Lemniscate elliptic functions explained

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.^[1]

x ^{2+y} ^{2} = x, ^[3] the lemniscate sine relates the arc length to the chord length of a lemniscate l (x ^{2+y} ^{2r){}} ^{2=x} ^{2-y} ^{2.}

The lemniscate functions have periods related to a number called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic), ratio of perimeter to diameter of a circle.

\{( 1 + i)\varpi, ( 1 - i)\varpi\}, ^[4] and are a special case of two Jacobi elliptic functions on that lattice, \operatorname{sl} z = \operatorname{sn}(z; i), \operatorname{cl} z = \operatorname{cd}(z; i) .

Similarly, the hyperbolic lemniscate sine and hyperbolic lemniscate cosine have a square period lattice with fundamental periods

l \{\sqrt 2 \varpi, \sqrt 2 \varpi i r \}. \wp (z;a, 0 ) .

Lemniscate sine and cosine functions

Definitions

The lemniscate functions and can be defined as the solution to the initial value problem:^[5]

(d)/(d z) \operatorname{sl} z = l ( 1 + \operatorname{sl} ^{2} z r )\operatorname{cl}z, (d)/(d z) \operatorname{cl} z = - l ( 1 + \operatorname{cl} ^{2} z r )\operatorname{sl}z, \operatorname{sl} 0 = 0 , \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

\ {\tfrac 12 \varpi, \tfrac 12 \varpi i, - \tfrac 12 \varpi, - \tfrac 12 \varpi i \ }\colon ^[6] z = \int ^{\operatorname{sl} _{0} z} (d t)/(\sqrt{ 1- t ^{4}) } = \int_^1\frac.

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:

(d)/(d z) \sin z = \cos z, (d)/(d z) \cos z = - \sin z, \sin 0 = 0 , \cos 0 = 1 ,

or as inverses of a map from the upper half-plane to a half-infinite strip with real part between

- \tfrac 12 \pi, \tfrac 12 \pi and positive imaginary part:z = \int ^{\sin z} _{0} (d t)/(\sqrt{ 1- t ^{2}) } = \int_^1\frac.

Relation to the lemniscate constant

See main article: Lemniscate constant.

The lemniscate functions have minimal real period, minimal imaginary period and fundamental complex periods

( 1+ i)\varpi and ( 1- i)\varpi for a constant called the lemniscate constant,^[7] \varpi = 2 \int ^{1 (d t)/(\sqrt{ 1- t ^{4})} _{0} } = 2.62205\ldots

The lemniscate functions satisfy the basic relation

\operatorname{cl}z = {\operatorname{sl}} l (\tfrac 12 \varpi - z r ), analogous to the relation \cos z = {\sin} l (\tfrac 12 \pi - z r ).

The lemniscate constant is a close analog of the circle constant , and many identities involving have analogues involving, as identities involving the trigonometric functions have analogues involving the lemniscate functions. For example, Viète's formula for can be written:

$$\backslash frac2\backslash pi\; =\; \backslash sqrt\backslash frac12\; \backslash cdot\; \backslash sqrt\; \backslash cdot\; \backslash sqrt\; \backslash cdots$$

An analogous formula for is:

$$\backslash frac2\backslash varpi\; =\; \backslash sqrt\backslash frac12\; \backslash cdot\; \backslash sqrt\; \backslash cdot\; \backslash sqrt\; \backslash cdots$$

The Machin formula for is $\backslash tfrac14\backslash pi\; =\; 4\; \backslash arctan\; \backslash tfrac15\; -\; \backslash arctan\; \backslash tfrac1,$ and several similar formulas for 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, including the following found by Gauss:

\tfrac 12 \varpi = 2 \operatorname{arcsl} \tfrac 12 + \operatorname{arcsl} \tfrac 7 { 23 }.

The lemniscate and circle constants were found by Gauss to be related to each-other by the arithmetic-geometric mean :

$$\backslash frac\backslash pi\backslash varpi\; =\; M$$

Argument identities

Zeros, poles and symmetries

The lemniscate functions and are even and odd functions, respectively,

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

At translations of

\tfrac 12 \varpi, and are exchanged, and at translations of \tfrac 12 i\varpi they are additionally rotated and reciprocated:^[8] \begin{aligned} {\operatorname{cl}} l (z \pm \tfrac 12 \varpi r ) & = \mp\operatorname{sl} z,& {\operatorname{cl}} l (z \pm \tfrac 12 i\varpi r ) & = (\mp i)/(\operatorname{sl) z} \\[ 6 mu] {\operatorname{sl}} l (z \pm \tfrac 12 \varpi r ) & = \pm\operatorname{cl} z,& {\operatorname{sl}} l (z \pm \tfrac 12 i\varpi r ) & = (\pm i)/(\operatorname{cl) z} \end{aligned}

Doubling these to translations by a unit-Gaussian-integer multiple of

\varpi (that is, \pm \varpi or \pm i\varpi ), negates each function, an involution:\begin{aligned} \operatorname{cl} (z + \varpi) & = \operatorname{cl} (z + i\varpi) = - \operatorname{cl} z \\[ 4 mu] \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

\varpi .^[9] That is, a displacement (a + bi)\varpi, with a + b = 2 k for integers,, and .\begin{aligned} {\operatorname{cl}} l (z + ( 1 + i)\varpi r ) & = {\operatorname{cl}} l (z + ( 1 - i)\varpi r ) = \operatorname{cl} z \\[ 4 mu] {\operatorname{sl}} l (z + ( 1 + i)\varpi r ) & = {\operatorname{sl}} l (z + ( 1 - i)\varpi r ) = \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)\varpi and ( 1 - i)\varpi . 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:

\begin{aligned} \operatorname{cl} \bar{z} & = \overline{\operatorname{cl} z} \\[ 6 mu] \operatorname{sl} \bar{z} & = \overline{\operatorname{sl} z} \\[ 4 mu] \operatorname{cl} iz & = (1)/(\operatorname{cl) z} \\[ 6 mu] \operatorname{sl} iz & = i \operatorname{sl} z \end{aligned}

The function has simple zeros at Gaussian integer multiples of, complex numbers of the form

a\varpi + b\varpi i for integers and . It has simple poles at Gaussian half-integer multiples of, complex numbers of the form l (a + \tfrac 12 r )\varpi + l (b + \tfrac 12 r )\varpi i , with residues ( -1 ) ^{a-b+1} i . The function is reflected and offset from the function, \operatorname{cl}z = {\operatorname{sl}} l (\tfrac 12 \varpi - z r ) . It has zeros for arguments l (a + \tfrac 12 r )\varpi + b\varpi i and poles for arguments a\varpi + l (b + \tfrac 12 r )\varpi i, with residues ( -1 ) ^{a-b} i.

