Lemniscate elliptic functions: Difference between revisions

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

== Lemniscate sine and cosine functions ==

=== Definitions ===

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

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

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

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

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

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

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

or as inverses of a map from the [[upper half-plane]] to a half-infinite strip with real part between <math>-\tfrac12\pi, \tfrac12\pi</math> and positive imaginary part:

: <math> z = \int_0^{\sin z}\frac{\mathrm{d}t}{\sqrt{1-t^2}} = \int_{\cos z}^1\frac{\mathrm{d}t}{\sqrt{1-t^2}}.</math>

=== Relation to the lemniscate constant ===

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

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

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

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== Argument identities ==

=== Zeros, poles, and symmetries ===

[[File:Lemniscate sine in the complex plane.svg|thumb|right|upright=1.3|<math>\operatorname{sl}</math> in the complex plane.<ref>Dark areas represent zeros, and bright areas represent poles. As the [[Argument (complex analysis)|argument]] of <math>\operatorname{sl}z</math> changes from <math>-\pi</math> (excluding <math>-\pi</math>) to <math>\pi</math>, the colors go through cyan, blue <math>(\operatorname{Arg}\approx -\pi/2)</math>, magneta, red <math>(\operatorname{Arg}\approx 0)</math>, orange, yellow <math>(\operatorname{Arg}\approx\pi/2)</math>, green, and back to cyan <math>(\operatorname{Arg}\approx\pi)</math>.</ref> In the picture, it can be seen that the fundamental periods <math>(1+i)\varpi</math> and <math>(1-i)\varpi</math> are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.]]

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

: <math>\begin{aligned}

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

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

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

: <math>\begin{aligned}

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

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

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

: <math>\begin{aligned}

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

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

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

: <math>\begin{aligned}

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

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

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

: <math>\begin{aligned}

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

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

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Also

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

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

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

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

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

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

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

=== Pythagorean-like identity ===

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

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

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

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

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

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

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

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

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

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

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

=== Derivatives and integrals ===

The derivatives are as follows:

: <math>\begin{aligned}

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

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

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

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

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

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

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

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

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

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

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

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

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=== Argument sum and multiple identities ===

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

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

: <math>\begin{aligned}

\operatorname{cl}(u+v)

&= \operatorname{cl}u\,\operatorname{cl}v \ominus \operatorname{sl}u\, \operatorname{sl}v

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

: <math>\begin{aligned}

\cos(u \pm v) &= \cos u\,\cos v \mp \sin u\,\sin v \\[6mu]

\sin(u \pm v) &= \sin u\,\cos v \pm \cos u\,\sin v

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

: <math>\begin{aligned}

\operatorname{cl}(x + iy)

&= \frac{\operatorname{cl}x - i \operatorname{sl}x\, \operatorname{sl}y\, \operatorname{cl}y}

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Gauss discovered that

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

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

Also

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

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

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

Bisection formulas:

: <math>

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

</math>

: <math>

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

</math>

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

: <math>

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

</math>

: <math>

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

</math>

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

: <math>

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

</math>

: <math>

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

</math>

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

=== Lemnatomic polynomials ===

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

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

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

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

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

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

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

and

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

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

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

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

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

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

: <math>\omega_5=\sqrt[4]{-13+6\sqrt{5}+2\sqrt{85-38\sqrt{5}}}</math>

: <math>\tilde{\omega}_5=\sqrt[4]{-13-6\sqrt{5}+2\sqrt{85+38\sqrt{5}}}</math><ref name="ReferenceA">The fourth root with the least positive [[Argument (complex analysis)|principal argument]] is chosen.</ref>

(an equivalent expression is given in the table below). Another example is<ref name="CH" />

: <math>\Lambda_{-1+2i}(x)=x^4-1+2i</math>

which is the minimal polynomial of <math>\omega_{-1+2i}</math> (and also of <math>\tilde{\omega}_{-1+2i}</math>) in <math>K[x].</math>

