Lemniscate elliptic functions#Inverse functions

The lemniscate functions {{math|sl}} and {{math|cl}} can be defined as the solution to the initial value problem:{{harvp|Robinson|2019a}} starts from this definition and thence derives other properties of the lemniscate functions.

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}z\}\; \backslash operatorname\{sl\}\; z\; =\; \backslash bigl(1\; +\; \backslash operatorname\{sl\}^2\; z\backslash bigr)\backslash operatorname\{cl\}z,\backslash \; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}z\}\; \backslash operatorname\{cl\}\; z\; =\; -\backslash bigl(1\; +\; \backslash operatorname\{cl\}^2\; z\backslash bigr)\backslash operatorname\{sl\}z,\backslash \; \backslash operatorname\{sl\}\; 0\; =\; 0,\backslash \; \backslash operatorname\{cl\}\; 0\; =\; 1,$

or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners $\backslash big\backslash \{\backslash tfrac12\backslash varpi,\; \backslash tfrac12\backslash varpi\; i,\; -\backslash tfrac12\backslash varpi,\; -\backslash tfrac12\backslash varpi\; i\backslash big\backslash \}\backslash colon$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].

:$z\; =\; \backslash int\_0^\{\backslash operatorname\{sl\}\; z\}\backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1-t^4\}\}\; =\; \backslash int\_\{\backslash operatorname\{cl\}\; z\}^1\backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1-t^4\}\}.$

Beyond that square, the functions can be analytically continued to the whole complex plane by a series of reflections.

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

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}z\}\; \backslash sin\; z\; =\; \backslash cos\; z,\backslash \; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}z\}\; \backslash cos\; z\; =\; -\backslash sin\; z,\backslash \; \backslash sin\; 0\; =\; 0,\backslash \; \backslash cos\; 0\; =\; 1,$

or as inverses of a map from the upper half-plane to a half-infinite strip with real part between $-\backslash tfrac12\backslash pi,\; \backslash tfrac12\backslash pi$ and positive imaginary part:

:$z\; =\; \backslash int\_0^\{\backslash sin\; z\}\backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1-t^2\}\}\; =\; \backslash int\_\{\backslash cos\; z\}^1\backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1-t^2\}\}.$

{{main|Lemniscate constant}}

File:Lemniscate constant as an integral.png

The lemniscate functions have minimal real period {{math|2ϖ}}, minimal imaginary period {{math|2ϖi}} and fundamental complex periods $(1+i)\backslash varpi$ and $(1-i)\backslash varpi$ for a constant {{math|ϖ}} called the lemniscate constant,{{harvp|Schappacher|1997}}. OEIS sequence [https://oeis.org/A062539 A062539] lists the lemniscate constant's decimal digits.

:$\backslash varpi\; =\; 2\backslash int\_0^1\backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1-t^4\}\}\; =\; 2.62205\backslash ldots$

The lemniscate functions satisfy the basic relation $\backslash operatorname\{cl\}z\; =\; \{\backslash operatorname\{sl\}\}\backslash bigl(\backslash tfrac12\backslash varpi\; -\; z\backslash bigr),$ analogous to the relation $\backslash cos\; z\; =\; \{\backslash sin\}\backslash bigl(\backslash tfrac12\backslash pi\; -\; z\backslash bigr).$

The lemniscate constant {{math|ϖ}} is a close analog of the pi, 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:

\frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots

An analogous formula for {{math|ϖ}} is:{{harvp|Levin|2006}}

\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12 \Bigg/ \!\sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12}} \cdots

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

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

\frac\pi\varpi = M{\left(1, \sqrt2\!~\right)}

File:Lemniscate sine in the complex plane.svg of $\backslash operatorname\{sl\}z$ changes from $-\backslash pi$ (excluding $-\backslash pi$) to $\backslash pi$, the colors go through cyan, blue $(\backslash operatorname\{Arg\}\backslash approx\; -\backslash pi/2)$, magneta, red $(\backslash operatorname\{Arg\}\backslash approx\; 0)$, orange, yellow $(\backslash operatorname\{Arg\}\backslash approx\backslash pi/2)$, green, and back to cyan $(\backslash operatorname\{Arg\}\backslash approx\backslash pi)$. In the picture, it can be seen that the fundamental periods $(1+i)\backslash varpi$ and $(1-i)\backslash varpi$ 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,

:$\backslash begin\{aligned\}$

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

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

\end{aligned}

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

:$\backslash begin\{aligned\}$

{\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}

Doubling these to translations by a unit-Gaussian-integer multiple of $\backslash varpi$ (that is, $\backslash pm\; \backslash varpi$ or $\backslash pm\; i\backslash varpi$), negates each function, an involution:

:$\backslash begin\{aligned\}$

\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}

As a result, both functions are invariant under translation by an even-Gaussian-integer multiple of $\backslash varpi$.The even Gaussian integers are the residue class of 0, modulo {{math|1 + i}}, the black squares on a checkerboard. That is, a displacement $(a\; +\; bi)\backslash varpi,$ with $a\; +\; b\; =\; 2k$ for integers {{math|a}}, {{math|b}}, and {{math|k}}.

:$\backslash begin\{aligned\}$

{\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}

This makes them elliptic functions (doubly periodic meromorphic functions in the complex plane) with a diagonal square period lattice of fundamental periods $(1\; +\; i)\backslash varpi$ and $(1\; -\; i)\backslash varpi$.{{harvp|Prasolov|Solovyev|1997}}; {{harvp|Robinson|2019a}} 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:

:$\backslash begin\{aligned\}$

\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}

The {{math|sl}} function has simple zeros at Gaussian integer multiples of {{math|ϖ}}, complex numbers of the form $a\backslash varpi\; +\; b\backslash varpi\; i$ for integers {{math|a}} and {{math|b}}. It has simple poles at Gaussian half-integer multiples of {{math|ϖ}}, complex numbers of the form $\backslash bigl(a\; +\; \backslash tfrac12\backslash bigr)\backslash varpi\; +\; \backslash bigl(b\; +\; \backslash tfrac12\backslash bigr)\backslash varpi\; i$, with residues $(-1)^\{a-b+1\}i$. The {{math|cl}} function is reflected and offset from the {{math|sl}} function, $\backslash operatorname\{cl\}z\; =\; \{\backslash operatorname\{sl\}\}\backslash bigl(\backslash tfrac12\backslash varpi\; -\; z\backslash bigr)$. It has zeros for arguments $\backslash bigl(a\; +\; \backslash tfrac12\backslash bigr)\backslash varpi\; +\; b\backslash varpi\; i$ and poles for arguments $a\backslash varpi\; +\; \backslash bigl(b\; +\; \backslash tfrac12\backslash bigr)\backslash varpi\; i,$ with residues $(-1)^\{a-b\}i.$

Also

:$\backslash operatorname\{sl\}z=\backslash operatorname\{sl\}w\backslash leftrightarrow\; z=(-1)^\{m+n\}w+(m+ni)\backslash varpi$

for some $m,n\backslash in\backslash mathbb\{Z\}$ and

:$\backslash operatorname\{sl\}((1\backslash pm\; i)z)=(1\backslash pm\; i)\backslash frac\{\backslash operatorname\{sl\}z\}\{\backslash operatorname\{sl\}\text{'}z\}.$

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

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

:$\backslash frac\{1\}\{\backslash operatorname\{sl\}z\}=\backslash sum\_\{(n,k)\backslash in\backslash mathbb\{Z\}^2\}\backslash frac\{(-1)^\{n+k\}\}\{z+n\backslash varpi+k\backslash varpi\; i\}$

:$\backslash operatorname\{sl\}z=-i\backslash sum\_\{(n,k)\backslash in\backslash mathbb\{Z\}^2\}\backslash frac\{(-1)^\{n+k\}\}\{z+(n+1/2)\backslash varpi\; +(k+1/2)\backslash varpi\; i\}.$

File:Algebraic curves (x² + y²) = a(1 - x²y²) for various values of a.png

The lemniscate functions satisfy a Pythagorean-like identity:

:$\backslash operatorname\{cl^2\}\; z\; +\; \backslash operatorname\{sl^2\}\; z\; +\; \backslash operatorname\{cl^2\}\; z\; \backslash ,\; \backslash operatorname\{sl^2\}\; z\; =\; 1$

As a result, the parametric equation $(x,\; y)\; =\; (\backslash operatorname\{cl\}\; t,\; \backslash operatorname\{sl\}\; t)$ parametrizes the quartic curve $x^2\; +\; y^2\; +\; x^2y^2\; =\; 1.$

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

:$\backslash bigl(1\; +\; \backslash operatorname\{cl^2\}\; z\backslash bigr)\; \backslash bigl(1+\backslash operatorname\{sl^2\}\; z\backslash bigr)\; =\; 2$

:$\backslash operatorname\{cl^2\}\; z\; =\; \backslash frac\{1\; -\; \backslash operatorname\{sl^2\}\; z\}\{1\; +\; \backslash operatorname\{sl^2\}\; z\},\backslash quad$

\operatorname{sl^2} z = \frac{1 - \operatorname{cl^2} z}{1 + \operatorname{cl^2} z}

Defining a tangent-sum operator as $a\; \backslash oplus\; b\; \backslash mathrel\{:=\}\; \backslash tan(\backslash arctan\; a\; +\; \backslash arctan\; b)\; =\; \backslash frac\{a+b\}\{1-ab\},$ gives:

:$\backslash operatorname\{cl^2\}\; z\; \backslash oplus\; \backslash operatorname\{sl^2\}\; z\; =\; 1.$

The functions $\backslash tilde\{\backslash operatorname\{cl\}\}$ and $\backslash tilde\{\backslash operatorname\{sl\}\}$ satisfy another Pythagorean-like identity:

:$\backslash left(\backslash int\_0^x\; \backslash tilde\{\backslash operatorname\{cl\}\}\backslash ,t\backslash ,\backslash mathrm\; dt\backslash right)^2+\backslash left(1-\backslash int\_0^x\; \backslash tilde\{\backslash operatorname\{sl\}\}\backslash ,t\backslash ,\backslash mathrm\; dt\backslash right)^2=1.$

The derivatives are as follows:

:$\backslash begin\{aligned\}$

\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}

:

\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}

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

:$\backslash frac\{\backslash mathrm\{d\}^2\}\{\backslash mathrm\{d\}z^2\}\backslash operatorname\{cl\}z\; =\; -2\backslash operatorname\{cl^3\}z$

:$\backslash frac\{\backslash mathrm\{d\}^2\}\{\backslash mathrm\{d\}z^2\}\backslash operatorname\{sl\}z\; =\; -2\backslash operatorname\{sl^3\}z$

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

:

\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}

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:{{harvp|Ayoub|1984}}; {{harvp|Prasolov|Solovyev|1997}}

: $\backslash operatorname\{sl\}(u+v)\; =\; \backslash frac\{\backslash operatorname\{sl\}u\backslash ,\backslash operatorname\{sl\text{'}\}v\; +\; \backslash operatorname\{sl\}v\backslash ,\backslash operatorname\{sl\text{'}\}u\}$

{1 + \operatorname{sl^2}u\, \operatorname{sl^2}v}

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 tangent-sum operator $a\; \backslash oplus\; b\; \backslash mathrel\{:=\}\; \backslash tan(\backslash arctan\; a\; +\; \backslash arctan\; b)$ and tangent-difference operator $a\; \backslash ominus\; b\; \backslash mathrel\{:=\}\; a\; \backslash oplus\; (-b),$ the argument sum and difference identities can be expressed as:{{harvp|Euler|1761}} [https://archive.org/details/novicommentariia06impe/page/79/ §44 p. 79], §47 pp. 80–81

:$\backslash begin\{aligned\}$

\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}

These resemble their trigonometric analogs:

:$\backslash begin\{aligned\}$

\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}

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

:$\backslash begin\{aligned\}$

\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}

Gauss discovered that

:$\backslash frac\{\backslash operatorname\{sl\}(u-v)\}\{\backslash operatorname\{sl\}(u+v)\}=\backslash frac\{\backslash operatorname\{sl\}((1+i)u)-\backslash operatorname\{sl\}((1+i)v)\}\{\backslash operatorname\{sl\}((1+i)u)+\backslash operatorname\{sl\}((1+i)v)\}$

where $u,v\backslash in\backslash mathbb\{C\}$ such that both sides are well-defined.