Also

\operatorname{sl}z = \operatorname{sl}w\leftrightarrow z = ( -1 ) ^{m+n} w + (m + ni)\varpi for some m,n\in Z and\operatorname{sl}(( 1 \pm i)z) = ( 1 \pm i) (\operatorname{sl)/(z}{\operatorname{sl}'z}.) The last formula is a special case of complex multiplication. Analogous formulas can be given for \operatorname{sl}((n + mi)z) where n + mi is any Gaussian integer – the function \operatorname{sl} has complex multiplication by Z [i] .

There are also infinite series reflecting the distribution of the zeros and poles of :^{[10] [11]}

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

Pythagorean-like identity

The lemniscate functions satisfy a Pythagorean-like identity:

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

As a result, the parametric equation

(x, y) = (\operatorname{cl} t, \operatorname{sl} t) parametrizes the quartic curve x ^{2} + y ^{2} + x ^{2y} ^{2} = 1 .

This identity can alternately be rewritten:^[12]

l ( 1 + \operatorname{cl ^{2}} z r ) l ( 1+ \operatorname{sl ^{2}} z r ) = 2 \operatorname{cl ^{2}} z = (1- \operatorname{sl ^{2})/(z}{ 1) + \operatorname{sl ^{2}} z}, \operatorname{sl ^{2}} z = (1- \operatorname{cl ^{2})/(z}{ 1) + \operatorname{cl ^{2}} z}

Defining a tangent-sum operator as

a ⊕ b l{:=} \tan(\arctan a + \arctan b) = (a + b)/(1- ab) , gives:\operatorname{cl ^{2}} z ⊕ \operatorname{sl ^{2}} z = 1 .

The functions

\tilde{\operatorname{cl}} and \tilde{\operatorname{sl}} satisfy another Pythagorean-like identity:\left(\int ^{x} _{0} \tilde{\operatorname{cl}}t dt\right) ^{x} _{0} \tilde{\operatorname{sl}}t dt\right) ^{2=1.}

Derivatives and integrals

The derivatives are as follows:

\begin{aligned} (d)/(d z) \operatorname{cl} z = \operatorname{cl'}z & = - l ( 1 + \operatorname{cl ^{2}} z r )\operatorname{sl}z =- (2 \operatorname{sl)/(z}{\operatorname{sl}) ^{2z+1}} \\ \operatorname{cl' ^{2}} z & = 1 - \operatorname{cl ^{4}} z \\[ 5 mu] (d)/(d z) \operatorname{sl} z = \operatorname{sl'}z & = l ( 1 + \operatorname{sl ^{2}} z r )\operatorname{cl}z = (2 \operatorname{cl)/(z}{\operatorname{cl}) ^{2z+1}\\ \operatorname{sl'} ^{2}} z & = 1 - \operatorname{sl ^{4}} z\end{aligned} \begin{align} (d)/(\tilde{\operatorname{cl}}z& =-2 \tilde{\operatorname{sl}}z\operatorname{cl}z - dz) (\tilde{\operatorname{sl)/() }\,z}\\

\frac\,\tilde\,z&=2\,\tilde\,z\,\operatornamez-\frac

\end

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

(d ^{2})/(d z ^{2}) \operatorname{cl}z = -2 \operatorname{cl ^{3}z} (d ^{2})/(d z ^{2}) \operatorname{sl}z = -2 \operatorname{sl ^{3}z}

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

\begin{align}\int\operatorname{cl} z d z & = \arctan \operatorname{sl} z + C\\ \int\operatorname{sl} z d z & = - \arctan \operatorname{cl} z + C\\ \int\tilde{\operatorname{cl}}z dz&= (\tilde{\operatorname{sl)/() }\,z}+C\\\int\tilde\,z\,\mathrm dz&=-\frac+C\end

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:

\operatorname{sl}(u + v) = (\operatorname{sl)/(u\operatorname{sl'}v) + \operatorname{sl}v\operatorname{sl'}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 and . Defining a tangent-sum operator

a ⊕ b l{:=} \tan(\arctan a + \arctan b) and tangent-difference operator a \ominus b l{:=} a ⊕ ( - b), the argument sum and difference identities can be expressed as:^[13] \begin{aligned} \operatorname{cl}(u + v) & = \operatorname{cl}u\operatorname{cl}v \ominus \operatorname{sl}u \operatorname{sl}v = (\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} \\[ 2 mu] \operatorname{cl}(u - v) & = \operatorname{cl}u\operatorname{cl}v ⊕ \operatorname{sl}u \operatorname{sl}v \\[ 2 mu] \operatorname{sl}(u + v) & = \operatorname{sl}u\operatorname{cl}v ⊕ \operatorname{cl}u\operatorname{sl}v = (\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} \\[ 2 mu] \operatorname{sl}(u - v) & = \operatorname{sl}u\operatorname{cl}v \ominus \operatorname{cl}u\operatorname{sl}v \end{aligned}

These resemble their trigonometric analogs:

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

\begin{aligned} \operatorname{cl}(x + iy) & = (\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} \\[ 4 mu] & = (\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 (\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) & = (\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 } \\[ 4 mu] & = (\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 (\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}}

Bisection formulas:

\operatorname{cl} ^{2} \tfrac 12 x = (1+ \operatorname{cl)/(x) \sqrt{ 1+ \operatorname{sl} ^{2x}}{\sqrt{1+\operatorname{sl}} ^{2x}+1}} \operatorname{sl} ^{2} \tfrac 12 x = (1- \operatorname{cl)/(x\sqrt{ 1+ \operatorname{sl}) ^{2x}}{\sqrt{1+\operatorname{sl}} ^{2x}+1}}

Duplication formulas:^[14]

\operatorname{cl} 2 x = (-1+2 \operatorname{cl)/(^{2x}) + \operatorname{cl} ^{4x}{1+2\operatorname{cl}} ^{2x} - \operatorname{cl} ^{4x}} \operatorname{sl} 2 x =2 \operatorname{sl}x\operatorname{cl}x (1+ \operatorname{sl)/(^{2x}{1+\operatorname{sl}}) ^{4x}}

Triplication formulas:

\operatorname{cl} 3 x = (-3 \operatorname{cl)/(x) + 6 \operatorname{cl} ^{5x} + \operatorname{cl} ^{9x}{1+6\operatorname{cl}} ^{4x} - 3 \operatorname{cl} ^{8x}} \operatorname{sl} 3 x = (\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

\operatorname{sl} 3 x . This phenomenon can be observed in multiplication formulas for \operatorname{sl}\beta x where \beta = m + ni whenever m,n\in Z and m + n is odd.^[15] wherexP _{\beta} (x ^{4 ) = \prod} _{\gamma |\beta} Λ _{\gamma} (x) andΛ _{\beta} (x) = \prod _{[\alpha]\in} /\beta l{O}) ^{x }(x - \operatorname{sl}\alpha\delta} _{\beta)} where \delta _{\beta} is any \beta -torsion generator (i.e. \delta _{\beta} \in ( 1 /\beta)L and [\delta _{\beta]\in} ( 1 /\beta)L/L generates ( 1 /\beta)L/L as an l{O} -module). Examples of \beta -torsion generators include 2 \varpi/\beta and ( 1+ i)\varpi/\beta . The polynomial Λ _{\beta} (x)\in l{O}[x] is called the \beta -th lemnatomic polynomial. It is monic and is irreducible over K . The lemnatomic polynomials are the "lemniscate analogs" of the cyclotomic polynomials,\Phi _{k(x)=\prod} ^{[a]\in ( Z /k Z ) ^{x}} (x - \zeta ^{a).} _{k}

The

\beta -th lemnatomic polynomial Λ _{\beta(x)} is the minimal polynomial of \operatorname{sl}\delta _{\beta} in K[x] . For convenience, let \omega _{\beta} = \operatorname{sl}( 2 \varpi/\beta) and \tilde{\omega} _{\beta} = \operatorname{sl}(( 1+ i)\varpi/\beta) . So for example, the minimal polynomial of \omega _{5} (and also of \tilde{\omega} _{5} ) in K[x] isΛ ^{16} _{5 (x) = x} +52 x ^{12} -26 x ^{8-12x} ^{4+1,} and\omega _{5=\sqrt[4]{-13+6\sqrt{5}+2\sqrt{85-38\sqrt{5}}}} \tilde{\omega} _{5=\sqrt[4]{-13-6\sqrt{5}+2\sqrt{85+38\sqrt{5}}}} ^[17] (an equivalent expression is given in the table below). Another example isΛ _{-1+2i} (x) = x ^{4-1+2i} which is the minimal polynomial of \omega _{-1+2i} (and also of \tilde{\omega} _{-1+2i} ) in K[x].

If

p is prime and \beta is positive and odd,^[18] then^[19] \operatorname{deg} Λ _{\beta} = \beta ^{2 \prod} _{p|\beta} \left( 1- (1)/(\right)\left( 1- p) (( -1 ) ^{(p-1)/2})/(p) \right) which can be compared to the cyclotomic analog\operatorname{deg}\Phi _{k} = k\prod _{p|k} \left( 1- (1)/(p) \right).

Specific values

Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into parts of equal length, using only basic arithmetic and square roots, if and only if is of the form

n = 2 ^{kp} _{1 p} _{2 …} p _{m} where is a non-negative integer and each (if any) is a distinct Fermat prime.

n	\operatorname{cl}n\varpi	\operatorname{sl}n\varpi
1	-1	0
\tfrac{ 5 }{ 6 }	- \sqrt[ 4 ]{ 2 \sqrt{ 3 } -3 }	\tfrac 12 l (\sqrt{ 3 } +1- \sqrt[ 4 ]{ 12 } r )
\tfrac{ 3 }{ 4 }	- \sqrt{\sqrt 2-1 }	\sqrt{\sqrt 2-1 }
\tfrac{ 2 }{ 3 }	- \tfrac 12 l (\sqrt{ 3 } +1- \sqrt[ 4 ]{ 12 } r )	\sqrt[ 4 ]{ 2 \sqrt{ 3 } -3 }
\tfrac{ 1 }{ 2 }	0	1
\tfrac{ 1 }{ 3 }	\tfrac 12 l (\sqrt{ 3 } +1- \sqrt[ 4 ]{ 12 } r )	\sqrt[ 4 ]{ 2 \sqrt{ 3 } -3 }
\tfrac{ 1 }{ 4 }	\sqrt{\sqrt 2-1 }	\sqrt{\sqrt 2-1 }
\tfrac{ 1 }{ 6 }	\sqrt[ 4 ]{ 2 \sqrt{ 3 } -3 }	\tfrac 12 l (\sqrt{ 3 } +1- \sqrt[ 4 ]{ 12 } r )