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

: <math>\operatorname{deg}\Lambda_{\beta}=\beta^2\prod_{p|\beta}\left(1-\frac{1}{p}\right)\left(1-\frac{(-1)^{(p-1)/2}}{p}\right)</math>

which can be compared to the cyclotomic analog

: <math>\operatorname{deg}\Phi_{k}=k\prod_{p|k}\left(1-\frac{1}{p}\right).</math>

=== Specific values ===

=== Specific values ===

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

Line 299 ⟶ 304:

| <math> 0</math>

|-

| <math> \tfrac{5}{6}</math>

| <math> -\sqrt[4]{2\sqrt{3}-3}</math>

| <math> \tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>

|-

| <math> \tfrac{3}{4}</math>

| <math> -\sqrt{\sqrt2-1}</math>

| <math> \sqrt{\sqrt2-1}</math>

|-

| <math> \tfrac{2}{3}</math>

Line 323 ⟶ 328:

| <math> \sqrt{\sqrt2-1}</math>

|-

| <math> \tfrac{1}{6}</math>

| <math> \sqrt[4]{2\sqrt{3}-3}</math>

| <math> \tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>

|}

== Relation to geometric shapes ==

=== Arc length of Bernoulli's lemniscate ===

[[File:The lemniscate sine and cosine related to the arclength of the lemniscate of Bernoulli.png|thumb|upright=1.8|The lemniscate sine and cosine relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.]]

[[File:The sine and cosine related to the arclength of the unit-diameter circle.png|thumb|upright=1.1|The trigonometric sine and cosine analogously relate the arc length of an arc of a unit-diameter circle to the distance of one endpoint from the origin.]]

Line 343 ⟶ 349:

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

: <math>\big(x(r), y(r)\big) = \biggl(\!\sqrt{\tfrac12r^2\bigl(1 + r^2\bigr)},\, \sqrt{\tfrac12r^2\bigl(1 - r^2\bigr)}\,\biggr).</math>

Using this [[Parametric equation|parametrization]] with <math>r \in [0, 1]</math> for a quarter of <math>\mathcal{L}</math>, the [[arc length]] from the origin to a point <math>\big(x(r), y(r)\big)</math> is:<ref>{{harvp|Euler|1761}}; {{harvp|Siegel|1969}}. {{harvp|Prasolov|Solovyev|1997}} use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same.</ref>

: <math>\begin{aligned}

&\int_0^r \sqrt{x'(t)^2 + y'(t)^2} \mathop{\mathrm{d}t} \\

& \quad {}= \int_0^r \sqrt{\frac{(1+2t^2)^2}{2(1+t^2)} + \frac{(1-2t^2)^2}{2(1-t^2)}} \mathop{\mathrm{d}t} \\[6mu]

Line 354 ⟶ 360:

Likewise, the arc length from <math>(1,0)</math> to <math>\big(x(r), y(r)\big)</math> is:

: <math>\begin{aligned}

&\int_r^1 \sqrt{x'(t)^2 + y'(t)^2} \mathop{\mathrm{d}t} \\

& \quad {}= \int_r^1 \frac{\mathrm{d}t}{\sqrt{1-t^4}} \\[6mu]

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

: <math>\big(x(r), y(r)\big) = \biggl(r^2,\, \sqrt{r^2\bigl(1-r^2\bigr)}\,\biggr).</math>

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

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

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

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

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

Line 377 ⟶ 383:

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

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

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

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

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

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

Line 392 ⟶ 398:

[[File:Rectangular elastica and lemniscatic sine.png|thumb|upright=1.3|The lemniscate sine relates the arc length to the x coordinate in the rectangular elastica.]]