Also

:$\backslash operatorname\{sl\}(u+v)\backslash operatorname\{sl\}(u-v)=\backslash frac\{\backslash operatorname\{sl\}^2u-\backslash operatorname\{sl\}^2v\}\{1+\backslash operatorname\{sl\}^2u\backslash operatorname\{sl\}^2v\}$

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

:$\backslash sin\; (u+v)\backslash sin\; (u-v)=\backslash sin^2u-\backslash sin^2v.$

Bisection formulas:

:$$

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

:$$

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

Duplication formulas:{{harvp|Euler|1761}} [https://archive.org/details/novicommentariia06impe/page/80 §46 p. 80]

:$$

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

:$$

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

Triplication formulas:

:$$

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

:$$

\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}}

Note the "reverse symmetry" of the coefficients of numerator and denominator of $\backslash operatorname\{sl\}3x$. This phenomenon can be observed in multiplication formulas for $\backslash operatorname\{sl\}\backslash beta\; x$ where $\backslash beta=m+ni$ whenever $m,n\backslash in\backslash mathbb\{Z\}$ and $m+n$ is odd.

Let $L$ be the lattice

:$L=\backslash mathbb\{Z\}(1+i)\backslash varpi\; +\backslash mathbb\{Z\}(1-i)\backslash varpi.$

Furthermore, let $K=\backslash mathbb\{Q\}(i)$, $\backslash mathcal\{O\}=\backslash mathbb\{Z\}[i]$, $z\backslash in\backslash mathbb\{C\}$, $\backslash beta=m+in$, $\backslash gamma=m\text{'}+in\text{'}$ (where $m,n,m\text{'},n\text{'}\backslash in\backslash mathbb\{Z\}$), $m+n$ be odd, $m\text{'}+n\text{'}$ be odd, $\backslash gamma\backslash equiv\; 1\backslash ,\backslash operatorname\{mod\}\backslash ,\; 2(1+i)$ and $\backslash operatorname\{sl\}\; \backslash beta\; z=M\_\backslash beta\; (\backslash operatorname\{sl\}z)$. Then

:$M\_\backslash beta\; (x)=i^\backslash varepsilon\; x\; \backslash frac\{P\_\backslash beta\; (x^4)\}\{Q\_\backslash beta\; (x^4)\}$

for some coprime polynomials $P\_\backslash beta\; (x),\; Q\_\backslash beta\; (x)\backslash in\; \backslash mathcal\{O\}[x]$

and some $\backslash varepsilon\backslash in\; \backslash \{0,1,2,3\backslash \}$In fact, $i^\backslash varepsilon=\backslash operatorname\{sl\}\backslash tfrac\{\backslash beta\backslash varpi\}\{2\}$. where

:$xP\_\backslash beta\; (x^4)=\backslash prod\_\{\backslash gamma\; |\backslash beta\}\backslash Lambda\_\backslash gamma\; (x)$

and

:$\backslash Lambda\_\backslash beta\; (x)=\backslash prod\_\{[\backslash alpha]\backslash in\; (\backslash mathcal\{O\}/\backslash beta\backslash mathcal\{O\})^\backslash times\}(x-\backslash operatorname\{sl\}\backslash alpha\backslash delta\_\backslash beta)$

where $\backslash delta\_\backslash beta$ is any $\backslash beta$-torsion generator (i.e. $\backslash delta\_\backslash beta\; \backslash in\; (1/\backslash beta)L$ and $[\backslash delta\_\backslash beta]\backslash in\; (1/\backslash beta)L/L$ generates $(1/\backslash beta)L/L$ as an $\backslash mathcal\{O\}$-module). Examples of $\backslash beta$-torsion generators include $2\backslash varpi/\backslash beta$ and $(1+i)\backslash varpi/\backslash beta$. The polynomial $\backslash Lambda\_\backslash beta\; (x)\backslash in\backslash mathcal\{O\}[x]$ is called the $\backslash beta$-th lemnatomic polynomial. It is monic and is irreducible over $K$. The lemnatomic polynomials are the "lemniscate analogs" of the cyclotomic polynomials,{{harvp|Cox|Hyde|2014}}

:$\backslash Phi\_k(x)=\backslash prod\_\{[a]\backslash in\; (\backslash mathbb\{Z\}/k\backslash mathbb\{Z\})^\backslash times\}(x-\backslash zeta\_k^a).$

The $\backslash beta$-th lemnatomic polynomial $\backslash Lambda\_\backslash beta(x)$ is the minimal polynomial of $\backslash operatorname\{sl\}\backslash delta\_\backslash beta$ in $K[x]$. For convenience, let $\backslash omega\_\{\backslash beta\}=\backslash operatorname\{sl\}(2\backslash varpi/\backslash beta)$ and $\backslash tilde\{\backslash omega\}\_\{\backslash beta\}=\backslash operatorname\{sl\}((1+i)\backslash varpi/\backslash beta)$. So for example, the minimal polynomial of $\backslash omega\_5$ (and also of $\backslash tilde\{\backslash omega\}\_5$) in $K[x]$ is

:$\backslash Lambda\_5(x)=x^\{16\}+52x^\{12\}-26x^8-12x^4+1,$

and{{harvp|Gómez-Molleda|Lario|2019}}

:$\backslash omega\_5=\backslash sqrt[4]\{-13+6\backslash sqrt\{5\}+2\backslash sqrt\{85-38\backslash sqrt\{5\}\}\}$

:$\backslash tilde\{\backslash omega\}\_5=\backslash sqrt[4]\{-13-6\backslash sqrt\{5\}+2\backslash sqrt\{85+38\backslash sqrt\{5\}\}\}$The fourth root with the least positive principal argument is chosen.

(an equivalent expression is given in the table below). Another example is

:$\backslash Lambda\_\{-1+2i\}(x)=x^4-1+2i$

which is the minimal polynomial of $\backslash omega\_\{-1+2i\}$ (and also of $\backslash tilde\{\backslash omega\}\_\{-1+2i\}$) in $K[x].$

If $p$ is prime and $\backslash beta$ is positive and odd,The restriction to positive and odd $\backslash beta$ can be dropped in $\backslash operatorname\{deg\}\backslash Lambda\_\backslash beta=\backslash left|(\backslash mathcal\{O\}/\backslash beta\backslash mathcal\{O\})^\backslash times\backslash right|$. then{{harvp|Cox|2013}} p. 142, Example 7.29(c)

:$\backslash operatorname\{deg\}\backslash Lambda\_\{\backslash beta\}=\backslash beta^2\backslash prod\_\{p|\backslash beta\}\backslash left(1-\backslash frac\{1\}\{p\}\backslash right)\backslash left(1-\backslash frac\{(-1)^\{(p-1)/2\}\}\{p\}\backslash right)$

which can be compared to the cyclotomic analog

:$\backslash operatorname\{deg\}\backslash Phi\_\{k\}=k\backslash prod\_\{p|k\}\backslash left(1-\backslash frac\{1\}\{p\}\backslash right).$

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 $n\; =\; 2^kp\_1p\_2\backslash cdots\; p\_m$ where {{math|k}} is a non-negative integer and each {{math|p_i}} (if any) is a distinct Fermat prime.{{harvp|Rosen|1981}}

File:The lemniscate sine and cosine related to the arclength of the lemniscate of Bernoulli.png

File:The sine and cosine related to the arclength of the unit-diameter circle.png

$\backslash mathcal\{L\}$, 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 characterized in at least three ways:

Angular characterization: Given two points $A$ and $B$ which are unit distance apart, let $B\text{'}$ be the reflection of $B$ about $A$. Then $\backslash mathcal\{L\}$ is the closure of the locus of the points $P$ such that $|APB-APB\text{'}|$ is a right angle.{{harvp|Eymard|Lafon|2004}} p. 200

Focal characterization: $\backslash mathcal\{L\}$ is the locus of points in the plane such that the product of their distances from the two focal points $F\_1\; =\; \backslash bigl(\{-\backslash tfrac1\backslash sqrt2\},0\backslash bigr)$ and $F\_2\; =\; \backslash bigl(\backslash tfrac1\backslash sqrt2,0\backslash bigr)$ is the constant $\backslash tfrac12$.

Explicit coordinate characterization: $\backslash mathcal\{L\}$ is a quartic curve satisfying the polar equation $r^2\; =\; \backslash cos\; 2\backslash theta$ or the Cartesian equation $\backslash bigl(x^2+y^2\backslash bigr)\{\}^2=x^2-y^2.$

The perimeter of $\backslash mathcal\{L\}$ is $2\backslash varpi$.And the area enclosed by $\backslash mathcal\{L\}$ is $1$, which stands in stark contrast to the unit circle (whose enclosed area is a non-constructible number).

The points on $\backslash mathcal\{L\}$ at distance $r$ from the origin are the intersections of the circle $x^2+y^2=r^2$ and the hyperbola $x^2-y^2=r^4$. The intersection in the positive quadrant has Cartesian coordinates:

:$\backslash big(x(r),\; y(r)\backslash big)\; =\; \backslash biggl(\backslash !\backslash sqrt\{\backslash tfrac12r^2\backslash bigl(1\; +\; r^2\backslash bigr)\},\backslash ,\; \backslash sqrt\{\backslash tfrac12r^2\backslash bigl(1\; -\; r^2\backslash bigr)\}\backslash ,\backslash biggr).$

Using this parametrization with $r\; \backslash in\; [0,\; 1]$ for a quarter of $\backslash mathcal\{L\}$, the arc length from the origin to a point $\backslash big(x(r),\; y(r)\backslash big)$ is:{{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.

:$\backslash begin\{aligned\}$

&\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}

Likewise, the arc length from $(1,0)$ to $\backslash big(x(r),\; y(r)\backslash big)$ is:

:$\backslash begin\{aligned\}$

&\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}

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 $(1,0)$, respectively.

Analogously, the circular sine and cosine functions relate the chord length to the arc length for the unit diameter circle with polar equation $r\; =\; \backslash cos\; \backslash theta$ or Cartesian equation $x^2\; +\; y^2\; =\; x,$ using the same argument above but with the parametrization:

:$\backslash big(x(r),\; y(r)\backslash big)\; =\; \backslash biggl(r^2,\backslash ,\; \backslash sqrt\{r^2\backslash bigl(1-r^2\backslash bigr)\}\backslash ,\backslash biggr).$

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

:$(x(s),y(s))=(\backslash cos\; s,\backslash sin\; s),$

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

:$(x(s),y(s))=\backslash left(\backslash frac\{\backslash operatorname\{cl\}s\}\{\backslash sqrt\{1+\backslash operatorname\{sl\}^2\; s\}\},\backslash frac\{\backslash operatorname\{sl\}s\backslash operatorname\{cl\}s\}\{\backslash sqrt\{1+\backslash operatorname\{sl\}^2\; s\}\}\backslash right)=\backslash left(\backslash tilde\{\backslash operatorname\{cl\}\}\backslash ,s,\backslash tilde\{\backslash operatorname\{sl\}\}\backslash ,s\backslash right).$

The notation $\backslash tilde\{\backslash operatorname\{cl\}\},\backslash ,\backslash tilde\{\backslash operatorname\{sl\}\}$ 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:{{harvp|Siegel|1969}}; {{harvp|Schappacher|1997}}

:$\backslash int\_0^z\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1\; -\; t^4\}\}\; =\; 2\; \backslash int\_0^u\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1\; -\; t^4\}\},\; \backslash quad\; \backslash text\{if\; \}$

z = \frac{2u\sqrt{1 - u^4}}{1 + u^4} \text{ and } 0\le u\le\sqrt{\sqrt{2}-1}.