Further values

n	\operatorname{cl}n\varpi	\operatorname{sl}n\varpi
\tfrac{ 3 }{ 7 }	\tanh l \{\tfrac{ 1 }{ 2 }\operatorname{arcoth} l [\tfrac{ 1 }{ 2 }\sqrt{ 2 \cos(\tfrac{ 3 }{ 14 }\pi)\cot(\tfrac{ 1 }{ 28 }\pi)} + \cos(\tfrac{ 1 }{ 7 }\pi) r ] r \}
\tfrac{ 5 }{ 12 }	\tfrac{ 1 }{ 2 }\sqrt[ 4 ]{ 8 }\left[\sin\left(\tfrac{ 5 }{ 24 }\pi\right) - \sqrt[ 4 ]{ 3 }\sin\left(\tfrac{ 1 }{ 24 }\pi\right)\right] l (\sqrt[ 4 ]{ 2 \sqrt{ 3 } + 3 } - 1 r )	\tfrac{ 1 }{ 2 }\sqrt[ 4 ]{ 8 }\left[\sin\left(\tfrac{ 5 }{ 24 }\pi\right) - \sqrt[ 4 ]{ 3 }\sin\left(\tfrac{ 1 }{ 24 }\pi\right)\right] l (\sqrt[ 4 ]{ 2 \sqrt{ 3 } + 3 } + 1 r )
\tfrac{ 2 }{ 5 }	\tfrac{ 1 }{ 2 }(\sqrt[ 4 ]{ 5 } -1 ) l (\sqrt{\sqrt{ 5 } +2 } -1 r )	2 \sqrt[ 4 ]{\sqrt{ 5 } - 2 }\sqrt{\sin(\tfrac{ 3 }{ 20 }\pi)\cos(\tfrac{ 1 }{ 20 }\pi)}
\tfrac{ 3 }{ 8 }	\sqrt{ l (\sqrt[ 4 ]{ 2 } -1 r ) l (\sqrt 2+1- \sqrt{ 2+ \sqrt 2 } r )}	\sqrt{ l (\sqrt[ 4 ]{ 2 } -1 r ) l (\sqrt 2+1+ \sqrt{ 2+ \sqrt 2 } r )}
\tfrac{ 5 }{ 14 }		\tanh l \{\tfrac{ 1 }{ 2 }\operatorname{arcoth} l [\tfrac{ 1 }{ 2 }\sqrt{ 2 \sin(\tfrac{ 1 }{ 7 }\pi)\cot(\tfrac{ 3 }{ 28 }\pi)} + \sin(\tfrac{ 1 }{ 14 }\pi) r ] r \}
\tfrac{ 3 }{ 10 }	2 \sqrt[ 4 ]{\sqrt{ 5 } - 2 }\sqrt{\sin(\tfrac{ 1 }{ 20 }\pi)\cos(\tfrac{ 3 }{ 20 }\pi)}	\tfrac 12 l (\sqrt[ 4 ]{ 5 } -1 r ) l (\sqrt{\sqrt{ 5 } +2 } +1 r )
\tfrac{ 2 }{ 7 }	\tanh l \{\tfrac{ 1 }{ 2 }\operatorname{arcoth} l [\tfrac{ 1 }{ 2 }\sqrt{ 2 \cos(\tfrac{ 1 }{ 14 }\pi)\tan(\tfrac{ 5 }{ 28 }\pi)} + \sin(\tfrac{ 3 }{ 14 }\pi) r ] r \}
\tfrac{ 3 }{ 14 }		\tanh l \{\tfrac{ 1 }{ 2 }\operatorname{arcoth} l [\tfrac{ 1 }{ 2 }\sqrt{ 2 \cos(\tfrac{ 1 }{ 14 }\pi)\tan(\tfrac{ 5 }{ 28 }\pi)} + \sin(\tfrac{ 3 }{ 14 }\pi) r ] r \}
\tfrac{ 1 }{ 5 }	\tfrac 12 l (\sqrt[ 4 ]{ 5 } -1 r ) l (\sqrt{\sqrt{ 5 } +2 } +1 r )	2 \sqrt[ 4 ]{\sqrt{ 5 } - 2 }\sqrt{\sin(\tfrac{ 1 }{ 20 }\pi)\cos(\tfrac{ 3 }{ 20 }\pi)}
\tfrac{ 1 }{ 7 }	\tanh l \{\tfrac{ 1 }{ 2 }\operatorname{arcoth} l [\tfrac{ 1 }{ 2 }\sqrt{ 2 \sin(\tfrac{ 1 }{ 7 }\pi)\cot(\tfrac{ 3 }{ 28 }\pi)} + \sin(\tfrac{ 1 }{ 14 }\pi) r ] r \}
\tfrac{ 1 }{ 8 }	\sqrt{ l (\sqrt[ 4 ]{ 2 } -1 r ) l (\sqrt 2+1+ \sqrt{ 2+ \sqrt 2 } r )}	\sqrt{ l (\sqrt[ 4 ]{ 2 } -1 r ) l (\sqrt 2+1- \sqrt{ 2+ \sqrt 2 } r )}
\tfrac{ 1 }{ 10 }	2 \sqrt[ 4 ]{\sqrt{ 5 } - 2 }\sqrt{\sin(\tfrac{ 3 }{ 20 }\pi)\cos(\tfrac{ 1 }{ 20 }\pi)}	\tfrac{ 1 }{ 2 }(\sqrt[ 4 ]{ 5 } -1 ) l (\sqrt{\sqrt{ 5 } +2 } -1 r )
\tfrac{ 1 }{ 12 }	\tfrac{ 1 }{ 2 }\sqrt[ 4 ]{ 8 }\left[\sin\left(\tfrac{ 5 }{ 24 }\pi\right) - \sqrt[ 4 ]{ 3 }\sin\left(\tfrac{ 1 }{ 24 }\pi\right)\right] l (\sqrt[ 4 ]{ 2 \sqrt{ 3 } + 3 } + 1 r )	\tfrac{ 1 }{ 2 }\sqrt[ 4 ]{ 8 }\left[\sin\left(\tfrac{ 5 }{ 24 }\pi\right) - \sqrt[ 4 ]{ 3 }\sin\left(\tfrac{ 1 }{ 24 }\pi\right)\right] l (\sqrt[ 4 ]{ 2 \sqrt{ 3 } + 3 } - 1 r )
\tfrac{ 1 }{ 14 }		\tanh l \{\tfrac{ 1 }{ 2 }\operatorname{arcoth} l [\tfrac{ 1 }{ 2 }\sqrt{ 2 \cos(\tfrac{ 3 }{ 14 }\pi)\cot(\tfrac{ 1 }{ 28 }\pi)} + \cos(\tfrac{ 1 }{ 7 }\pi) r ] r \}

Relation to geometric shapes

Arc length of Bernoulli's lemniscate

l{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' be the reflection of B about A . Then l{L} is the closure of the locus of the points P such that |APB - APB'| is a right angle.^[20]

Focal characterization:

l{L} is the locus of points in the plane such that the product of their distances from the two focal points F _{1} = l ({ - \tfrac 1 \sqrt 2 }, 0 r ) and F _{2} = l (\tfrac 1 \sqrt 2 , 0 r ) is the constant \tfrac 12 .

Explicit coordinate characterization:

l{L} is a quartic curve satisfying the polar equation r ^{2} = \cos 2 \theta or the Cartesian equation l (x ^{2+y} ^{2r){}} ^{2=x} ^{2-y} ^{2.}

The perimeter of

l{L} is 2 \varpi .