The inverse lemniscate sine also describes the arc length {{tmath|s}} relative to the {{tmath|x}} coordinate of the rectangular [[Elastica theory|elastica]].<ref>{{harvp|Euler|1786}}; {{harvp|Sridharan|2004}}; {{harvp|Levien|2008}}</ref> This curve has {{tmath|y}} coordinate and arc length:

: <math>y = \int_x^1 \frac{t^2\mathop{\mathrm{d}t}}{\sqrt{1 - t^4}},\quad s = \operatorname{arcsl} x = \int_0^x \frac{\mathrm{d}t}{\sqrt{1 - t^4}}</math>

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

=== Elliptic characterization ===

[[File:The lemniscate elliptic functions and an ellipse.jpg|thumb|The lemniscate elliptic functions and an ellipse]]

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

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

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

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

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

== Series Identities ==

Line 409 ⟶ 415:

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

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

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

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

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

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

: <math>a_{13}=-\tfrac{2}{12\cdot 13}(a_9a_1a_1+a_1a_9a_1+a_1a_1a_9+a_5a_5a_1+a_5a_1a_5+a_1a_5a_5)=-\tfrac{11}{15600}.</math>

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

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

where

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

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

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

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

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

if <math>n</math> is odd.

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

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

where

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

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

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

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

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

where

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

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

=== Ramanujan's cos/cosh identity ===

Ramanujan's famous cos/cosh identity states that if

: <math>R(s)=\frac{\pi}{\varpi\sqrt{2}}\sum_{n\in\mathbb{Z}}\frac{\cos (2n\pi s/\varpi)}{\cosh n\pi},</math>

then<ref name="OEIS_sl_tilde">{{cite web | url=https://oeis.org/A193543 | title=A193543 - Oeis }}</ref>

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

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

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

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

and

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

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

=== Continued fractions ===

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

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

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

=== Methods of computation ===

Line 489 ⟶ 495:

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

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

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

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

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

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

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

The lemniscate functions can be computed more rapidly by

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

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

where

: <math>\begin{aligned}

\theta_1(x,e^{-\pi})&=\sum_{n\in\mathbb{Z}}(-1)^{n+1}e^{-\pi (n+1/2+x/\pi)^2}=\sum_{n\in\mathbb{Z}} (-1)^n e^{-\pi (n+1/2)^2}\sin ((2n+1)x),\\

\theta_2(x,e^{-\pi})&=\sum_{n\in\mathbb{Z}}(-1)^n e^{-\pi (n+x/\pi)^2}=\sum_{n\in\mathbb{Z}} e^{-\pi (n+1/2)^2}\cos ((2n+1)x),\\

Line 514 ⟶ 520:

Fourier series for the logarithm of the lemniscate sine:

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

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

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

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

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

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

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

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

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

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

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

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

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

Line 538 ⟶ 544:

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

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

where

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

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

Line 549 ⟶ 555:

It can be easily seen (using uniform and [[Absolute convergence|absolute]] convergence arguments to justify [[Interchange of limiting operations|interchanging of limiting operations]]) that

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

(where <math>\mathrm{H}_n</math> are the Hurwitz numbers defined in [[#Hurwitz numbers|Lemniscate elliptic functions § Hurwitz numbers]]) and

: <math>\frac{N'(z)}{N(z)}=(1+i)\frac{M'((1+i)z)}{M((1+i)z)}-\frac{M'(z)}{M(z)}.</math>

Therefore

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

It is known that

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

Then from

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

and

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

we get

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

Hence

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

Therefore

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

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

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

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

Line 573 ⟶ 579:

Let

: <math>f(z)=\frac{M(z)}{N(z)}=\frac{(1+i)M(z)^2}{M((1+i)z)},</math>

with patches at removable singularities.

The shifting formulas

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

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

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

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

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

{{Collapse bottom}}

Line 585 ⟶ 591:

[[File:The M function in the complex plane.png|thumb|The <math>M</math> function in the complex plane. The complex argument is represented by varying hue.]]

[[File:The N function in the complex plane.png|thumb|The <math>N</math> function in the complex plane. The complex argument is represented by varying hue.]]