File:Lemniscate 15-gon.png

Later mathematicians generalized this result. Analogously to the constructible polygons in the circle, the lemniscate can be divided into {{math|n}} sections of equal arc length using only straightedge and compass if and only if {{math|n}} is of the form $n\; =\; 2^kp\_1p\_2\backslash cdots\; p\_m$ where {{math|k}} is a non-negative integer and each {{math|p_i}} (if any) is a distinct Fermat prime.Such numbers are OEIS sequence A003401. The "if" part of the theorem was proved by Niels Abel in 1827–1828, and the "only if" part was proved by Michael Rosen in 1981.{{harvp|Abel|1827–1828}}; {{harvp|Rosen|1981}}; {{harvp|Prasolov|Solovyev|1997}} Equivalently, the lemniscate can be divided into {{math|n}} sections of equal arc length using only straightedge and compass if and only if $\backslash varphi\; (n)$ is a power of two (where $\backslash varphi$ 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 $r\_j=\backslash operatorname\{sl\}\backslash dfrac\{2j\backslash varpi\}\{n\}$. Then the {{math|n}}-division points for $\backslash mathcal\{L\}$ are the points

:$\backslash left(r\_j\backslash sqrt\{\backslash tfrac12\backslash bigl(1+r\_j^2\backslash bigr)\},\backslash \; (-1)^\{\backslash left\backslash lfloor\; 4j/n\backslash right\backslash rfloor\}\; \backslash sqrt\{\backslash tfrac12r\_j^2\backslash bigl(1-r\_j^2\backslash bigr)\}\backslash right),\backslash quad\; j\backslash in\backslash \{1,2,\backslash ldots\; ,n\backslash \}$

where $\backslash lfloor\backslash cdot\backslash rfloor$ is the floor function. See below for some specific values of $\backslash operatorname\{sl\}\backslash dfrac\{2\backslash varpi\}\{n\}$.

File:Rectangular elastica and lemniscatic sine.png

The inverse lemniscate sine also describes the arc length {{math|s}} relative to the {{math|x}} coordinate of the rectangular elastica.{{harvp|Euler|1786}}; {{harvp|Sridharan|2004}}; {{harvp|Levien|2008}} This curve has {{math|y}} coordinate and arc length:

:$y\; =\; \backslash int\_x^1\; \backslash frac\{t^2\backslash mathop\{\backslash mathrm\{d\}t\}\}\{\backslash sqrt\{1\; -\; t^4\}\},\backslash quad\; s\; =\; \backslash operatorname\{arcsl\}\; x\; =\; \backslash int\_0^x\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1\; -\; t^4\}\}$

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.

File:The lemniscate elliptic functions and an ellipse.jpg

Let $C$ be a point on the ellipse $x^2+2y^2=1$ in the first quadrant and let $D$ be the projection of $C$ on the unit circle $x^2+y^2=1$. The distance $r$ between the origin $A$ and the point $C$ is a function of $\backslash varphi$ (the angle $BAC$ where $B=(1,0)$; equivalently the length of the circular arc $BD$). The parameter $u$ is given by

:$u=\backslash int\_0^\{\backslash varphi\}r(\backslash theta)\backslash ,\; \backslash mathrm\; d\backslash theta=\backslash int\_0^\{\backslash varphi\}\backslash frac\{\backslash mathrm\; d\backslash theta\}\{\backslash sqrt\{1+\backslash sin^2\backslash theta\}\}.$

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

:$\backslash operatorname\{cl\}u=\backslash overline\{AF\},\; \backslash quad\; \backslash operatorname\{sl\}u=\backslash overline\{DE\},$

:$\backslash tilde\{\backslash operatorname\{cl\}\}\backslash ,\; u=\backslash overline\{AF\}\backslash overline\{AC\},\; \backslash quad\; \backslash tilde\{\backslash operatorname\{sl\}\}\backslash ,\; u=\backslash overline\{AF\}\backslash overline\{FC\}.$

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

:

where the coefficients $a\_n$ are determined as follows:

:$n\backslash not\backslash equiv\; 1\backslash pmod\; 4\backslash implies\; a\_n=0,$

:$a\_1=1,\backslash ,\; \backslash forall\; n\backslash in\backslash mathbb\{N\}\_0:\backslash ,a\_\{n+2\}=-\backslash frac\{2\}\{(n+1)(n+2)\}\backslash sum\_\{i+j+k=n\}a\_ia\_ja\_k$

where $i+j+k=n$ stands for all three-term compositions of $n$. For example, to evaluate $a\_\{13\}$, it can be seen that there are only six compositions of $13-2=11$ that give a nonzero contribution to the sum: $11=9+1+1=1+9+1=1+1+9$ and $11=5+5+1=5+1+5=1+5+5$, so

:$a\_\{13\}=-\backslash tfrac\{2\}\{12\backslash cdot\; 13\}(a\_9a\_1a\_1+a\_1a\_9a\_1+a\_1a\_1a\_9+a\_5a\_5a\_1+a\_5a\_1a\_5+a\_1a\_5a\_5)=-\backslash tfrac\{11\}\{15600\}.$

The expansion can be equivalently written as{{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}}

:

where

:$p\_\{n+2\}=-12\backslash sum\_\{j=0\}^n\backslash binom\{2n+2\}\{2j+2\}p\_\{n-j\}\backslash sum\_\{k=0\}^j\; \backslash binom\{2j+1\}\{2k+1\}p\_k\; p\_\{j-k\},\backslash quad\; p\_0=1,\backslash ,\; p\_1=0.$

The power series expansion of $\backslash tilde\{\backslash operatorname\{sl\}\}$ at the origin is

:

where $\backslash alpha\_n=0$ if $n$ is even and

:$\backslash alpha\_n=\backslash sqrt\{2\}\backslash frac\{\backslash pi\}\{\backslash varpi\}\backslash frac\{(-1)^\{(n-1)/2\}\}\{n!\}\backslash sum\_\{k=1\}^\{\backslash infty\}\backslash frac\{(2k\backslash pi/\backslash varpi)^\{n+1\}\}\{\backslash cosh\; k\backslash pi\},\backslash quad\; \backslash left|\backslash alpha\_n\backslash right|\backslash sim\; 2^\{n+5/2\}\backslash frac\{n+1\}\{\backslash varpi^\{n+2\}\}$

if $n$ is odd.

The expansion can be equivalently written as{{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

:

where

:$s\_\{n+2\}=3\; s\_\{n+1\}\; +24\; \backslash sum\_\{j=0\}^n\; \backslash binom\{2n+2\}\{2j+2\}\; s\_\{n-j\}\; \backslash sum\_\{k=0\}^j\; \backslash binom\{2j+1\}\{2k+1\}\; s\_k\; s\_\{j-k\},\backslash quad\; s\_0=1,\backslash ,\; s\_1=3,$

:$t\_\{n+2\}=3\; t\_\{n+1\}+3\; \backslash sum\_\{j=0\}^n\; \backslash binom\{2n+2\}\{2j+2\}\; t\_\{n-j\}\; \backslash sum\_\{k=0\}^j\; \backslash binom\{2j+1\}\{2k+1\}\; t\_k\; t\_\{j-k\},\backslash quad\; t\_0=1,\backslash ,\; t\_1=3.$

For the lemniscate cosine,{{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

:

:

where

:$r\_\{n+2\}=3\; \backslash sum\_\{j=0\}^n\; \backslash binom\{2n+2\}\{2j+2\}\; r\_\{n-j\}\; \backslash sum\_\{k=0\}^j\; \backslash binom\{2j+1\}\{2k+1\}\; r\_k\; r\_\{j-k\},\backslash quad\; r\_0=1,\backslash ,\; r\_1=0,$

:$q\_\{n+2\}=\backslash tfrac\{3\}\{2\}\; q\_\{n+1\}+6\; \backslash sum\_\{j=0\}^n\; \backslash binom\{2n+2\}\{2j+2\}\; q\_\{n-j\}\; \backslash sum\_\{k=0\}^j\; \backslash binom\{2j+1\}\{2k+1\}\; q\_k\; q\_\{j-k\},\backslash quad\; q\_0=1,\; \backslash ,q\_1=\backslash tfrac\{3\}\{2\}.$

Ramanujan's famous cos/cosh identity states that if

:$R(s)=\backslash frac\{\backslash pi\}\{\backslash varpi\backslash sqrt\{2\}\}\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}\backslash frac\{\backslash cos\; (2n\backslash pi\; s/\backslash varpi)\}\{\backslash cosh\; n\backslash pi\},$

then{{cite web | url=https://oeis.org/A193543 | title=A193543 - Oeis }}

:

There is a close relation between the lemniscate functions and $R(s)$. Indeed,{{cite web | url=https://oeis.org/A289695 | title=A289695 - Oeis }}

:

:

and

:$R(s)=\backslash frac\{1\}\{\backslash sqrt\{1+\backslash operatorname\{sl\}^2\; s\}\},\backslash quad\; \backslash left|\backslash operatorname\{Im\}s\backslash right$

|<\frac{\varpi}{2}.

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

:$\backslash int\_0^\backslash infty\; e^\{-tz\backslash sqrt\{2\}\}\backslash operatorname\{cl\}t\backslash ,\; \backslash mathrm\; dt=\backslash cfrac\{1/\backslash sqrt\{2\}\}\{z+\backslash cfrac\{a\_1\}\{z+\backslash cfrac\{a\_2\}\{z+\backslash cfrac\{a\_3\}\{z+\backslash ddots\}\}\}\},\backslash quad\; a\_n=\backslash frac\{n^2\}\{4\}((-1)^\{n+1\}+3)$

:$\backslash int\_0^\backslash infty\; e^\{-tz\backslash sqrt\{2\}\}\backslash operatorname\{sl\}t\backslash operatorname\{cl\}t\; \backslash ,\; \backslash mathrm\; dt=\backslash cfrac\{1/2\}\{z^2+b\_1-\backslash cfrac\{a\_1\}\{z^2+b\_2-\backslash cfrac\{a\_2\}\{z^2+b\_3-\backslash ddots\}\}\},\backslash quad\; a\_n=n^2(4n^2-1),\backslash ,\; b\_n=3(2n-1)^2$

{{quote box

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

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

| $a\_0\; \backslash leftarrow\; 1;$ $b\_0\; \backslash leftarrow\; \backslash tfrac\{1\}\{\backslash sqrt2\};$ $c\_0\; \backslash leftarrow\backslash sqrt\{\backslash tfrac12\}$

| for each $n\backslash ge\; 1$ do

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

| $a\_n\; \backslash leftarrow\; \backslash tfrac12(a\_\{n-1\}+b\_\{n-1\});$ $b\_n\; \backslash leftarrow\; \backslash sqrt\{a\_\{n-1\}b\_\{n-1\}\};$ $c\_n\; \backslash leftarrow\; \backslash tfrac12(a\_\{n-1\}-b\_\{n-1\})$

| if then

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

| $N\; \backslash leftarrow\; n;$ break

}}

}}

| $\backslash phi\_N\; \backslash leftarrow\; 2^N\; a\_N\; \backslash sqrt2x$

| for each {{math|n}} from {{math|N}} to {{math|0}} do

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

| $\backslash phi\_\{n-1\}\; \backslash leftarrow\; \backslash tfrac12\backslash left(\backslash phi\_n\; +\; \{\backslash arcsin\}\{\backslash left(\backslash frac\{c\_n\}\{a\_n\}\backslash sin\; \backslash phi\_n\backslash right)\}\backslash right)$

}}

| return $\backslash frac\{\backslash sin\; \backslash phi\_0\}\{\backslash sqrt\{2-\backslash sin^2\backslash phi\_0\}\}$