The points on

l{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:( x(r), y(r) ) = l (\sqrt{\tfrac 12 r ^{2l(1} + r ^{2r)},} \sqrt{\tfrac 12 r ^{2l(1} - r ^{2r)}r).}

Using this parametrization with

r \in [ 0 , 1 ] for a quarter of l{L} , the arc length from the origin to a point ( x(r), y(r) ) is:^[21] \begin{aligned} &\int ^{r} _{0} \sqrt{x'(t) ^{2} + y'(t) ^{2}} d t \\ & {} = \int ^{r} _{0} \sqrt{ (( 1+2 t ^{2)} ^{2})/(2 ( 1+ t ^{2)}) + (( 1-2 t ^{2)} ^{2})/(2 ( 1- t ^{2)}) } \mathop \\[6mu]& \quad = \int_0^r \frac \\[6mu]& \quad = \operatorname r.\end

Likewise, the arc length from

( 1 , 0 ) to ( x(r), y(r) ) is:\begin{aligned} &\int ^{1} _{r} \sqrt{x'(t) ^{2} + y'(t) ^{2}} d t \\ & {} = \int ^{1} _{r} (d t)/(\sqrt{ 1- t ^{4}) } \\[6mu]& \quad = \operatorname r = \tfrac12\varpi - \operatorname r.\end

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 = \cos \theta or Cartesian equation x ^{2} + y ^{2} = x, using the same argument above but with the parametrization:( x(r), y(r) ) = l (r ^{2,} \sqrt{r ^{2l(1-r} ^{2r)}r).} x ^{2+y} ^{2=1} is parametrized in terms of the arc length s from the point ( 1 , 0 ) by(x(s),y(s)) = (\cos s,\sin s), l{L} is parametrized in terms of the arc length s from the point ( 1 , 0 ) by^[22] (x(s),y(s)) = \left( (\operatorname{cl)/(s}{\sqrt{ 1+ \operatorname{sl}) ^{2} s}}, (\operatorname{sl)/(s\operatorname{cl}s}{\sqrt{ 1+ \operatorname{sl}) ^{2} s}}\right) = \left(\tilde{\operatorname{cl}}s,\tilde{\operatorname{sl}}s\right).

The notation

\tilde{\operatorname{cl}},\tilde{\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:

\int ^{z} _{0} (d t)/(\sqrt{ 1- t ^{4}) } = 2 \int_0^u \frac, \quad \textz = \frac \text 0\le u\le\sqrt.

Later mathematicians generalized this result. Analogously to the constructible polygons in the circle, the lemniscate can be divided into sections of equal arc length using only straightedge and compass if and only if is of the form

n = 2 ^{kp} _{1 p} _{2 …} p _{m} where is a non-negative integer and each (if any) is a distinct Fermat prime.^[23] 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.^[24] Equivalently, the lemniscate can be divided into sections of equal arc length using only straightedge and compass if and only if \varphi (n) is a power of two (where \varphi is Euler's totient function). The lemniscate is not assumed to be already drawn; the theorem refers to constructing the division points only.

Let

r _{j=\operatorname{sl}\dfrac{2j\varpi}{n}} . Then the -division points for l{L} are the points\left(r _{j\sqrt{\tfrac12l(1+r} ^{2 r )}, ( -1 )} ^{j} ^{\left\lfloor} \sqrt{\tfrac 12 r ^{2 r )}\right),} _{j} j\in\{ 1 , 2 ,\ldots ,n\}

where

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

Arc length of rectangular elastica

The inverse lemniscate sine also describes the arc length relative to the coordinate of the rectangular elastica.^[25] This curve has coordinate and arc length:

y = \int ^{1} _{x} (t ^{2dt})/(\sqrt{ 1- t ^{4}) },\quad s = \operatorname x = \int_0^x \frac

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

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 \varphi (the angle BAC where B = ( 1 , 0 ) ; equivalently the length of the circular arc BD ). The parameter u is given byu = \int ^{\varphi} _{0} r(\theta) d\theta=\int ^{\varphi} _{0} (d\theta)/(\sqrt{ 1+ \sin ^{2\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\operatorname{cl}u = \overline{AF}, \operatorname{sl}u = \overline{DE}, \tilde{\operatorname{cl}} u = \overline{AF}\overline{AC}, \tilde{\operatorname{sl}} u = \overline{AF}\overline{FC}.

Series Identities

Power series

The power series expansion of the lemniscate sine at the origin is^[26]

\operatorname{sl}z = \sum ^{inf ty} _{n =0} a _{n} z ^{n = z -12 (z ^{5})/(5 !)} +3024 (z ^{9})/(-4390848 9 !) (z ^{13})/(13 !) + … , |z|< \tfrac{\varpi}{\sqrt{ 2 }} where the coefficients a _{n} are determined as follows:n\not\equiv 1 \pmod 4 \implies a _{n=0,} a _{1=1,} \forall n\in N _{0:a} _{n+2} =- (2)/((n +1 )(n +2 )) \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 , soa _{13} =- \tfrac{ 2 }{ 12 ⋅ 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}.}

The expansion can be equivalently written as^[27]

\operatorname{sl}z = \sum ^{inf ty} _{n =0} p _{2n} (z ^{4n+1})/(( 4 n +1 )!) , \left|z\right|< (\varpi)/(\sqrt{ 2) }wherep _{n+2} =-12 \sum ^{n\binom{ 2 n +2 }{ 2 j +2 }p} _{n - j} \sum ^{j} _{k =0} \binom{ 2 j +1 }{ 2 k +1 }p _{k} p _{j-k} , p _{0=1,} p _{1=0.}

The power series expansion of

\tilde{\operatorname{sl}} at the origin is\tilde{\operatorname{sl}}z = \sum ^{inf ty} _{n =0} \alpha _{n} z ^{n = z -9 (z ^{3})/(3 !)} +153 (z ^{5})/(-4977 5 !) (z ^{7})/(7 !) + … , \left|z\right|< (\varpi)/(2) where \alpha _{n=0} if n is even and\alpha _{n = \sqrt{ 2 } (\pi)/(\varpi)} (( -1 ) ^{(n-1)/2})/(n!) \sum ^{inf ty} _{k =1} (( 2 k\pi/\varpi) ^{n+1})/(\cosh k\pi) , \left|\alpha _{n\right|\sim} 2 ^{n+5/2} (n +1)/(\varpi ^{n+2}) if n is odd.