: <math>N(z)=\frac{M((1+i)z)}{(1+i)M(z)},\quad z\notin \varpi\mathbb{Z}[i]</math>

and

: <math>N(2z)=M(z)^4+N(z)^4.</math>

Thanks to a certain theorem<ref>More precisely, if for each <math>k</math>, <math display="inline">\lim_{n\to\infty} a_k(n)</math> exists and there is a convergent series <math display="inline">\sum_{k=1}^\infty M_k</math> of nonnegative real numbers such that <math>\left|a_k(n)\right|\le M_k</math> for all <math>n\in\mathbb{N}</math> and <math>1\le k\le n</math>, then

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

: <math>M(3z)=d_9M(z)^9+d_5M(z)^5N(z)^4+d_1 M(z)N(z)^8</math>

for some constants <math>d_1,d_5,d_9</math>, from

: <math>3z-2\frac{(3z)^5}{5!}-36\frac{(3z)^9}{9!}+\operatorname{O}(z^{13})=d_9x^9+d_5x^5y^4+d_1xy^8,</math>

where

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

we have

: <math>\{d_1,d_5,d_9\}=\{3,-6,-1\}.</math>

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

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

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

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

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

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

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

where

: <math>\begin{align}a_1(\beta)&=1,\\

a_5(\beta)&=\frac{\beta^4-\beta\overline{\beta}}{12},\\

a_9(\beta)&=\frac{-\beta^8-70\beta^5\overline{\beta}+336\beta^4+35\beta^2\overline{\beta}^2-300\beta\overline{\beta}}{10080}\end{align}</math>

and so on.</ref>

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

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

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

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

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

Then the lemniscate cosine can be written as

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

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

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

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

Furthermore, the identities

: <math>M(2z)=2 M(z) N(z) S(z) T(z),</math>

: <math>S(2z)=S(z)^4-2M(z)^4,</math>

: <math>T(2z)=T(z)^4-2M(z)^4</math>

and the Pythagorean-like identities

: <math>M(z)^2+S(z)^2=N(z)^2,</math>

: <math>M(z)^2+N(z)^2=T(z)^2</math>

hold for all <math>z\in\mathbb{C}</math>.

The quasi-addition formulas

: <math>M(z+w)M(z-w)=M(z)^2N(w)^2-N(z)^2M(w)^2,</math>

: <math>N(z+w)N(z-w)=N(z)^2N(w)^2+M(z)^2M(w)^2</math>

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

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

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

so

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

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

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

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

: <math>M(z)M''(z)=M'(z)^2-N(z)^2,</math>

: <math>N(z)N''(z)=N'(z)^2+M(z)^2</math>

where <math>z\in\mathbb{C}</math>. Both <math>M</math> and <math>N</math> satisfy the differential equation<ref>Gauss (1866), p. 408</ref>

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

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

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

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

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

An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see [[#Methods of computation|Lemniscate elliptic functions § Methods of computation]]); the relation between <math>M,N</math> and <math>\theta_1,\theta_3</math> is

: <math>M(z)=2^{-1/4}e^{\pi z^2/(2\varpi^2)}\sqrt{\frac{\pi}{\varpi}}\theta_1\left(\frac{\pi z}{\varpi},e^{-\pi}\right),</math>

: <math>N(z)=2^{-1/4}e^{\pi z^2/(2\varpi^2)}\sqrt{\frac{\pi}{\varpi}}\theta_3\left(\frac{\pi z}{\varpi},e^{-\pi}\right)</math>

where <math>z\in\mathbb{C}</math>.

== Relation to other functions ==

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

The lemniscate functions are closely related to the [[Weierstrass elliptic function]] <math>\wp(z; 1, 0)</math> (the "lemniscatic case"), with invariants {{tmath|1= g_2 = 1}} and {{tmath|1= g_3 = 0}}. This lattice has fundamental periods <math>\omega_1 = \sqrt{2}\varpi,</math> and <math>\omega_2 = i\omega_1</math>. The associated constants of the Weierstrass function are <math>e_1=\tfrac12,\ e_2=0,\ e_3=-\tfrac12.</math>

Line 668 ⟶ 676:

The square of the lemniscate sine can be represented as

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

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

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

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

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

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

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

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

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

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

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

Line 690 ⟶ 698:

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

: <math>

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

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

Line 696 ⟶ 704:

For example:

: <math>\begin{aligned}

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

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

Line 708 ⟶ 716:

\end{aligned}</math>

== Inverse functions ==

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

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

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

: <math>\operatorname{arcsl}x=x\,{}_2F_1\bigl(\tfrac12,\tfrac14;\tfrac54;x^4\bigr)</math>

which can be easily seen by using the [[binomial series]].