}}

This is effectively using the arithmetic-geometric mean and is based on Landen's transformations.{{harvp|Carlson|2010}} [https://dlmf.nist.gov/19.8 §19.8]

| fontsize = 90%

| qalign = left

}}

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

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

:$\backslash operatorname\{sl\}\backslash left(\backslash frac\{\backslash varpi\}\{\backslash pi\}x\backslash right)=\backslash frac\{\backslash pi\}\{\backslash varpi\}\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}\; \backslash frac\{(-1)^n\}\{\backslash cosh\; (x-(n+1/2)\backslash pi)\},\backslash quad\; x\backslash in\backslash mathbb\{C\}$

:$\backslash frac\{1\}\{\backslash operatorname\{sl\}(\backslash varpi\; x/\backslash pi)\}\; =\; \backslash frac\backslash pi\backslash varpi\; \backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}\backslash frac\{(-1)^n\}\{\{\backslash sinh\}\; \{\backslash left(x-n\backslash pi\backslash right)\}\}=\backslash frac\backslash pi\backslash varpi\; \backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}\backslash frac\{(-1)^n\}\{\backslash sin\; (x-n\backslash pi\; i)\},\backslash quad\; x\backslash in\backslash mathbb\{C\}$

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

:

:

:

The lemniscate functions can be computed more rapidly by

:

\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}

where

:$\backslash begin\{aligned\}$

\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}

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

Fourier series for the logarithm of the lemniscate sine:

:

The following series identities were discovered by Ramanujan:{{harvp|Berndt|1994}} p. 247, 248, 253

:

:

The functions $\backslash tilde\{\backslash operatorname\{sl\}\}$ and $\backslash tilde\{\backslash operatorname\{cl\}\}$ analogous to $\backslash sin$ and $\backslash cos$ on the unit circle have the following Fourier and hyperbolic series expansions:{{harvp|Reinhardt|Walker|2010a}} [https://dlmf.nist.gov/22.11.E1 §22.11.E1]

:

:

:$\backslash tilde\{\backslash operatorname\{sl\}\}\backslash ,s=\backslash frac\{\backslash pi^2\}\{\backslash varpi^2\backslash sqrt\{2\}\}\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}\backslash frac\{\backslash sinh\; (\backslash pi\; (n+s/\backslash varpi))\}\{\backslash cosh^2\; (\backslash pi\; (n+s/\backslash varpi))\},\backslash quad\; s\backslash in\backslash mathbb\{C\}$

:$\backslash tilde\{\backslash operatorname\{cl\}\}\backslash ,s=\backslash frac\{\backslash pi^2\}\{\backslash varpi^2\backslash sqrt\{2\}\}\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}\backslash frac\{(-1)^n\}\{\backslash cosh^2\; (\backslash pi\; (n+s/\backslash varpi))\},\backslash quad\; s\backslash in\backslash mathbb\{C\}$

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

:$\backslash mathrm\{sl\}\backslash Bigl(\backslash frac\backslash varpi\backslash pi\; x\backslash Bigr)\; =\; 2e^\{-\backslash pi/4\}\backslash sin\; x\backslash prod\_\{n\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1-2e^\{-2n\backslash pi\}\backslash cos\; 2x+e^\{-4n\backslash pi\}\}\{1+2e^\{-(2n-1)\backslash pi\}\backslash cos\; 2x+e^\{-(4n-2)\backslash pi\}\},\backslash quad\; x\backslash in\backslash mathbb\{C\}$

:$\backslash mathrm\{cl\}\backslash Bigl(\backslash frac\backslash varpi\backslash pi\; x\backslash Bigr)\; =\; 2e^\{-\backslash pi/4\}\backslash cos\; x\backslash prod\_\{n\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1+2e^\{-2n\backslash pi\}\backslash cos\; 2x+e^\{-4n\backslash pi\}\}\{1-2e^\{-(2n-1)\backslash pi\}\backslash cos\; 2x+e^\{-(4n-2)\backslash pi\}\},\backslash quad\; x\backslash in\backslash mathbb\{C\}$

A similar formula involving the $\backslash operatorname\{sn\}$ function can be given.{{harvp|Borwein|Borwein|1987}}

Since the lemniscate sine is a meromorphic function in the whole complex plane, it can be written as a ratio of entire functions. Gauss showed that {{math|sl}} has the following product expansion, reflecting the distribution of its zeros and poles:{{harvp|Eymard|Lafon|2004}} p. 227.

:$\backslash operatorname\{sl\}z=\backslash frac\{M(z)\}\{N(z)\}$

where

:$M(z)=z\backslash prod\_\{\backslash alpha\}\backslash left(1-\backslash frac\{z^4\}\{\backslash alpha^4\}\backslash right),\backslash quad\; N(z)=\backslash prod\_\{\backslash beta\}\backslash left(1-\backslash frac\{z^4\}\{\backslash beta^4\}\backslash right).$

Here, $\backslash alpha$ and $\backslash beta$ denote, respectively, the zeros and poles of {{math|sl}} which are in the quadrant $\backslash operatorname\{Re\}z>0,\backslash operatorname\{Im\}z\backslash ge\; 0$. A proof can be found in.{{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}} 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.More precisely, suppose $\backslash \{a\_n\backslash \}$ is a sequence of bounded complex functions on a set $S$, such that $\backslash sum\backslash left|a\_n(z)\backslash right|$ converges uniformly on $S$. If $\backslash \{n\_1,n\_2,n\_3,\backslash ldots\backslash \}$ is any permutation of $\backslash \{1,2,3,\backslash ldots\backslash \}$, then $\backslash prod\_\{n=1\}^\backslash infty\; (1+a\_n(z))=\backslash prod\_\{k=1\}^\backslash infty\; (1+a\_\{n\_k\}(z))$ for all $z\backslash in\; S$. The theorem in question then follows from the fact that there exists a bijection between the natural numbers and $\backslash alpha$'s (resp. $\backslash beta$'s).

{{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 arguments to justify interchanging of limiting operations) that

:

(where $\backslash mathrm\{H\}\_n$ are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers) and

:$\backslash frac\{N\text{'}(z)\}\{N(z)\}=(1+i)\backslash frac\{M\text{'}((1+i)z)\}\{M((1+i)z)\}-\backslash frac\{M\text{'}(z)\}\{M(z)\}.$

Therefore

:

It is known that

:

Then from

:$\backslash frac\{\backslash mathrm\; d\}\{\backslash mathrm\; dz\}\backslash frac\{\backslash operatorname\{sl\}\text{'}z\}\{\backslash operatorname\{sl\}z\}=-\backslash frac\{1\}\{\backslash operatorname\{sl\}^2z\}-\backslash operatorname\{sl\}^2z$

and

:$\backslash operatorname\{sl\}^2z=\backslash frac\{1\}\{\backslash operatorname\{sl\}^2z\}-\backslash frac\{(1+i)^2\}\{\backslash operatorname\{sl\}^2((1+i)z)\}$

we get

:

Hence

:

Therefore

:$\backslash operatorname\{sl\}z=C\backslash frac\{M(z)\}\{N(z)\}$

for some constant $C$ for but this result holds for all $z\backslash in\backslash mathbb\{C\}$ by analytic continuation. Using

:$\backslash lim\_\{z\backslash to\; 0\}\backslash frac\{\backslash operatorname\{sl\}z\}\{z\}=1$

gives $C=1$ which completes the proof. $\backslash blacksquare$

Proof by Liouville's theorem

Let

:$f(z)=\backslash frac\{M(z)\}\{N(z)\}=\backslash frac\{(1+i)M(z)^2\}\{M((1+i)z)\},$

with patches at removable singularities.

The shifting formulas

:$M(z+2\backslash varpi)=e^\{2\backslash frac\{\backslash pi\}\{\backslash varpi\}(z+\backslash varpi)\}M(z),\backslash quad\; M(z+2\backslash varpi\; i)=e^\{-2\backslash frac\{\backslash pi\}\{\backslash varpi\}(iz-\backslash varpi)\}M(z)$

imply that $f$ is an elliptic function with periods $2\backslash varpi$ and $2\backslash varpi\; i$, just as $\backslash operatorname\{sl\}$.

It follows that the function $g$ defined by

:$g(z)=\backslash frac\{\backslash operatorname\{sl\}z\}\{f(z)\},$

when patched, is an elliptic function without poles. By Liouville's theorem, it is a constant. By using $\backslash operatorname\{sl\}z=z+\backslash operatorname\{O\}(z^5)$, $M(z)=z+\backslash operatorname\{O\}(z^5)$ and $N(z)=1+\backslash operatorname\{O\}(z^4)$, this constant is $1$, which proves the theorem. $\backslash blacksquare$

{{Collapse bottom}}

Gauss conjectured that $\backslash ln\; N(\backslash varpi)=\backslash pi/2$ (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”.{{harvp|Bottazzini|Gray|2013}} p. 58 Gauss expanded the products for $M$ and $N$ as infinite series (see below). He also discovered several identities involving the functions $M$ and $N$, such as

File:The M function in the complex plane.png

File:The N function in the complex plane.png

:$N(z)=\backslash frac\{M((1+i)z)\}\{(1+i)M(z)\},\backslash quad\; z\backslash notin\; \backslash varpi\backslash mathbb\{Z\}[i]$

and

:$N(2z)=M(z)^4+N(z)^4.$

Thanks to a certain theoremMore precisely, if for each $k$, $\backslash lim\_\{n\backslash to\backslash infty\}\; a\_k(n)$ exists and there is a convergent series $\backslash sum\_\{k=1\}^\backslash infty\; M\_k$ of nonnegative real numbers such that $\backslash left|a\_k(n)\backslash right|\backslash le\; M\_k$ for all $n\backslash in\backslash mathbb\{N\}$ and $1\backslash le\; k\backslash le\; n$, then

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash sum\_\{k=1\}^n\; a\_k(n)=\backslash sum\_\{k=1\}^\backslash infty\; \backslash lim\_\{n\backslash to\backslash infty\}a\_k(n).$ on splitting limits, we are allowed to multiply out the infinite products and collect like powers of $z$. Doing so gives the following power series expansions that are convergent everywhere in the complex plane:Alternatively, it can be inferred that these expansions exist just from the analyticity of $M$ and $N$. 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 $\backslash sum\_\{\backslash alpha\}\backslash frac\{1\}\{\backslash alpha^4\}=-\backslash ,\backslash text\{the\; coefficient\; of\}\backslash ,z^5$ in the $M$ series, and infinitely many others.{{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 $\backslash varphi^\{17\}$ should be $\backslash tfrac\{107\}\{7\backslash ,410\backslash ,154\backslash ,752\backslash ,000\}$, not $\backslash tfrac\{107\}\{207\backslash ,484\backslash ,333\backslash ,056\backslash ,000\}$.If $M(z)=\backslash sum\_\{n=0\}^\backslash infty\; a\_nz^\{n+1\}$, then the coefficients $a\_n$ are given by the recurrence $a\_\{n+1\}=-\backslash frac\{1\}\{n+1\}\backslash sum\_\{k=0\}^n\; 2^\{n-k+1\}\; a\_k\; \backslash frac\{\backslash mathrm\{H\}\_\{n-k+1\}\}\{(n-k+1)!\}$ with $a\_0=1$ where $\backslash mathrm\{H\}\_n$ are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers.The power series expansions of $M$ and $N$ are useful for finding a $\backslash beta$-division polynomial for the $\backslash beta$-division of the lemniscate $\backslash mathcal\{L\}$ (where $\backslash beta=m+ni$ where $m,n\backslash in\backslash mathbb\{Z\}$ such that $m+n$ is odd). For example, suppose we want to find a $3$-division polynomial. Given that

:$M(3z)=d\_9M(z)^9+d\_5M(z)^5N(z)^4+d\_1\; M(z)N(z)^8$

for some constants $d\_1,d\_5,d\_9$, from

:$3z-2\backslash frac\{(3z)^5\}\{5!\}-36\backslash frac\{(3z)^9\}\{9!\}+\backslash operatorname\{O\}(z^\{13\})=d\_9x^9+d\_5x^5y^4+d\_1xy^8,$

where

:$x=z-2\backslash frac\{z^5\}\{5!\}-36\backslash frac\{z^9\}\{9!\}+\backslash operatorname\{O\}(z^\{13\}),\backslash quad\; y=1+2\backslash frac\{z^4\}\{4!\}-4\backslash frac\{z^8\}\{8!\}+\backslash operatorname\{O\}(z^\{12\}),$

we have

:$\backslash \{d\_1,d\_5,d\_9\backslash \}=\backslash \{3,-6,-1\backslash \}.$

Therefore, a $3$-division polynomial is

:$-X^9-6X^5+3X$

(meaning one of its roots is $\backslash operatorname\{sl\}(2\backslash varpi/3)$).