The expansion can be equivalently written as^[28]

\tilde{\operatorname{sl}} z = \sum ^{inf ty} _{n =0} (( -1 ) ^{n})/(2 ^{n+1}) \left(\sum ^{n} _{l =0} 2 ^{l} \binom{ 2 n +2 }{ 2 l +1 } s _{l} t _{n-l} \right) (z ^{2n+1})/(( 2 n +1 )!) , \left|z\right|< (\varpi)/(2) wheres _{n+2} =3 s _{n+1} +24 \sum ^{n} _{j =0} \binom{ 2 n +2 }{ 2 j +2 } s _{n-j} \sum ^{j} _{k =0} \binom{ 2 j +1 }{ 2 k +1 } s _{k} s _{j-k} , s _{0=1,} s _{1=3,} t _{n+2} =3 t _{n+1} +3 \sum ^{n} _{j =0} \binom{ 2 n +2 }{ 2 j +2 } t _{n-j} \sum ^{j} _{k =0} \binom{ 2 j +1 }{ 2 k +1 } t _{k} t _{j-k} , t _{0=1,} t _{1=3.}

For the lemniscate cosine,^[29]

\operatorname{cl}{z} =1- \sum ^{inf ty} _{n =0} ( -1 ) ^{n} \left(\sum ^{n} _{l =0} 2 ^{l} \binom{ 2 n +2 }{ 2 l +1 } q _{l} r _{n-l} \right) (z ^{2n+2})/(=1-2 ( 2 n +2 )!) (z ^{2})/(+12 2 !) (z ^{4})/(-216 4 !) (z ^{6})/(6 !) + … , \left|z\right|< (\varpi)/(2) , \tilde{\operatorname{cl}}z = \sum ^{inf ty} _{n =0} ( -1 ) ^{n} 2 ^{n} q _{n} (z ^{2n})/(=1-3 ( 2 n)!) (z ^{2})/(+33 2 !) (z ^{4})/(-819 4 !) (z ^{6})/(6 !) + … , \left|z\right|< (\varpi)/(2) wherer _{n+2} =3 \sum ^{n} _{j =0} \binom{ 2 n +2 }{ 2 j +2 } r _{n-j} \sum ^{j} _{k =0} \binom{ 2 j +1 }{ 2 k +1 } r _{k} r _{j-k} , r _{0=1,} r _{1=0,} q _{n+2} = \tfrac{ 3 }{ 2 } q _{n+1} +6 \sum ^{n} _{j =0} \binom{ 2 n +2 }{ 2 j +2 } q _{n-j} \sum ^{j} _{k =0} \binom{ 2 j +1 }{ 2 k +1 } q _{k} q _{j-k} , q _{0=1,} q _{1=\tfrac{3}{2}.}

Ramanujan's cos/cosh identity

Ramanujan's famous cos/cosh identity states that if

R(s) = (\pi)/(\varpi\sqrt{ 2) }\sum_\frac,then^[30] R(s) ^{-2} + R(is) ^{-2} =2 , \left|\operatorname{Re}s\right|< (\varpi)/(2) ,\left|\operatorname{Im}s\right|< (\varpi)/(2) . There is a close relation between the lemniscate functions and R(s) . Indeed,^[31] \tilde{\operatorname{sl}}s =- (d)/(ds) R(s) \left|\operatorname{Im}s\right|< (\varpi)/(2) \tilde{\operatorname{cl}}s = (d)/(ds) \sqrt{ 1- R(s) ^{2},} \left|\operatorname{Re}s - (\varpi)/(\right|< 2) (\varpi)/(,\left|\operatorname{Im}s\right|< 2) (\varpi)/(2) andR(s) = (1)/(\sqrt{ 1+ \operatorname{sl) ^{2} s}}, \left|\operatorname{Im}s\right |< (\varpi)/(2) .

Continued fractions

For

z\in C \setminus\{ 0 \} :^[32] \int ^{inf ty} _{0} e ^{-tz\sqrt{2} }\operatornamet\, \mathrm dt=\cfrac,\quad a_n=\frac((-1)^+3)\int ^{inf ty} _{0} e ^{-tz\sqrt{2} }\operatornamet\operatornamet \, \mathrm dt=\cfrac,\quad a_n=n^2(4n^2-1),\, b_n=3(2n-1)^2

Methods of computation

Several methods of computing

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

A hyperbolic series method:^{[33] [34] [35]}

\operatorname{sl}\left( (\varpi)/(x\right) = \pi) (\pi)/(\varpi) \sum _{n\inZ} (( -1 ) ^{n})/(\cosh (x - (n +1 / 2 )\pi)) , x\in C (1)/(\operatorname{sl) (\varpi x/\pi)} = (\pi\varpi)/(\sum) _{n\inZ} (( -1 ) ^{n})/({\left(x - n\pi\right)}} = {\sinh) (\pi\varpi)/(\sum) _{n\inZ} (( -1 ) ^{n})/(\sin (x - n\pi i)) , x\in C

Fourier series method:^[36]

\operatorname{sl} l ( (\varpi)/(x r ) = \pi) (2 \pi)/(\varpi) \sum ^{inf ty} _{n =0} (( -1 ) ^{n\sin} (( 2 n +1 )x))/(\cosh ((n +1 / 2 )\pi)) , \left|\operatorname{Im}x\right|< (\pi)/(2) \operatorname{cl}\left( (\varpi)/(x\right) = \pi) (2 \pi)/(\varpi) \sum ^{inf ty} _{n =0} (\cos (( 2 n +1 )x))/(, \left|\operatorname{Im}x\right|< \cosh ((n +1 / 2 )\pi)) (\pi)/(2) (1)/(\operatorname{sl) (\varpi x/\pi)} = (\pi)/(\left( \varpi) (1)/(\sin x) -4 \sum ^{inf ty} _{n =0} (\sin (( 2 n +1 )x))/(e ^{(2n+1)\pi} +1) \right), \left|\operatorname{Im}x\right|<\pi

The lemniscate functions can be computed more rapidly by

\begin{align}\operatorname{sl} l ( (\varpi\pi)/(x r )&) = ({\theta _{1})/(\left(x, e ^{-\pi} \right)) },\quad x\in\mathbb\\\operatorname\Bigl(\frac\varpi\pi x\Bigr)&=\frac,\quad x\in\mathbb\endwhere\begin{aligned} \theta ^{- \pi} _{1 (x,e} )& = \sum _{n\inZ} ( -1 ) ^{n+1} e ^{- \pi (n +1 / 2+ x/\pi) ^{2}} = \sum _{n\inZ} ( -1 ) ^{n} e ^{- \pi (n +1 / 2 ) ^{2}} \sin (( 2 n +1 )x),\\ \theta ^{- \pi} _{2 (x,e} )& = \sum _{n\inZ} ( -1 ) ^{n} e ^{- \pi (n + x/\pi) ^{2}} = \sum _{n\inZ} e ^{- \pi (n +1 / 2 ) ^{2}} \cos (( 2 n +1 )x),\\ \theta ^{- \pi} _{3 (x,e} )& = \sum _{n\inZ} e ^{- \pi (n + x/\pi) ^{2}} = \sum _{n\inZ} e ^{- \pi n ^{2}} \cos 2 nx,\\ \theta ^{- \pi} _{4 (x,e} )& = \sum _{n\inZ} e ^{- \pi (n +1 / 2+ x/\pi) ^{2}} = \sum _{n\inZ} ( -1 ) ^{n} e ^{- \pi n ^{2}} \cos 2 nx\end{aligned}

are the Jacobi theta functions.^[37]