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

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

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

Line 727 ⟶ 735:

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

: <math>\begin{aligned}

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

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

Line 734 ⟶ 742:

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

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

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

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

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

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

=== Expression using elliptic integrals ===

Line 750 ⟶ 758:

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

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

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

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

: <math>\begin{aligned}

\operatorname{arcsl} x = {}&\frac{2+\sqrt2}{2}E\left({\arcsin}{\frac{(\sqrt2+1)x}{\sqrt{1+x^2}+1}};(\sqrt2-1)^2\right) \\[5mu]

&\ \ - E\left({\arcsin}{\frac{\sqrt2x}{\sqrt{1+x^2}}};\frac{1}{\sqrt2}\right) + \frac{x\sqrt{1-x^2}}{\sqrt2(1+x^2+\sqrt{1+x^2})}

Line 763 ⟶ 771:

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

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

=== Use in integration ===

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

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

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

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

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

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

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

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

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

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

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

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

== Hyperbolic lemniscate functions ==

=== Fundamental information ===

[[File:The hyperbolic lemniscate sine and cosine functions of a real variable.png|thumb|upright=1.3|The hyperbolic lemniscate sine (red) and hyperbolic lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric tangent (pale dashed red).]]

[[File:Slh in the complex plane.png|thumb|The hyperbolic lemniscate sine in the complex plane. Dark areas represent zeros and bright areas represent poles. The complex argument is represented by varying hue.]]

Line 799 ⟶ 810:

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

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

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

Line 805 ⟶ 816:

The complete integral has the value:

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

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

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

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

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

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

Line 819 ⟶ 830:

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

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

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

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

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

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

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

: <math>\operatorname{slh}z

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

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

Line 838 ⟶ 849:

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

: <math>\sinh z

= -i \sin (iz)

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

Line 844 ⟶ 855:

=== Relation to quartic Fermat curve ===

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

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

Line 851 ⟶ 864:

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

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

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

Line 857 ⟶ 870:

The hyperbolic lemniscate sine satisfies the argument addition identity:

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

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

Line 875 ⟶ 888:

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

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

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

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

In addition, those relations to the [[Jacobi elliptic function]]s are valid:

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

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

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

: {|class = "wikitable"

|

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

Line 897 ⟶ 910:

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

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

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

: <math>x(w)^4 + y(w)^4 = 1 </math>

: <math>\frac{\mathrm{d}}{\mathrm{d}w} x(w) = -y(w)^3 </math>

: <math>\frac{\mathrm{d}}{\mathrm{d}w} y(w) = x(w)^3 </math>

: <math>x(w = 0) = 1 </math>

: <math>y(w = 0) = 0 </math>

The solutions to this system of equations are as follows:

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

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

The following therefore applies to the quotient:

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

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

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

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

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

Line 923 ⟶ 937:

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

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

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

Line 930 ⟶ 944:

The following derivation applies to this:

: <math>\frac{\mathrm{d}}{\mathrm{d}s} \arctan(s) = \frac{1}{s^2 + 1} </math>

To determine the derivation of the areasinus lemniscatus hyperbolicus, the comparison of the infinitesimally small triangular areas for the same diagonal in the superellipse and the unit circle is set up below. Because the summation of the infinitesimally small triangular areas describes the area dimensions. In the case of the superellipse in the picture, half of the area concerned is shown in green. Because of the quadratic ratio of the areas to the lengths of triangles with the same infinitesimally small angle at the origin of the coordinates, the following formula applies:

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

==== Second proof: integral formation and area subtraction ====

Line 939 ⟶ 953:

In the picture shown, the area tangent lemniscatus hyperbolicus assigns the height of the intersection of the diagonal and the curved line to twice the green area. The green area itself is created as the difference integral of the superellipse function from zero to the relevant height value minus the area of the adjacent triangle:

: <math>\text{atlh}(v) = 2\biggl(\int_{0}^{v} \sqrt[4]{1 - w^4} \mathrm{d}w\biggr) - v\sqrt[4]{1 - v^4} </math>

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

The following transformation applies:

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

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

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

: <math>= \frac{1}{(x^4 + 1)^{5/4}} \biggl[1 - \biggl(\frac{x}{\sqrt[4]{x^4 + 1}}\biggr)^4\biggr]^{-3/4} = \frac{1}{(x^4 + 1)^{5/4}} \biggl(\frac{1}{x^4 + 1}\biggr )^{-3/4} = \frac{1}{\sqrt{x^4 + 1}} </math>

=== Specific values ===

Line 954 ⟶ 968:

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

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

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

: <math>

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

</math>

: <math>

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

</math>

: <math>

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

</math>

: <math>

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

</math>

: <math>

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

</math>

: <math>

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

</math>

: <math>

\operatorname{slh}\,\left(\frac{2\varpi}{5\sqrt{2}}\right) = \frac{1}{2\sqrt[4]{2}}(\sqrt{5}+1)\sqrt{\sqrt[4]{20}-\sqrt{\sqrt{5}+1}} = 2\sqrt[4]{\sqrt{5} + 2}\sqrt{\sin(\tfrac{1}{20}\pi)\sin(\tfrac{3}{20}\pi)}

</math>

: <math>

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

</math>

: <math>

\operatorname{slh}\,\left(\frac{4\varpi}{5\sqrt{2}}\right) = \frac{1}{2\sqrt[4]{2}}(\sqrt{5}+1)\sqrt{\sqrt[4]{20}+\sqrt{\sqrt{5}+1}} = 2\sqrt[4]{\sqrt{5} + 2}\sqrt{\cos(\tfrac{1}{20}\pi)\cos(\tfrac{3}{20}\pi)}

</math>

: <math>

\operatorname{slh}\,\left(\frac{\varpi}{6\sqrt{2}}\right) = \frac{1}{2}(\sqrt{2\sqrt{3}+3}+1)(1-\sqrt[4]{2\sqrt{3}-3})

</math>

: <math>

\operatorname{slh}\,\left(\frac{5\varpi}{6\sqrt{2}}\right) = \frac{1}{2}(\sqrt{2\sqrt{3}+3}+1)(1+\sqrt[4]{2\sqrt{3}-3})

</math>

Line 994 ⟶ 1,008:

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

{| class="wikitable"