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

:$X^n=1$

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

:$\backslash sum\_\{n=0\}^\{(\backslash beta\backslash overline\{\backslash beta\}-1)/4\}a\_\{4n+1\}(\backslash beta)X^\{\backslash beta\backslash overline\{\backslash beta\}-4n\}$

where

:

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}

and so on.

:$M(z)=z-2\backslash frac\{z^5\}\{5!\}-36\backslash frac\{z^9\}\{9!\}+552\backslash frac\{z^\{13\}\}\{13!\}+\backslash cdots,\backslash quad\; z\backslash in\backslash mathbb\{C\}$

:$N(z)=1+2\backslash frac\{z^4\}\{4!\}-4\backslash frac\{z^8\}\{8!\}+408\backslash frac\{z^\{12\}\}\{12!\}+\backslash cdots,\backslash quad\; z\backslash in\backslash mathbb\{C\}.$

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

We define $S$ and $T$ by

:$S(z)=N\backslash left(\backslash frac\{z\}\{1+i\}\backslash right)^2-iM\backslash left(\backslash frac\{z\}\{1+i\}\backslash right)^2,\backslash quad\; T(z)=S(iz).$

Then the lemniscate cosine can be written as

:$\backslash operatorname\{cl\}z=\backslash frac\{S(z)\}\{T(z)\}$

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

:$S(z)=1-\backslash frac\{z^2\}\{2!\}-\backslash frac\{z^4\}\{4!\}-3\backslash frac\{z^6\}\{6!\}+17\backslash frac\{z^8\}\{8!\}-9\backslash frac\{z^\{10\}\}\{10!\}+111\backslash frac\{z^\{12\}\}\{12!\}+\backslash cdots,\backslash quad\; z\backslash in\backslash mathbb\{C\}$

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

Furthermore, the identities

:$M(2z)=2\; M(z)\; N(z)\; S(z)\; T(z),$

:$S(2z)=S(z)^4-2M(z)^4,$

:$T(2z)=T(z)^4-2M(z)^4$

and the Pythagorean-like identities

:$M(z)^2+S(z)^2=N(z)^2,$

:$M(z)^2+N(z)^2=T(z)^2$

hold for all $z\backslash in\backslash mathbb\{C\}$.

The quasi-addition formulas

:$M(z+w)M(z-w)=M(z)^2N(w)^2-N(z)^2M(w)^2,$

:$N(z+w)N(z-w)=N(z)^2N(w)^2+M(z)^2M(w)^2$

(where $z,w\backslash in\backslash mathbb\{C\}$) imply further multiplication formulas for $M$ and $N$ by recursion.For example, by the quasi-addition formulas, the duplication formulas and the Pythagorean-like identities, we have

:$M(3z)=-M(z)^9-6M(z)^5N(z)^4+3M(z)N(z)^8,$

:$N(3z)=N(z)^9+6M(z)^4N(z)^5-3M(z)^8N(z),$

so

:$\backslash operatorname\{sl\}3z=\backslash frac\{-M(z)^9-6M(z)^5N(z)^4+3M(z)N(z)^8\}\{N(z)^9+6M(z)^4N(z)^5-3M(z)^8N(z)\}.$

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

:$\backslash operatorname\{sl\}3z=\backslash frac\{-\backslash operatorname\{sl\}^9z-6\backslash operatorname\{sl\}^5z+3\backslash operatorname\{sl\}z\}\{1+6\backslash operatorname\{sl\}^4z-3\backslash operatorname\{sl\}^8z\}.$

Gauss' $M$ and $N$ satisfy the following system of differential equations:

:$M(z)M\text{'}\text{'}(z)=M\text{'}(z)^2-N(z)^2,$

:$N(z)N\text{'}\text{'}(z)=N\text{'}(z)^2+M(z)^2$

where $z\backslash in\backslash mathbb\{C\}$. Both $M$ and $N$ satisfy the differential equationGauss (1866), p. 408

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

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

:$M(z)=z\backslash exp\backslash left(-\backslash int\_0^z\backslash int\_0^w\; \backslash left(\backslash frac\{1\}\{\backslash operatorname\{sl\}^2v\}-\backslash frac\{1\}\{v^2\}\backslash right)\backslash ,\; \backslash mathrm\; dv\backslash ,\backslash mathrm\; dw\backslash right),$

:$N(z)=\backslash exp\backslash left(\backslash int\_0^z\backslash int\_0^w\; \backslash operatorname\{sl\}^2v\backslash ,\backslash mathrm\; dv\backslash ,\backslash mathrm\; dw\backslash right)$

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 Lemniscate elliptic functions § Methods of computation); the relation between $M,N$ and $\backslash theta\_1,\backslash theta\_3$ is

:$M(z)=2^\{-1/4\}e^\{\backslash pi\; z^2/(2\backslash varpi^2)\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{\backslash varpi\}\}\backslash theta\_1\backslash left(\backslash frac\{\backslash pi\; z\}\{\backslash varpi\},e^\{-\backslash pi\}\backslash right),$

:$N(z)=2^\{-1/4\}e^\{\backslash pi\; z^2/(2\backslash varpi^2)\}\backslash sqrt\{\backslash frac\{\backslash pi\}\{\backslash varpi\}\}\backslash theta\_3\backslash left(\backslash frac\{\backslash pi\; z\}\{\backslash varpi\},e^\{-\backslash pi\}\backslash right)$

where $z\backslash in\backslash mathbb\{C\}$.

The lemniscate functions are closely related to the Weierstrass elliptic function $\backslash wp(z;\; 1,\; 0)$ (the "lemniscatic case"), with invariants {{math|g₂ {{=}} 1}} and {{math|g₃ {{=}} 0}}. This lattice has fundamental periods $\backslash omega\_1\; =\; \backslash sqrt\{2\}\backslash varpi,$ and $\backslash omega\_2\; =\; i\backslash omega\_1$. The associated constants of the Weierstrass function are $e\_1=\backslash tfrac12,\backslash \; e\_2=0,\backslash \; e\_3=-\backslash tfrac12.$

The related case of a Weierstrass elliptic function with {{math|g₂ {{=}} a}}, {{math|g₃ {{=}} 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 $\backslash wp\; (z;-1,0)$ is called the "pseudolemniscatic case".{{harvp|Robinson|2019a}}

The square of the lemniscate sine can be represented as

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

where the second and third argument of $\backslash wp$ denote the lattice invariants {{math|g₂}} and {{math|g₃}}. The lemniscate sine is a rational function in the Weierstrass elliptic function and its derivative:{{harvp|Eymard|Lafon|2004}} p. 234

:$\backslash operatorname\{sl\}z=-2\backslash frac\{\backslash wp\; (z;-1,0)\}\{\backslash wp\; \text{'}(z;-1,0)\}.$

The lemniscate functions can also be written in terms of Jacobi elliptic functions. The Jacobi elliptic functions $\backslash operatorname\{sn\}$ and $\backslash operatorname\{cd\}$ with positive real elliptic modulus have an "upright" rectangular lattice aligned with real and imaginary axes. Alternately, the functions $\backslash operatorname\{sn\}$ and $\backslash operatorname\{cd\}$ with modulus {{math|i}} (and $\backslash operatorname\{sd\}$ and $\backslash operatorname\{cn\}$ with modulus $1/\backslash sqrt\{2\}$) have a square period lattice rotated 1/8 turn.{{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}}The identity $\backslash operatorname\{cl\}\; z\; =\; \{\backslash operatorname\{cn\}\}\backslash left(\backslash sqrt2z;\backslash tfrac\{1\}\{\backslash sqrt2\}\backslash right)$ can be found in {{harvp|Greenhill|1892}} p. 33.

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

:$\backslash operatorname\{cl\}\; z\; =\; \backslash operatorname\{cd\}(z;i)=\; \backslash operatorname\{dn\}(z;\backslash sqrt\{2\})=\{\backslash operatorname\{cn\}\}\backslash left(\backslash sqrt2z;\backslash tfrac\{1\}\{\backslash sqrt2\}\backslash right)$

where the second arguments denote the elliptic modulus $k$.

The functions $\backslash tilde\{\backslash operatorname\{sl\}\}$ and $\backslash tilde\{\backslash operatorname\{cl\}\}$ can also be expressed in terms of Jacobi elliptic functions:

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

:$\backslash tilde\{\backslash operatorname\{cl\}\}\backslash ,z=\backslash operatorname\{cd\}(z;i)\backslash operatorname\{nd\}(z;i)=\backslash operatorname\{dn\}(z;\backslash sqrt\{2\})\backslash operatorname\{cn\}(z;\backslash sqrt\{2\})=\backslash operatorname\{cn\}\backslash left(\backslash sqrt\{2\}z;\backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\backslash right)\backslash operatorname\{dn\}\backslash left(\backslash sqrt\{2\}z;\backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\backslash right).$

{{Unreferenced section |date=August 2024}}

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

:$$

\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)}}

For example:

:$\backslash begin\{aligned\}$

&{\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}

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

:$\backslash operatorname\{arcsl\}\; x\; =\; \backslash int\_0^x\; \backslash frac\{\backslash mathrm\; dt\}\{\backslash sqrt\{1-t^4\}\}.$

It can also be represented by the hypergeometric function:

:$\backslash operatorname\{arcsl\}x=x\backslash ,\{\}\_2F\_1\backslash bigl(\backslash tfrac12,\backslash tfrac14;\backslash tfrac54;x^4\backslash bigr)$

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:

:$\backslash operatorname\{arccl\}\; x\; =\; \backslash int\_\{x\}^\{1\}\; \backslash frac\{\backslash mathrm\; dt\}\{\backslash sqrt\{1-t^4\}\}\; =\; \backslash tfrac12\backslash varpi\; -\; \backslash operatorname\{arcsl\}x$

For {{math|x}} in the interval $-1\; \backslash leq\; x\; \backslash leq\; 1$, $\backslash operatorname\{sl\}\backslash operatorname\{arcsl\}\; x\; =\; x$ and $\backslash operatorname\{cl\}\backslash operatorname\{arccl\}\; x\; =\; x$

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

:$\backslash begin\{aligned\}$

{\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}

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

:$\backslash operatorname\{aslh\}(x)\; =\; \backslash int\_\{0\}^\{x\}\; \backslash frac\{1\}\{\backslash sqrt\{y^4\; +\; 1\}\}\; \backslash mathrm\{d\}y\; =\; \backslash tfrac\{1\}\{2\}F\backslash left(2\backslash arctan\; x;\; \backslash tfrac\{1\}\{\backslash sqrt2\}\backslash right)$

:$\backslash operatorname\{aclh\}(x)\; =\; \backslash int\_\{x\}^\{\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{y^4\; +\; 1\}\}\; \backslash mathrm\{d\}y\; =\; \backslash tfrac12\; F\backslash left(2\backslash arccot\; x;\; \backslash tfrac\{1\}\{\backslash sqrt2\}\backslash right)$

:$\backslash operatorname\{aclh\}(x)\; =\; \backslash frac\{\backslash varpi\}\{\backslash sqrt\{2\}\}\; -\; \backslash operatorname\{aslh\}(x)$