Fourier series for the logarithm of the lemniscate sine:

ln \operatorname{sl}\left( (\varpi\pi)/(2- x\right) =ln) (\pi)/(4) +ln \sin x +2 \sum ^{inf ty} _{n =1} (( -1 ) ^{n} \cos 2 nx)/(n(e ^{n\pi} + ( -1 ) ^{n)}) , \left|\operatorname{Im}x\right|< (\pi)/(2)

The following series identities were discovered by Ramanujan:^[38]

(\varpi ^{2})/(\pi ^{2\operatorname{sl}) ^{2(\varpi} x/\pi)} = (1)/(- \sin ^{2x}) (1)/(\pi) -8 \sum ^{inf ty} _{n =1} (n\cos 2 nx)/(e ^{2n\pi} -1) , \left|\operatorname{Im}x\right|<\pi \arctan\operatorname{sl} l ( (\varpi\pi)/(x r ) =2 \sum) ^{inf ty} ^{n =0} (\sin(( 2 n +1 )x))/(( 2 n +1 )\cosh ((n +1 / 2 )\pi)) , \left|\operatorname{Im}x\right|< (\pi)/(2)

The functions

\tilde{\operatorname{sl}} and \tilde{\operatorname{cl}} analogous to \sin and \cos on the unit circle have the following Fourier and hyperbolic series expansions:^[39] \tilde{\operatorname{sl}}s =2 \sqrt{ 2 } (\pi ^{2})/(\varpi ^{2}) \sum ^{inf ty (n\sin ( 2 n\pi s/\varpi))/(\cosh n\pi)} _{n =1} , \left|\operatorname{Im}s\right|< (\varpi)/(2) \tilde{\operatorname{cl}}s = \sqrt{ 2 } (\pi ^{2})/(\varpi ^{2}) \sum ^{inf ty} _{n =0} (( 2 n +1 )\cos (( 2 n +1 )\pi s/\varpi))/(\sinh ((n +1 / 2 )\pi)) , \left|\operatorname{Im}s\right|< (\varpi)/(2) \tilde{\operatorname{sl}}s = (\pi ^{2})/(\varpi ^{2\sqrt{2}) }\sum_\frac,\quad s\in\mathbb\tilde{\operatorname{cl}}s = (\pi ^{2})/(\varpi ^{2\sqrt{2}) }\sum_\frac,\quad s\in\mathbb

Two other fast computation methods use the following sum and product series:

The following identities come from product representations of the theta functions:

sl l ( (\varpi\pi)/(x r )) = 2 e ^{-\pi/4} \sin x\prod ^{inf ty} _{n =1} (1-2 e ^{-2n\pi} \cos 2 x + e ^{-4n\pi})/(1+2 e ^{-(2n-1)\pi} \cos 2 x + e ^{-(4n-2)\pi}) , x\in C cl l ( (\varpi\pi)/(x r )) = 2 e ^{-\pi/4} \cos x\prod ^{inf ty} _{n =1} (1+2 e ^{-2n\pi} \cos 2 x + e ^{-4n\pi})/(1-2 e ^{-(2n-1)\pi} \cos 2 x + e ^{-(4n-2)\pi}) , x\in C

A similar formula involving the

\operatorname{sn} function can be given.

In the same pattern following sum formulas can be set up with the help of the tangent duplication theorem:

sl l ( (\varpi\pi)/(x r )) = f l ( (4 \pi)/(\varpi\sin) x\sum ^{inf ty} _{n =1} (\cosh[( 2 n -1 )\pi])/(\cosh ^{2[(2n-1)\pi]} - \cos ^{2} x) r ) cl l ( (\varpi\pi)/(x r )) = f l ( (4 \pi)/(\varpi\cos) x\sum ^{inf ty} _{n =1} (\cosh[( 2 n -1 )\pi])/(\cosh ^{2[(2n-1)\pi]} - \sin ^{2} x) r )

where

f(x) = \tan( 2 \arctan x) = 2 x / ( 1 - x ^{2).}

The lemniscate functions as a ratio of entire functions

Since the lemniscate sine is a meromorphic function in the whole complex plane, it can be written as a ratio of entire functions. Gauss showed that has the following product expansion, reflecting the distribution of its zeros and poles:^[40]

\operatorname{sl}z = (M(z))/(N(z))

where

M(z) = z\prod _{\alpha} \left( 1- (z ^{4})/(\alpha ^{4}) \right), N(z) = \prod _{\beta} \left( 1- (z ^{4})/(\beta ^{4}) \right).

Here,

\alpha and \beta denote, respectively, the zeros and poles of which are in the quadrant \operatorname{Re}z> 0 ,\operatorname{Im}z\ge 0 . A proof can be found in.^{[40] [41]} 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.^[42] It can be easily seen (using uniform and absolute convergence arguments to justify interchanging of limiting operations) that(M'(z))/(M(z)) =- \sum ^{inf ty} _{n =0} 2 ^{4n} H _{4n} (z ^{4n-1})/(( 4 n)!) , \left|z\right|<\varpi (where H _{n} are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers) and(N'(z))/(= ( 1+ i) N(z)) (M'(( 1+ i)z))/(- M(( 1+ i)z)) (M'(z))/(M(z)) . Therefore(N'(z))/(N(z)) = \sum ^{inf ty} _{n =0} 2 ^{4n} ( 1- ( -1 ) ^{n} 2 ^{2n} ) H _{4n} (z ^{4n-1})/(( 4 n)!) , \left|z\right|< (\varpi)/(\sqrt{ 2) }.It is known that(1)/(\operatorname{sl) ^{inf ty} ^{n =0} 2 ^{4n} ( 4 n -1 ) H _{4n} (z ^{4n-2})/(( 4 n)!) , \left|z\right|<\varpi. Then from(d)/(dz) (\operatorname{sl)/('z}{\operatorname{sl}z} =- (1)/(\operatorname{sl)) ^{2z}-\operatorname{sl}} ^{2z} and\operatorname{sl} ^{2 z = (1)/(\operatorname{sl)} ^{2 z} - (( 1+ i) ^{2})/(\operatorname{sl)} ^{2((1+i)z)}} we get(\operatorname{sl)/('z}{\operatorname{sl}z} =- \sum) ^{inf ty} ^{n =0} 2 ^{4n} ( 2- ( -1 ) ^{n} 2 ^{2n} ) H _{4n} (z ^{4n-1})/(( 4 n)!) , \left|z\right|< (\varpi)/(\sqrt{ 2) }.Hence(\operatorname{sl)/(- 'z}{\operatorname{sl}z} = (M'(z))/(M(z))) (N'(z))/(N(z)) , \left|z\right|< (\varpi)/(\sqrt{ 2) }.Therefore\operatorname{sl}z = C (M(z))/(N(z)) for some constant C for \left|z\right|<\varpi/\sqrt{ 2 } but this result holds for all z\in C by analytic continuation. Using\lim _{z\to} (\operatorname{sl)/(z}{z} =1) gives C =1 which completes the proof. \blacksquare Gauss conjectured that ln N(\varpi) = \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”.^[43] 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 asN(z) = (M(( 1+ i)z))/(( 1+ i)M(z)) , z\notin \varpi Z [i] andN( 2 z) = M(z) ^{4+N(z)} ^{4.}