| !<math> 0z</math>

| !<math> \sigmaoperatorname{clh} z</math>

| !<math> \inftyoperatorname{slh} z</math>

!<math> \operatorname{ctlh} z = \cos_{4} z</math>

!<math> \operatorname{tlh} z = \sin_{4} z</math>

|-

! |<math>z 0</math>

! |<math> \operatorname{clh} zinfty</math>

! |<math> \operatorname{slh} z0</math>

! |<math> \operatorname{ctlh} z = \cos_{4} z1</math>

! |<math> \operatorname{tlh} z = \sin_{4} z0</math>

|-

| <math> 0{\tfrac14}\sigma</math>

| <math> \infty1</math>

| <math> 01</math>

| <math> 1\big/\sqrt[4]{2}</math>

| <math> 01\big/\sqrt[4]{2}</math>

|-

| <math> {\tfrac14tfrac12}\sigma</math>

| <math> 10</math>

| <math> 1\infty</math>

| <math> 1\big/\sqrt[4]{2}0</math>

| <math> 1\big/\sqrt[4]{2}</math>

|-

| <math> {\tfrac12tfrac34}\sigma</math>

| <math> 0-1</math>

| <math> \infty-1</math>

| <math> 0-1\big/\sqrt[4]{2}</math>

| <math> 1\big/\sqrt[4]{2}</math>

|-

| <math> {\tfrac34}\sigma</math>

| <math> -1\infty</math>

| <math> -10</math>

| <math> -1\big/\sqrt[4]{2}</math>

| <math> 1\big/\sqrt[4]{2}0</math>

|-

| <math> \sigma</math>

| <math> \infty</math>

| <math> 0</math>

| <math> -1</math>

| <math> 0</math>

|}

Line 1,037 ⟶ 1,050:

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

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

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

Then the following identities can be established,

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

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

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

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

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

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

: <math>\text{slh}\bigl[\tfrac{1}{2}\text{aslh}(x)\bigr] = \frac{\sqrt{2}x}{\sqrt{x^2 + 1 + \sqrt{x^4 + 1}} + \sqrt{\sqrt{x^4 + 1} - x^2 + 1}} </math>

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

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

: <math>\text{arcsl}(x) = \sqrt{2}\,\text{aslh}\bigl(\frac{\sqrt{2}x}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\bigr) </math>

In addition, the following formulas are valid for all real values <math>x \in \R</math>:

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

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

These identities follow from the last-mentioned formula:

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

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

Hence, their 4th powers again equal one,

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

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

: <math>\text{sl}[\tfrac{1}{2}\sqrt{2}\,\text{aclh}(x)] = \text{cl}[\tfrac{1}{2}\sqrt{2}\,\text{aslh}(x)] = \sqrt{\sqrt{x^4 + 1} - x^2} </math>

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

=== Coordinate Transformations ===

Line 1,079 ⟶ 1,092:

This is the [[polar coordinates|cylindrical coordinate transformation]] in the Gaussian bell curve function:

: <math>\biggl[\int_{0}^{\infty} \exp(-x^2) \,\mathrm{d}x\biggr]^2 = \int_{0}^{\infty} \int_{0}^{\infty} \exp(-y^2-z^2) \,\mathrm{d}y \,\mathrm{d}z = </math>

: <math>= \int_{0}^{\pi/2} \int_{0}^{\infty} \det\begin{bmatrix} \partial/\partial r\,\,r\cos(\phi) & \partial/\partial \phi\,\,r\cos(\phi) \\ \partial/\partial r\,\,r\sin(\phi) & \partial/\partial \phi\,\, r\sin(\phi) \end{bmatrix} \exp\bigl\{-\bigl[r\cos(\phi)\bigr]^2-\bigl[r\sin(\phi)\bigr]^2\bigr\} \,\mathrm{d}r \,\mathrm{d}\phi = </math>

: <math>= \int_{0}^{\pi/2} \int_{0}^{\infty} r\exp(-r^2) \,\mathrm{d}r \,\mathrm{d}\phi = \int_{0}^{\pi/2} \frac{1}{2} \,\mathrm{d}\phi = \frac{\pi }{4} </math>

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

: <math>\biggl[\int_{0}^{\infty} \exp(-x^4) \,\mathrm{d}x\biggr]^2 = \int_{0}^{\infty} \int_{0}^{\infty} \exp(-y^4-z^4) \,\mathrm{d}y \,\mathrm{d}z = </math>

: <math>= \int_{0}^{\varpi/\sqrt{2}} \int_{0}^{\infty} \det\begin{bmatrix} \partial/\partial r\,\,r\,\text{ctlh}(\phi) & \partial/\partial \phi\,\,r\,\text{ctlh}(\phi) \\ \partial/\partial r\,\,r\, \text{tlh}(\phi) & \partial/\partial \phi\,\,r\,\text{tlh}(\phi) \end{bmatrix} \exp\bigl\{-\bigl[r\,\text{ctlh}(\phi)\bigr]^4-\bigl[r\,\text{tlh}(\phi)\bigr]^4\bigr\} \,\mathrm{d}r \,\mathrm{d }\phi = </math>