:$\backslash operatorname\{aslh\}(x)\; =\; \backslash sqrt\{2\}\backslash operatorname\{arcsl\}\backslash left(x\; \backslash Big/\; \backslash sqrt\{\backslash textstyle\; 1\; +\; \backslash sqrt\{x^4\; +\; 1\}\}\; \backslash right)$

:$\backslash operatorname\{arcsl\}(x)\; =\; \backslash sqrt\{2\}\backslash operatorname\{aslh\}\backslash left(x\; \backslash Big/\; \backslash sqrt\{\backslash textstyle\; 1\; +\; \backslash sqrt\{1\; -\; x^4\}\}\backslash right)$

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}}

:$\backslash operatorname\{arcsl\}\; x\; =\; \backslash frac\{1\}\{\backslash sqrt2\}F\backslash left(\{\backslash arcsin\}\{\backslash frac\{\backslash sqrt2x\}\{\backslash sqrt\{1+x^2\}\}\};\backslash frac\{1\}\{\backslash sqrt2\}\backslash right)$

:$\backslash operatorname\{arcsl\}\; x\; =\; 2(\backslash sqrt2-1)F\backslash left(\{\backslash arcsin\}\{\backslash frac\{(\backslash sqrt2+1)x\}\{\backslash sqrt\{1+x^2\}+1\}\};(\backslash sqrt2-1)^2\backslash right)$

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

:$\backslash begin\{aligned\}$

\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}

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

:$\backslash operatorname\{arccl\}\; x\; =\; \backslash frac\{1\}\{\backslash sqrt2\}F\backslash left(\backslash arccos\; x;\backslash frac\{1\}\{\backslash sqrt2\}\backslash right)$

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

:$\backslash int\backslash frac\{1\}\{\backslash sqrt\{1-x^4\}\}\backslash ,\backslash mathrm\; dx=\backslash operatorname\{arcsl\}\; x$

:$\backslash int\backslash frac\{1\}\{\backslash sqrt\{(x^2+1)(2x^2+1)\}\}\backslash ,\backslash mathrm\; dx=\{\backslash operatorname\{arcsl\}\}\{\backslash frac\{x\}\{\backslash sqrt\{x^2+1\}\}\}$

:$\backslash int\backslash frac\{1\}\{\backslash sqrt\{x^4+6x^2+1\}\}\backslash ,\backslash mathrm\; dx=\{\backslash operatorname\{arcsl\}\}\{\backslash frac\{\backslash sqrt2x\}\{\backslash sqrt\{\backslash sqrt\{x^4+6x^2+1\}+x^2+1\}\}\}$

:$\backslash int\backslash frac\{1\}\{\backslash sqrt\{x^4+1\}\}\backslash ,\backslash mathrm\; dx=\{\backslash sqrt2\backslash operatorname\{arcsl\}\}\{\backslash frac\{x\}\{\backslash sqrt\{\backslash sqrt\{x^4+1\}+1\}\}\}$

:$\backslash int\backslash frac\{1\}\{\backslash sqrt[4]\{(1-x^4)^3\}\}\backslash ,\backslash mathrm\; dx=\{\backslash sqrt2\backslash operatorname\{arcsl\}\}\{\backslash frac\{x\}\{\backslash sqrt\{1+\backslash sqrt\{1-x^4\}\}\}\}$

:$\backslash int\backslash frac\{1\}\{\backslash sqrt[4]\{(x^4+1)^3\}\}\backslash ,\backslash mathrm\; dx=\{\backslash operatorname\{arcsl\}\}\{\backslash frac\{x\}\{\backslash sqrt[4]\{x^4+1\}\}\}$

:$\backslash int\backslash frac\{1\}\{\backslash sqrt[4]\{(1-x^2)^3\}\}\backslash ,\backslash mathrm\; dx=\{2\backslash operatorname\{arcsl\}\}\{\backslash frac\{x\}\{1+\backslash sqrt\{1-x^2\}\}\}$

:$\backslash int\backslash frac\{1\}\{\backslash sqrt[4]\{(x^2+1)^3\}\}\backslash ,\backslash mathrm\; dx=\{2\backslash operatorname\{arcsl\}\}\{\backslash frac\{x\}\{\backslash sqrt\{x^2+1\}+1\}\}$

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

:$\backslash int\backslash sqrt\{\backslash operatorname\{sech\}\; x\}\backslash ,\backslash mathrm\; dx=\{2\backslash operatorname\{arcsl\}\}\backslash tanh\; \backslash tfrac12x$

:$\backslash int\backslash sqrt\{\backslash sec\; x\}\backslash ,\backslash mathrm\; dx=\{2\backslash operatorname\{arcsl\}\}\backslash tan\; \backslash tfrac12x$

File:The hyperbolic lemniscate sine and cosine functions of a real variable.png

File:Slh in the complex plane.png

For convenience, let $\backslash sigma=\backslash sqrt\{2\}\backslash varpi$. $\backslash sigma$ is the "squircular" analog of $\backslash pi$ (see below). The decimal expansion of $\backslash sigma$ (i.e. $3.7081\backslash ldots${{cite OEIS|A175576}}) appears in entry 34e of chapter 11 of Ramanujan's second notebook.{{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

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

:$z\; \backslash mathrel\{\backslash overset\{*\}\{=\}\}\; \backslash int\_0^\{\backslash operatorname\{slh\}\; z\}\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1\; +\; t^4\}\}\; =\; \backslash int\_\{\backslash operatorname\{clh\}\; z\}^\backslash infty\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{1\; +\; t^4\}\}$

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

The complete integral has the value:

:$\backslash int\_0^\backslash infty\; \backslash frac\{\backslash mathrm\{d\}t\}\{\backslash sqrt\{t^4\; +\; 1\}\}\; =\; \backslash tfrac14\; \backslash Beta\backslash bigl(\backslash tfrac14,\; \backslash tfrac14\backslash bigr)\; =\; \backslash frac\{\backslash sigma\}\{2\}\; =\; 1.85407\backslash ;46773\backslash ;01371\backslash ldots$

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

:$\backslash operatorname\{slh\}\; z\; =\; \{\backslash operatorname\{clh\}\}\{\backslash Bigl(\backslash frac\{\backslash sigma\}\{2\}\; -\; z\; \backslash Bigr)\}$

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

:$\backslash operatorname\{slh\}z\backslash ,\backslash operatorname\{clh\}z\; =\; 1$

The functions $\backslash operatorname\{slh\}$ and $\backslash operatorname\{clh\}$ have a square period lattice with fundamental periods $\backslash \{\backslash sigma,\backslash sigma\; i\backslash \}$.

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

:$\backslash operatorname\{slh\}\backslash bigl(\backslash sqrt2\; z\backslash bigr)\; =\; \backslash frac\{(1+\backslash operatorname\{cl\}^2\; z)\backslash operatorname\{sl\}z\}\{\backslash sqrt2\backslash operatorname\{cl\}z\}$

:$\backslash operatorname\{clh\}\backslash bigl(\backslash sqrt2\; z\backslash bigr)\; =\; \backslash frac\{(1\; +\; \backslash operatorname\{sl\}^2\; z)\backslash operatorname\{cl\}z\}\{\backslash sqrt2\backslash operatorname\{sl\}z\}$

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

:$\backslash operatorname\{slh\}z\; =\; \backslash frac\{\backslash operatorname\{sn\}(z;1/\backslash sqrt2)\}\{\backslash operatorname\{cd\}(z;1/\backslash sqrt2)\}$

:$\backslash operatorname\{clh\}z\; =\; \backslash frac\{\backslash operatorname\{cd\}(z;1/\backslash sqrt2)\}\{\backslash operatorname\{sn\}(z;1/\backslash sqrt2)\}$

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

:$\backslash operatorname\{slh\}z$

= \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} }

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

:$\backslash sinh\; z$

= -i \sin (iz)

= \frac{\sin\left(\sqrt[2]{-1}z\right) }{ \sqrt[2]{-1}}

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

File:Superellipse chamfered square.svg

In a quartic Fermat curve $x^4\; +\; y^4\; =\; 1$ (sometimes called a squircle) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle $x^2\; +\; y^2\; =\; 1$ (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 $x\; =\; 1$.{{harvp|Levin|2006}}; {{harvp|Robinson|2019b}} Just as $\backslash pi$ is the area enclosed by the circle $x^2+y^2=1$, the area enclosed by the squircle $x^4+y^4=1$ is $\backslash sigma$. Moreover,

:$M(1,1/\backslash sqrt\{2\})=\backslash frac\{\backslash pi\}\{\backslash sigma\}$

where $M$ is the arithmetic–geometric mean.

The hyperbolic lemniscate sine satisfies the argument addition identity:

:$\backslash operatorname\{slh\}(a+b)\; =\; \backslash frac\{\backslash operatorname\{slh\}a\backslash operatorname\{slh\}\text{'}b\; +\; \backslash operatorname\{slh\}b\backslash operatorname\{slh\}\text{'}a\}\{1-\backslash operatorname\{slh\}^2a\backslash ,\backslash operatorname\{slh\}^2b\}$

When $u$ is real, the derivative and the original antiderivative of $\backslash operatorname\{slh\}$ and $\backslash operatorname\{clh\}$ can be expressed in this way:

:

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:

:$\backslash text\{tlh\}(\backslash sqrt\{2\}\backslash ,u)\; =\; \backslash sin\_\{4\}(\backslash sqrt\{2\}\backslash ,u)\; =\; \backslash operatorname\{sl\}(u)\backslash sqrt\{\backslash frac\{\backslash operatorname\{cl\}^2\; u+1\}\{\backslash operatorname\{sl\}^2\; u+\backslash operatorname\{cl\}^2\; u\}\}$

:$\backslash text\{ctlh\}(\backslash sqrt\{2\}\backslash ,u)\; =\; \backslash cos\_\{4\}(\backslash sqrt\{2\}\backslash ,u)\; =\; \backslash operatorname\{cl\}(u)\backslash sqrt\{\backslash frac\{\backslash operatorname\{sl\}^2\; u+1\}\{\backslash operatorname\{sl\}^2\; u+\backslash operatorname\{cl\}^2\; u\}\}$

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 functions are valid:

:$\backslash text\{tlh\}(u)\; =\; \backslash frac\{\backslash text\{sn\}(u;\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})\}\{\backslash sqrt[4]\{\backslash text\{cd\}(u;\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})^4\; +\; \backslash text\{sn\}(u;\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})^4\}\}$

:$\backslash text\{ctlh\}(u)\; =\; \backslash frac\{\backslash text\{cd\}(u;\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})\}\{\backslash sqrt[4]\{\backslash text\{cd\}(u;\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})^4\; +\; \backslash text\{sn\}(u;\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\})^4\}\}$

When $u$ is real, the derivative and quarter period integral of $\backslash operatorname\{tlh\}$ and $\backslash operatorname\{ctlh\}$ can be expressed in this way:

:

File:Quartic Fermat curve.png

The horizontal and vertical coordinates of this superellipse are dependent on twice the enclosed area w = 2A, so the following conditions must be met:

:$x(w)^4\; +\; y(w)^4\; =\; 1$

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}w\}\; x(w)\; =\; -y(w)^3$

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}w\}\; y(w)\; =\; x(w)^3$

:$x(w\; =\; 0)\; =\; 1$

:$y(w\; =\; 0)\; =\; 0$

The solutions to this system of equations are as follows:

:$x(w)\; =\; \backslash operatorname\{cl\}(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}w)\; [\backslash operatorname\{sl\}(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}w)^2+1]^\{1/2\}\; [\backslash operatorname\{sl\}(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}w)^2+\backslash operatorname\{cl\}(\backslash tfrac\{\; 1\}\{2\}\backslash sqrt\{2\}w)^2]^\{-1/2\}$

:$y(w)\; =\; \backslash operatorname\{sl\}(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}w)\; [\backslash operatorname\{cl\}(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}w)^2+1]^\{1/2\}\; [\backslash operatorname\{sl\}(\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}w)^2+\backslash operatorname\{cl\}(\backslash tfrac\{\; 1\}\{2\}\backslash sqrt\{2\}w)^2]^\{-1/2\}$

The following therefore applies to the quotient:

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

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

:$x(w)\; =\; \backslash text\{ctlh\}(w)$

:$y(w)\; =\; \backslash text\{tlh\}(w)$

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.

