |last1=Levin |first1=Aaron |title=A Geometric Interpretation of an Infinite Product for the Lemniscate Constant |journal=The American Mathematical Monthly |date=2006 |volume=113 |issue=6 |pages=510–520 |doi=10.1080/00029890.2006.11920331 |jstor=27641976 |url=https://web.archive.org/web/20041220213524id_/http://math.berkeley.edu:80/~adlevin/Lemniscate.pdf }}
* {{cite journal |last1=Lindqvist |first1=Peter |last2=Peetre |first2=Jaak |authorlink2=Jaak Peetre |title=Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals |journal=The American Mathematical Monthly |date=2001 |volume=108 |issue=5 |pages=403–410 |doi=10.1080/00029890.2001.11919766 |url=https://people.math.osu.edu/lang.162/book/LiPe3.pdf  |archive-url=https://web.archive.org/web/20220528124044/https://people.math.osu.edu/lang.162/book/LiPe3.pdf |archive-date=2022-05-28 }}
* {{cite book |last1=Markushevich |first1=Aleksei Ivanovich |authorlink1=Aleksei Ivanovich Markushevich |title=The Remarkable Sine Functions |date= 1966 |publisher=Elsevier |url=https://archive.org/details/markushevich-the-remarkable-sine-functions/ }}
* {{cite book |last1=Markushevich |first1=Aleksei Ivanovich |title=Introduction to the Classical Theory of Abelian Functions |series=Translations of Mathematical Monographs |date=1992 |volume=96 |publisher=American Mathematical Society |doi=10.1090/mmono/096 |isbn=978-0-8218-4164-8 }}
*{{cite tech report | last1=McGregor |first1= John L. |date=2005 |title=C-CAM: Geometric Aspects and Dynamical Formulation |institution=[[CSIRO Oceans and Atmosphere|CSIRO Atmospheric Research]] |number=70 |url=https://publications.csiro.au/rpr/pub?pid=procite:eb33ff33-c6d6-4779-8b82-867b2ca6e112 }}
* {{cite book |last1=McKean |first1=Henry |authorlink1=Henry McKean |last2=Moll |first2=Victor |author-link2=Victor Hugo Moll |date=1999 |title=Elliptic Curves: Function Theory, Geometry, Arithmetic |publisher=Cambridge |isbn=9780521582285 |url=https://archive.org/details/ellipticcurvesfu0000mcke |url-access=limited }}
* {{cite book |last1=Milne-Thomson |first1=Louis Melville |authorlink1=L. M. Milne-Thomson |date=1964 |contribution=16. Jacobian Elliptic Functions and Theta Functions |pages=567–585 |contribution-url=https://archive.org/details/handbookofmathem00abra/page/567/ |editor1-first=Milton |editor1-last=Abramowitz |editor1-link=Milton Abramowitz |editor2-first=Irene Ann |editor2-last=Stegun |editor2-link=Irene Stegun |title=Handbook of Mathematical Functions |title-link=Abramowitz and Stegun |publisher=National Bureau of Standards }}
* {{cite journal |last1=Neuman |first1=Edward |authorlink1=Edward Neuman |year=2007 |title=On Gauss lemniscate functions and lemniscatic mean |journal=Mathematica Pannonica |volume=18 |issue=1 |pages=77–94 |url=http://mathematica-pannonica.ttk.pte.hu/articles/mp18-1/MP18-1(2007)pp077-094.pdf }}
* {{cite journal |last1=Nishimura |first1=Ryo |title=New properties of the lemniscate function and its transformation |journal=Journal of Mathematical Analysis and Applications |date=2015 |volume=427 |issue=1 |pages=460–468 |doi=10.1016/j.jmaa.2015.02.066 |doi-access=free }}
* {{cite journal |last1=Ogawa |first1=Takuma |title=Similarities between the trigonometric function and the lemniscate function from arithmetic view point |journal=Tsukuba Journal of Mathematics |date=2005 |volume=29 |issue=1 |doi=10.21099/tkbjm/1496164894 |doi-access=free }}