: <math>= \int_{0}^{\varpi/\sqrt{2}} \int_{0}^{\infty} r\exp(-r^4) \,\mathrm{d}r \,\mathrm{d}\phi = \int_{0}^{\varpi/\sqrt{2}} \frac{\sqrt{\pi}}{4} \,\mathrm{d}\phi = \frac{\varpi\sqrt{\pi}}{4\sqrt{2}} </math>

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

Line 1,099 ⟶ 1,112:

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

=== Hurwitz numbers ===

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

: <math>

\mathrm{B}_n

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

Line 1,109 ⟶ 1,122:

and appear in

: <math>

\sum_{k\in\mathbb{Z}\setminus\{0\}}\frac{1}{k^{2n}}

= (-1)^{n-1}\mathrm{B}_{2n}\frac{(2\pi)^{2n}}{(2n)!}=2\zeta (2n),\quad n\ge 1

Line 1,119 ⟶ 1,132:

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

: <math>

\mathrm{H}_n

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

Line 1,126 ⟶ 1,139:

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

: <math>

\sum_{z\in\mathbb{Z}[i]\setminus\{0\}}\frac{1}{z^{4n}}

= \mathrm{H}_{4n}\frac{(2\varpi)^{4n}}{(4n)!}

Line 1,134 ⟶ 1,147:

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

: <math>\displaystyle \begin{array}{ll}

\displaystyle\sum_{n=1}^\infty\dfrac{n^k}{e^{2\pi n}-1} = \begin{cases}

\dfrac{1}{24}-\dfrac{1}{8\pi} & {\text{if}}\ k=1 \\

Line 1,144 ⟶ 1,157:

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

: <math>

\mathrm{H}_{4n}

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

Line 1,150 ⟶ 1,163:

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

: <math>\mathrm{H}_8=\frac{3}{10},\,\mathrm{H}_{12}=\frac{567}{130},\,\mathrm{H}_{16}=\frac{43\,659}{170},\,\ldots</math>

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

: <math>\operatorname{denom}\mathrm{H}_{4n}=\prod_{(p-1)|4n}p</math>

where <math>p\in\mathbb{P}</math> such that <math>p\not\equiv 3\,(\text{mod}\,4),</math>

just as

: <math>\operatorname{denom}\mathrm{B}_{2n}=\prod_{(p-1)|2n}p</math>

where <math>p\in\mathbb{P}</math> (by the [[von Staudt–Clausen theorem]]).

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

: <math>

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

</math>

Line 1,167 ⟶ 1,180:

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

: <math>

\mathrm{H}_{4n}-\frac{1}{2}-\sum_{(p-1)|4n}\frac{\nu (p)^{4n/(p-1)}}{p}

\mathrel{\overset{\text{def}}{=}} \mathrm{G}_n\in\mathbb{Z},\quad n\ge 1.

Line 1,178 ⟶ 1,191:

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

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

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

: <math>\begin{align}

\operatorname{sl}^2z

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

Line 1,198 ⟶ 1,211:

Analogously, in terms of the Bernoulli numbers:

: <math>

\frac{1}{\sinh^2 z}

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

Line 1,204 ⟶ 1,217:

</math>

=== A quartic analog of the Legendre symbol ===

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

to be <math>1</math> if <math>a</math> is a quartic residue (mod <math>p</math>) and define it to be <math>-1</math> if <math>a</math> is not a quartic residue (mod <math>p</math>).

Line 1,210 ⟶ 1,223:

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

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

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

This theorem is analogous to

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

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

== World map projections ==

[[File:Peirce Quincuncial Projection 1879.jpg|thumb|upright=1.3|"The World on a Quincuncial Projection", from {{harvp|Peirce|1879}}.]]

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

Line 1,223 ⟶ 1,237:

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

== See also ==

* [[Elliptic function]]

** [[Abel elliptic functions]]

Line 1,330 ⟶ 1,344:

== External links ==

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