{{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:

:$D(s)\; =\; \backslash sqrt\{\backslash biggl(\backslash frac\{1\}\{\backslash sqrt[4]\{s^4\; +\; 1\}\}\backslash biggr)^2\; +\; \backslash biggl(\backslash frac\{s\}\{\backslash sqrt[4]\{s^4\; +\; 1\}\}\backslash biggr)^2\}\; =\; \backslash frac\{\backslash sqrt\{s^2\; +\; 1\}\}\{\backslash sqrt[4]\{s^4\; +\; 1\}\}$

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:

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}s\}\; \backslash arctan(s)\; =\; \backslash frac\{1\}\{s^2\; +\; 1\}$

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:

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}s\}\; \backslash text\{aslh\}(s)\; =\; \backslash biggl[\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}s\}\; \backslash arctan(s)\backslash biggr]\; D(s)^2\; =\; \backslash frac\{1\}\{s^2\; +\; 1\}D(s)^2\; =\; \backslash frac\{1\}\{s^2\; +\; 1\}\backslash biggl(\backslash frac\{\backslash sqrt\{s^2\; +\; 1\}\}\{\backslash sqrt[4]\{s^4\; +\; 1\}\}\backslash biggr)^2\; =\; \backslash frac\{1\}\{\backslash sqrt\{s^4\; +\; 1\}\}$

{{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:

:$\backslash text\{atlh\}(v)\; =\; 2\backslash biggl(\backslash int\_\{0\}^\{v\}\; \backslash sqrt[4]\{1\; -\; w^4\}\; \backslash mathrm\{d\}w\backslash biggr)\; -\; v\backslash sqrt[4]\{1\; -\; v^4\}$

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}v\}\; \backslash text\{atlh\}(v)\; =\; 2\backslash sqrt[4]\{1\; -\; v^4\}\; -\; \backslash biggl(\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}v\}\; v\backslash sqrt[4]\{1\; -\; v^4\}\backslash biggr)\; =\; \backslash frac\{1\}\{(1\; -\; v^4)^\{3/4\}\}$

The following transformation applies:

:$\backslash text\{aslh\}(x)\; =\; \backslash text\{atlh\}\backslash biggl(\backslash frac\{x\}\{\backslash sqrt[4]\{x^4\; +\; 1\}\}\backslash biggr)$

And so, according to the chain rule, this derivation holds:

:$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash text\{aslh\}(x)\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash text\{atlh\}\backslash biggl(\backslash frac\{x\}\{\backslash sqrt[4]\{x^4\; +\; 1\}\}\backslash biggr)\; =\; \backslash biggl(\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\; \backslash frac\; \{x\}\{\backslash sqrt[4]\{x^4\; +\; 1\}\}\backslash biggr)\; \backslash biggl[1\; -\; \backslash biggl(\backslash frac\{x\}\{\backslash sqrt[4]\{x^4\; +\; 1\}\}\backslash biggr)^4\backslash biggr]^\{-3/4\}\; =$

:$=\; \backslash frac\{1\}\{(x^4\; +\; 1)^\{5/4\}\}\; \backslash biggl[1\; -\; \backslash biggl(\backslash frac\{x\}\{\backslash sqrt[4]\{x^4\; +\; 1\}\}\backslash biggr)^4\backslash biggr]^\{-3/4\}\; =\; \backslash frac\{1\}\{(x^4\; +\; 1)^\{5/4\}\}\; \backslash biggl(\backslash frac\{1\}\{x^4\; +\; 1\}\backslash biggr\; )^\{-3/4\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{x^4\; +\; 1\}\}$

{{Unreferenced section |date=August 2024}}

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

:$\backslash int\_0^\backslash infty\; \backslash frac\{\backslash operatorname\{d\}t\}\{\backslash sqrt\{t^4\; +\; 1\}\}\; =\; \backslash tfrac14\; \backslash Beta\backslash bigl(\backslash tfrac14,\; \backslash tfrac14\backslash bigr)\; =\; \backslash frac\{\backslash varpi\}\{\backslash sqrt2\}\; =\; \backslash frac\{\backslash sigma\}\{2\}\; =\; 1.85407\backslash ldots$

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

:$$

\operatorname{slh}\,\left(\frac{\varpi}{2\sqrt{2}}\right) = 1

:$$

\operatorname{slh}\,\left(\frac{\varpi}{3\sqrt{2}}\right) = \frac{1}{\sqrt[4]{3}}\sqrt[4]{2\sqrt{3}-3}

:$$

\operatorname{slh}\,\left(\frac{2\varpi}{3\sqrt{2}}\right) = \sqrt[4]{2\sqrt{3}+3}

:$$

\operatorname{slh}\,\left(\frac{\varpi}{4\sqrt{2}}\right) = \frac{1}{\sqrt[4]{2}}(\sqrt{\sqrt{2}+1}-1)

:$$

\operatorname{slh}\,\left(\frac{3\varpi}{4\sqrt{2}}\right) = \frac{1}{\sqrt[4]{2}}(\sqrt{\sqrt{2}+1}+1)

:$$

\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)}

:$$

\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)}

:$$

\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)}

:$$

\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)}

:$$

\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})

:$$

\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})

That table shows the most important values of the Hyperbolic Lemniscate Tangent and Cotangent functions:

{{Unreferenced section |date=August 2024}}

Given the hyperbolic lemniscate tangent ($\backslash operatorname\{tlh\}$) and hyperbolic lemniscate cotangent ($\backslash operatorname\{ctlh\}$). Recall the hyperbolic lemniscate area functions from the section on inverse functions,

:$\backslash operatorname\{aslh\}(x)\; =\; \backslash int\_\{0\}^\{x\}\; \backslash frac\{1\}\{\backslash sqrt\{y^4\; +\; 1\}\}\; \backslash mathrm\{d\}y$

:$\backslash operatorname\{aclh\}(x)\; =\; \backslash int\_\{x\}^\{\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{y^4\; +\; 1\}\}\; \backslash mathrm\{d\}y$

Then the following identities can be established,

:$\backslash text\{tlh\}\backslash bigl[\backslash text\{aslh\}(x)\backslash bigr]\; =\; \backslash text\{ctlh\}\backslash bigl[\backslash text\{aclh\}(x)\backslash bigr]\; =\; \backslash frac\{x\}\{\backslash sqrt[4]\{x^4\; +\; 1\}\}$

:$\backslash text\{ctlh\}\backslash bigl[\backslash text\{aslh\}(x)\backslash bigr]\; =\; \backslash text\{tlh\}\backslash bigl[\backslash text\{aclh\}(x)\backslash bigr]\; =\; \backslash frac\{1\}\{\backslash sqrt[4]\{x^4\; +\; 1\}\}$

hence the 4th power of $\backslash operatorname\{tlh\}$ and $\backslash operatorname\{ctlh\}$ for these arguments is equal to one,

:$\backslash text\{tlh\}\backslash bigl[\backslash text\{aslh\}(x)\backslash bigr]^4\; +\; \backslash text\{ctlh\}\backslash bigl[\backslash text\{aslh\}(x)\backslash bigr]^4=1$

:$\backslash text\{tlh\}\backslash bigl[\backslash text\{aclh\}(x)\backslash bigr]^4\; +\; \backslash text\{ctlh\}\backslash bigl[\backslash text\{aclh\}(x)\backslash bigr]^4=1$

so a 4th power version of the Pythagorean theorem. The bisection theorem of the hyperbolic sinus lemniscatus reads as follows:

:$\backslash text\{slh\}\backslash bigl[\backslash tfrac\{1\}\{2\}\backslash text\{aslh\}(x)\backslash bigr]\; =\; \backslash frac\{\backslash sqrt\{2\}x\}\{\backslash sqrt\{x^2\; +\; 1\; +\; \backslash sqrt\{x^4\; +\; 1\}\}\; +\; \backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; -\; x^2\; +\; 1\}\}$

This formula can be revealed as a combination of the following two formulas:

:$\backslash mathrm\{aslh\}(x)\; =\; \backslash sqrt\{2\}\backslash ,\backslash text\{arcsl\}\backslash bigl[x(\backslash sqrt\{x^4\; +\; 1\}\; +\; 1)^\{-1/2\}\backslash bigr]$

:$\backslash text\{arcsl\}(x)\; =\; \backslash sqrt\{2\}\backslash ,\backslash text\{aslh\}\backslash bigl(\backslash frac\{\backslash sqrt\{2\}x\}\{\backslash sqrt\{1\; +\; x^2\}\; +\; \backslash sqrt\{1\; -\; x^2\}\}\backslash bigr)$

In addition, the following formulas are valid for all real values $x\; \backslash in\; \backslash R$:

:$\backslash text\{slh\}\backslash bigl[\backslash tfrac\{1\}\{2\}\backslash text\{aclh\}(x)\backslash bigr]\; =\; \backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; x^2\; -\; \backslash sqrt\{2\}x\backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; x^2\}\}\; =\; \backslash bigl(\backslash sqrt\{x^4\; +\; 1\}\; -\; x^2\; +\; 1\backslash bigr)\; ^\{-1/2\}\backslash bigl(\backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; 1\}\; -\; x\backslash bigr)$

:$\backslash text\{clh\}\backslash bigl[\backslash tfrac\{1\}\{2\}\backslash text\{aclh\}(x)\backslash bigr]\; =\; \backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; x^2\; +\; \backslash sqrt\{2\}x\backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; x^2\}\}\; =\; \backslash bigl(\backslash sqrt\{x^4\; +\; 1\}\; -\; x^2\; +\; 1\backslash bigr)^\; \{-1/2\}\backslash bigl(\backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; 1\}\; +\; x\backslash bigr)$

These identities follow from the last-mentioned formula:

:$\backslash text\{tlh\}[\backslash tfrac\{1\}\{2\}\backslash text\{aclh\}(x)]^2\; =\; \backslash tfrac\{1\}\{2\}\backslash sqrt\{2-2\backslash sqrt\{2\}\backslash ,x\backslash sqrt\{\backslash sqrt\{x^4+1\}-x^2\}\}\; =\; \backslash bigl(2x^2\; +\; 2\; +\; 2\backslash sqrt\{x^4\; +\; 1\}\backslash bigr)^\{-1\; /2\}\backslash bigl(\backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; 1\}\; -\; x\backslash bigr)$

:$\backslash text\{ctlh\}[\backslash tfrac\{1\}\{2\}\backslash text\{aclh\}(x)]^2\; =\; \backslash tfrac\{1\}\{2\}\backslash sqrt\{2+2\backslash sqrt\{2\}\backslash ,x\backslash sqrt\{\backslash sqrt\{x^4+1\}-x^2\}\}\; =\; \backslash bigl(2x^2\; +\; 2\; +\; 2\backslash sqrt\{x^4\; +\; 1\}\backslash bigr)^\{-1\; /2\}\backslash bigl(\backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; +\; 1\}\; +\; x\backslash bigr)$

Hence, their 4th powers again equal one,

:$\backslash text\{tlh\}\backslash bigl[\backslash tfrac\{1\}\{2\}\backslash text\{aclh\}(x)\backslash bigr]^4\; +\; \backslash text\{ctlh\}\backslash bigl[\backslash tfrac\{1\}\{2\}\backslash text\{aclh\}(x)\backslash bigr]^4=1$

The following formulas for the lemniscatic sine and lemniscatic cosine are closely related:

:$\backslash text\{sl\}[\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}\backslash ,\backslash text\{aclh\}(x)]\; =\; \backslash text\{cl\}[\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}\backslash ,\backslash text\{aslh\}(x)]\; =\; \backslash sqrt\{\backslash sqrt\{x^4\; +\; 1\}\; -\; x^2\}$

:$\backslash text\{sl\}[\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}\backslash ,\backslash text\{aslh\}(x)]\; =\; \backslash text\{cl\}[\backslash tfrac\{1\}\{2\}\backslash sqrt\{2\}\backslash ,\backslash text\{aclh\}(x)]\; =\; x\backslash bigl(\backslash sqrt\{x^4\; +\; 1\}\; +\; 1\backslash bigr)^\{-1/2\}$

{{Unreferenced section |date=August 2024}}

Analogous to the determination of the improper integral in the Gaussian bell curve function, the coordinate transformation of a general cylinder can be used to calculate the integral from 0 to the positive infinity in the function $f(x)=\; \backslash exp(-x^4)$ integrated in relation to x. In the following, the proofs of both integrals are given in a parallel way of displaying.