* {{cite journal |last1=Peirce |first1=Charles Sanders |authorlink1=Charles Sanders Peirce |date=1879 |title=A Quincuncial Projection of the Sphere |journal=American Journal of Mathematics |volume=2 |issue=4 |pages=394–397 |doi=10.2307/2369491 |doi-access=free |jstor=2369491 |jstor-access=free |url=https://archive.org/details/sim_american-journal-of-mathematics_1879_2/page/n403/mode/2up}}
* {{cite book |last=Popescu-Pampu |first=Patrick |date=2016 |title=What is the Genus? |series=Lecture Notes in Mathematics |volume=2162 |publisher=Springer |doi=10.1007/978-3-319-42312-8 |isbn=978-3-319-42311-1 }}
* {{cite book |last1=Prasolov |first1=Viktor |last2=Solovyev |first2=Yuri |date=1997 |chapter=4. Abel's Theorem on Division of Lemniscate |title=Elliptic functions and elliptic integrals |series=Translations of Mathematical Monographs |volume=170 |publisher=American Mathematical Society. |doi=10.1090/mmono/170 |isbn=978-0-8218-0587-9 }}
* {{cite journal| last1=Rančić |first1=Miodrag |last2=Purser |first2=R. James | last3=Mesinger |first3=Fedor |date=1996 |title=A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates |journal=Quarterly Journal of the Royal Meteorological Society |volume=122 |issue=532 |pages=959–982 |doi=10.1002/qj.49712253209 |bibcode=<!-- useless bibcode 1996QJRMS.122..959R --> }}
* {{cite book |last1=Reinhardt |first1=William P. |last2=Walker |first2=Peter L. |year=2010a |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=22. Jacobian Elliptic Functions |chapter-url=https://dlmf.nist.gov/22 |title-link=Digital Library of Mathematical Functions }}
* {{cite book |last1=Reinhardt |first1=William P. |last2=Walker |first2=Peter L. |year=2010b |display-editors=1 |editor1-last=Olver |editor1-first=Frank |editor2-last=Lozier |editor2-first=Daniel |editor3-last=Boisvert |editor3-first=Ronald |editor4-last=Clark |editor4-first=Charles |title=NIST Handbook of Mathematical Functions |publisher=Cambridge |chapter=23. Weierstrass Elliptic and Modular Functions |chapter-url=https://dlmf.nist.gov/23 |title-link=Digital Library of Mathematical Functions }}
* {{citation |mode=cs1 |last1=Robinson |first1=Paul L. |title=The Lemniscatic Functions |date=2019a |arxiv=1902.08614 |type=Preprint |url=https://arxiv.org/abs/1903.07147 }}
* {{citation |mode=cs1 |last1=Robinson |first1=Paul L. |title= The Elliptic Functions in a First-Order System |date=2019b |arxiv=1903.07147 |type=Preprint |url=https://arxiv.org/abs/1903.07147}}
* {{cite journal |last1=Rosen |first1=Michael |authorlink1=Michael Rosen (mathematician) |title=Abel's Theorem on the Lemniscate |journal=The American Mathematical Monthly |date=1981 |volume=88 |issue=6 |pages=387–395 |doi=10.1080/00029890.1981.11995279  |jstor=2321821 }}
* {{cite book |last1=Roy |first1=Ranjan |title=Elliptic and Modular Functions from Gauss to Dedekind to Hecke |publisher=Cambridge University Press |page=28 |year=2017 |isbn=978-1-107-15938-9}}
* {{cite book  |chapter=Some milestones of lemniscatomy  |last1=Schappacher |first1=Norbert |author-link1=Norbert Schappacher | date= 1997 |editor1-last=Sertöz |editor1-first=S. |title=Algebraic Geometry |type=Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey |publisher=Marcel Dekker |pages=257–290 | chapter-url=http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf}}
* {{cite journal |last1=Schneider |first1=Theodor |authorlink1=Theodor Schneider |date=1937 |title=Arithmetische Untersuchungen elliptischer Integrale |trans-title=Arithmetic investigations of elliptic integrals |language=de |journal=Mathematische Annalen |volume=113 |issue=1 |pages=1–13|doi=10.1007/BF01571618 |url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002278537 }}
* {{cite journal |last1=Schwarz |first1= Hermann Amandus | authorlink1= Hermann Schwarz| date=1869 |title=Ueber einige Abbildungsaufgaben |trans-title=About some mapping problems |language=de |journal=Crelle's Journal |volume=70 |pages= 105–120 |url=https://archive.org/details/sim_journal-fuer-die-reine-und-angewandte-mathematik_1869_70/page/105 |doi=10.1515/crll.1869.70.105 }} 
* {{cite book |last1=Siegel |first1=Carl Ludwig |authorlink1=Carl Ludwig Siegel |title=Topics in Complex Function Theory, Vol. I |chapter=1. Elliptic Functions |pages=1–89 |publisher=Wiley-Interscience |date=1969 |isbn=0-471-60844-0}}
* {{cite thesis |last1=Snape |first1=Jamie |year=2004 |chapter=Bernoulli's Lemniscate |pages=50–56 |title=Applications of Elliptic Functions in Classical and Algebraic Geometry |chapter-url=https://www.jamiesnape.io/publications/mmath/ |publisher=University of Durham}}
* {{cite book |last1=Southard |first1=Thomas H. |date=1964 |contribution=18. Weierstrass Elliptic and Related Functions |pages=627–683 |contribution-url=https://archive.org/details/handbookofmathem00abra/page/627/ |editor1-first=Milton |editor1-last=Abramowitz |editor1-link=Milton Abramowitz |editor2-first=Irene Ann |editor2-last=Stegun |editor2-link=Irene Stegun |title=Handbook of Mathematical Functions |title-link=Abramowitz and Stegun |publisher=National Bureau of Standards }}
* {{wikicite
|ref={{harvid|Sridharan|2004}}
|reference=
[[Ramaiyengar Sridharan|Sridharan, Ramaiyengar]] (2004) "Physics to Mathematics: from Lintearia to Lemniscate". ''Resonance''. <br />[https://www.ias.ac.in/public/Volumes/reso/009/04/0021-0029.pdf "Part I"]. '''9''' (4): 21–29. [[doi (identifier)|doi]]:[https://doi.org/10.1007%2FBF02834853 10.1007/BF02834853]. <br />[https://www.ias.ac.in/public/Volumes/reso/009/06/0011-0020.pdf "Part II: Gauss and Landen's Work"]. '''9''' (6): 11–20. [[doi (identifier)|doi]]:[https://doi.org/10.1007%2FBF02839214 10.1007/BF02839214].
}}
* {{cite journal |last1=Todd |first1=John |authorlink1=John (Jack) Todd |title=The lemniscate constants |journal=Communications of the ACM |date=1975 |volume=18 |issue=1 |pages=14–19 |doi=10.1145/360569.360580 |doi-access=free }}
* {{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1920 |orig-date=1st ed. 1902 |chapter=22.8 The lemniscate functions |pages=524–528 |chapter-url=https://archive.org/details/courseofmodernan00whit/page/524/ |edition=3rd |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge }}
* {{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1927 |orig-date=4th ed. 1927 |chapter=21 The theta functions |pages=469–470 |edition=4th |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge }}
{{refend}}

== 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}(\sqrt{2}t)=\frac{\cos_4^2(t)-\sin_4^2(t)}{\cos_4^2(t)+\sin_4^2(t)}</math>


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[[Category:Modular forms]]
[[Category:Elliptic functions]]