This is the cylindrical coordinate transformation in the Gaussian bell curve function:

:$\backslash biggl[\backslash int\_\{0\}^\{\backslash infty\}\; \backslash exp(-x^2)\; \backslash ,\backslash mathrm\{d\}x\backslash biggr]^2\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash exp(-y^2-z^2)\; \backslash ,\backslash mathrm\{d\}y\; \backslash ,\backslash mathrm\{d\}z\; =$

:

:$=\; \backslash int\_\{0\}^\{\backslash pi/2\}\; \backslash int\_\{0\}^\{\backslash infty\}\; r\backslash exp(-r^2)\; \backslash ,\backslash mathrm\{d\}r\; \backslash ,\backslash mathrm\{d\}\backslash phi\; =\; \backslash int\_\{0\}^\{\backslash pi/2\}\; \backslash frac\{1\}\{2\}\; \backslash ,\backslash mathrm\{d\}\backslash phi\; =\; \backslash frac\{\backslash pi\; \}\{4\}$

And this is the analogous coordinate transformation for the lemniscatory case:

:$\backslash biggl[\backslash int\_\{0\}^\{\backslash infty\}\; \backslash exp(-x^4)\; \backslash ,\backslash mathrm\{d\}x\backslash biggr]^2\; =\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash exp(-y^4-z^4)\; \backslash ,\backslash mathrm\{d\}y\; \backslash ,\backslash mathrm\{d\}z\; =$

:

:$=\; \backslash int\_\{0\}^\{\backslash varpi/\backslash sqrt\{2\}\}\; \backslash int\_\{0\}^\{\backslash infty\}\; r\backslash exp(-r^4)\; \backslash ,\backslash mathrm\{d\}r\; \backslash ,\backslash mathrm\{d\}\backslash phi\; =\; \backslash int\_\{0\}^\{\backslash varpi/\backslash sqrt\{2\}\}\; \backslash frac\{\backslash sqrt\{\backslash pi\}\}\{4\}\; \backslash ,\backslash mathrm\{d\}\backslash phi\; =\; \backslash frac\{\backslash varpi\backslash sqrt\{\backslash pi\}\}\{4\backslash sqrt\{2\}\}$

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 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.

In algebraic number theory, every finite abelian extension of the Gaussian rationals $\backslash mathbb\{Q\}(i)$ is a subfield of $\backslash mathbb\{Q\}(i,\backslash omega\_n)$ for some positive integer $n$.{{harvp|Cox|2012}} p. 508, 509 This is analogous to the Kronecker–Weber theorem for the rational numbers $\backslash mathbb\{Q\}$ which is based on division of the circle – in particular, every finite abelian extension of $\backslash mathbb\{Q\}$ is a subfield of $\backslash mathbb\{Q\}(\backslash zeta\_n)$ for some positive integer $n$. Both are special cases of Kronecker's Jugendtraum, which became Hilbert's twelfth problem.

The field $\backslash mathbb\{Q\}(i,\backslash operatorname\{sl\}(\backslash varpi\; /n))$ (for positive odd $n$) is the extension of $\backslash mathbb\{Q\}(i)$ generated by the $x$- and $y$-coordinates of the $(1+i)n$-torsion points on the elliptic curve $y^2=4x^3+x$.

The Bernoulli numbers $\backslash mathrm\{B\}\_n$ can be defined by

:$$

\mathrm{B}_n

= \lim_{z\to 0}\frac{\mathrm d^n}{\mathrm dz^n}\frac{z}{e^z-1},\quad n\ge 0

and appear in

:$$

\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

where $\backslash zeta$ is the Riemann zeta function.

The Hurwitz numbers $\backslash mathrm\{H\}\_n,$ named after Adolf Hurwitz, are the "lemniscate analogs" of the Bernoulli numbers. They can be defined by{{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—206Equivalently, $\backslash mathrm\{H\}\_n=-\backslash lim\_\{z\backslash to\; 0\}\backslash frac\{\backslash mathrm\; d^n\}\{\backslash mathrm\; dz^n\}\; \backslash left(\backslash frac\{(1+i)z/2\}\{\backslash operatorname\{sl\}((1+i)z/2)\}+\backslash frac\{z\}\{2\}\backslash mathcal\{E\}\backslash left(\backslash frac\{z\}\{2\};i\backslash right)\backslash right)$

where $n\backslash ge\; 4$ and $\backslash mathcal\{E\}(\backslash cdot;i)$ is the Jacobi epsilon function with modulus $i$.

:$$

\mathrm{H}_n

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

where $\backslash zeta\; (\backslash cdot;1/4,0)$ is the Weierstrass zeta function with lattice invariants $1/4$ and $0$. They appear in

:$$

\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

where $\backslash mathbb\{Z\}[i]$ are the Gaussian integers and $G\_\{4n\}$ are the Eisenstein series of weight $4n$, and in

:$\backslash displaystyle\; \backslash begin\{array\}\{ll\}$

\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}

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

:$$

\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

and $\backslash mathrm\{H\}\_n=0$ if $n$ is not a multiple of $4$.The Bernoulli numbers can be determined by an analogous recurrence: $\backslash mathrm\{B\}\_\{2n\}=-\backslash frac\{1\}\{2n+1\}\backslash sum\_\{k=1\}^\{n-1\}\backslash binom\{2n\}\{2k\}\backslash mathrm\{B\}\_\{2k\}\backslash mathrm\{B\}\_\{2(n-k)\}$ where $n\backslash ge\; 2$ and $\backslash mathrm\{B\}\_2=1/6$. This yields

:$\backslash mathrm\{H\}\_8=\backslash frac\{3\}\{10\},\backslash ,\backslash mathrm\{H\}\_\{12\}=\backslash frac\{567\}\{130\},\backslash ,\backslash mathrm\{H\}\_\{16\}=\backslash frac\{43\backslash ,659\}\{170\},\backslash ,\backslash ldots$

Also{{cite journal |last1=Katz |first1=Nicholas M. |date=1975 |title=The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers |url=https://link.springer.com/article/10.1007/BF02547966 |journal=Mathematische Annalen |volume=216 |issue=1 |pages=1–4}} See eq. (9)

:$\backslash operatorname\{denom\}\backslash mathrm\{H\}\_\{4n\}=\backslash prod\_\{(p-1)|4n\}p$

where $p\backslash in\backslash mathbb\{P\}$ such that $p\backslash not\backslash equiv\; 3\backslash ,(\backslash text\{mod\}\backslash ,4),$

just as

:$\backslash operatorname\{denom\}\backslash mathrm\{B\}\_\{2n\}=\backslash prod\_\{(p-1)|2n\}p$

where $p\backslash in\backslash mathbb\{P\}$ (by the von Staudt–Clausen theorem).

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

:$$

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

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

:$$

\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.

The sequence of the integers $\backslash mathrm\{G\}\_n$ starts with $0,-1,5,253,\backslash ldots\; .$

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

Some authors instead define the Hurwitz numbers as $\backslash mathrm\{H\}\_n\text{'}=\backslash mathrm\{H\}\_\{4n\}$.

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

:$\backslash begin\{align\}$

\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}

Analogously, in terms of the Bernoulli numbers:

:$$

\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.

Let $p$ be a prime such that $p\backslash equiv\; 1\backslash ,(\backslash text\{mod\}\backslash ,4)$. A quartic residue (mod $p$) is any number congruent to the fourth power of an integer. Define $\backslash left(\backslash tfrac\{a\}\{p\}\backslash right)\_4$

to be $1$ if $a$ is a quartic residue (mod $p$) and define it to be $-1$ if $a$ is not a quartic residue (mod $p$).

If $a$ and $p$ are coprime, then there exist numbers $p\text{'}\backslash in\backslash mathbb\{Z\}[i]$ (see{{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 $\backslash varphi=\backslash operatorname\{sl\}$ and $\backslash omega=2\backslash varpi$. for these numbers) such that{{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 |url=https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.full }}

:$\backslash left(\backslash frac\{a\}\{p\}\backslash right)\_4=\backslash prod\_\{p\text{'}\}\; \backslash frac\{\backslash operatorname\{sl\}(2\backslash varpi\; ap\text{'}/p)\}\{\backslash operatorname\{sl\}(2\backslash varpi\; p\text{'}/p)\}.$

This theorem is analogous to

:$\backslash left(\backslash frac\{a\}\{p\}\backslash right)=\backslash prod\_\{n=1\}^\{\backslash frac\{p-1\}\{2\}\}\backslash frac\{\backslash sin\; (2\backslash pi\; a\; n/p)\}\{\backslash sin\; (2\backslash pi\; n/p)\}$

where $\backslash left(\backslash tfrac\{\backslash cdot\}\{\backslash cdot\}\backslash right)$ is the Legendre symbol.

File:Peirce Quincuncial Projection 1879.jpg

The Peirce quincuncial projection, designed by Charles Sanders Peirce of the US Coast Survey in the 1870s, is a world map projection based on the inverse lemniscate sine of stereographically projected points (treated as complex numbers).{{harvp|Peirce|1879}}. {{harvp|Guyou|1887}} and {{harvp|Adams|1925}} introduced 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.

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 conics, the spherical analog of planar ellipses and hyperbolas.{{harvp|Adams|1925}} 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.{{harvp|Adams|1925}}; {{harvp|Lee|1976}}. Because many partial differential equations can be effectively solved by conformal mapping, this map from sphere to cube is convenient for atmospheric modeling.{{harvp|Rančić|Purser|Mesinger|1996}}; {{harvp|McGregor|2005}}.

• 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

{{Reflist|30em}}

• {{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 |archive-url=https://ghostarchive.org/varchive/youtube/20211219/gjtTcyWL0NA |archive-date=2021-12-19 |url-status=live|series=Stand-up Maths |date=2021 |network=YouTube }}{{cbignore}}

• {{Cite episode |last=Ramalingam |first=Muthu Veerappan|title=Bernoulli Lemniscate and the Squircle $||$ A remarkable Geometric fun fact!!? |url=https://www.youtube.com/watch?v=mAzIE5OkqWE |archive-url=https://ghostarchive.org/varchive/mAzIE5OkqWE |archive-date=2024-11-10 |url-status=live|series=Act of Learning |date=2023 |network=YouTube}}{{cbignore}} Relation shown in the video amounts to $\backslash operatorname\{cl\}(\backslash sqrt\{2\}t)=\backslash frac\{\backslash cos\_4^2(t)-\backslash sin\_4^2(t)\}\{\backslash cos\_4^2(t)+\backslash sin\_4^2(t)\}$

{{refbegin|30em}}

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• {{wikicite

|ref={{harvid|Sridharan|2004}}

|reference=

Sridharan, Ramaiyengar (2004) "Physics to Mathematics: from Lintearia to Lemniscate". Resonance.
[https://www.ias.ac.in/public/Volumes/reso/009/04/0021-0029.pdf "Part I"]. 9 (4): 21–29. doi:[https://doi.org/10.1007%2FBF02834853 10.1007/BF02834853].
[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:[https://doi.org/10.1007%2FBF02839214 10.1007/BF02839214].

}}

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• {{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 }}

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Category:Modular forms

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