Thanks to a certain theorem^[44] 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:^{[45] [46] [47] [48]} M(z) = z -2 (z ^{5})/(-36 5 !) (z ^{9})/(+552 9 !) (z ^{13})/(13 !) + … , z\in C N(z) =1+2 (z ^{4})/(-4 4 !) (z ^{8})/(+408 8 !) (z ^{12})/(12 !) + … , z\in C .

This can be contrasted with the power series of

\operatorname{sl} which has only finite radius of convergence (because it is not entire).

We define

S and T byS(z) = N\left( (z)/(1+ i) \right) ^{2- iM\left( (z)/(1+ i)} \right) ^{2,} T(z) = S(iz). Then the lemniscate cosine can be written as\operatorname{cl}z = (S(z))/(T(z)) where^[49] S(z) =1- (z ^{2})/(- 2 !) (z ^{4})/(-3 4 !) (z ^{6})/(+17 6 !) (z ^{8})/(-9 8 !) (z ^{10})/(+111 10 !) (z ^{12})/(12 !) + … , z\in C T(z) =1+ (z ^{2})/(- 2 !) (z ^{4})/(+3 4 !) (z ^{6})/(+17 6 !) (z ^{8})/(+9 8 !) (z ^{10})/(+111 10 !) (z ^{12})/(12 !) + … , z\in C .

Furthermore, the identities

M( 2 z) =2 M(z) N(z) S(z) T(z), S( 2 z) = S(z) ^{4-2M(z)} ^{4,} T( 2 z) = T(z) ^{4-2M(z)} ^{4} and the Pythagorean-like identitiesM(z) ^{2+S(z)} ^{2=N(z)} ^{2,} M(z) ^{2+N(z)} ^{2=T(z)} ^{2} hold for all z\in C .

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 theta functions and the above functions are not equivalent).

Relation to other functions

Relation to Weierstrass and Jacobi elliptic functions

\wp(z; 1 , 0 ) (the "lemniscatic case"), with invariants and . This lattice has fundamental periods \omega _{1} = \sqrt{ 2 }\varpi, and \omega _{2} = i\omega _{1} . The associated constants of the Weierstrass function are e _{1=\tfrac12, e} _{2=0, e} _{3=-\tfrac12.}

The related case of a Weierstrass elliptic function with, 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: and . The period parallelogram is either a square or a rhombus. The Weierstrass elliptic function

\wp (z; -1 , 0 ) is called the "pseudolemniscatic case".

The square of the lemniscate sine can be represented as

\operatorname{sl} ^{2} z = (1)/(= \wp (z; 4 , 0 )) (i)/(= { -2 \wp}{\left(\sqrt 2 z + (i -1 ) 2 \wp (( 1- i)z; -1 , 0 )) (\varpi)/(\sqrt 2) ; 1 , 0 \right)}

where the second and third argument of

\wp denote the lattice invariants and . The lemniscate sine is a rational function in the Weierstrass elliptic function and its derivative:^[50] \operatorname{sl}z =-2 (\wp (z; -1 , 0 ))/(\wp '(z; -1 , 0 )) .

The lemniscate functions can also be written in terms of Jacobi elliptic functions. The Jacobi elliptic functions

\operatorname{sn} and \operatorname{cd} with positive real elliptic modulus have an "upright" rectangular lattice aligned with real and imaginary axes. Alternately, the functions \operatorname{sn} and \operatorname{cd} with modulus (and \operatorname{sd} and \operatorname{cn} with modulus 1 /\sqrt{ 2 } ) have a square period lattice rotated 1/8 turn.^{[51] [52]} \operatorname{sl} z = \operatorname{sn}(z;i) = \operatorname{sc}(z;\sqrt{ 2 }) = {\tfrac 1 {\sqrt 2 }\operatorname{sd}}\left(\sqrt 2 z;\tfrac{ 1 }{\sqrt 2 }\right) \operatorname{cl} z = \operatorname{cd}(z;i) = \operatorname{dn}(z;\sqrt{ 2 }) = {\operatorname{cn}}\left(\sqrt 2 z;\tfrac{ 1 }{\sqrt 2 }\right)

where the second arguments denote the elliptic modulus

k .

The functions

\tilde{\operatorname{sl}} and \tilde{\operatorname{cl}} can also be expressed in terms of Jacobi elliptic functions:\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), \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).

Relation to the modular lambda function

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

\prod ^{n} _{k =1} {\operatorname{sl}}{\left( (2 k -1)/(2 n +1) (\varpi)/(\right)} = \sqrt[ 8 ]{ 2) (λ (( 2 n +1 )i))/(1-λ (( 2 n +1 )i)) }

For example:

\begin{aligned} &{\operatorname{sl}} l (\tfrac 1 { 14 }\varpi r ){\operatorname{sl}} l (\tfrac 3 { 14 }\varpi r ){\operatorname{sl}} l (\tfrac 5 { 14 }\varpi r ) \\[ 7 mu] & {} = \sqrt[ 8 ]{ (λ ( 7 i))/(1-λ ( 7 i)) } = \Bigl(\Bigl(\tfrac\sqrt+\tfrac\sqrt+1\Bigr)\Bigr)\\[7mu]&\quad = \frac 2 \\[18mu]& \bigl(\tfrac1\varpi\bigr)\, \bigl(\tfrac3\varpi\bigr)\,\bigl(\tfrac5\varpi\bigr)\,\bigl(\tfrac7\varpi\bigr) \\[-3mu]&\quad = \sqrt[8]= \Biggl(\frac\pi4 - \Biggl(\frac\Biggr)\Biggr)\end

Inverse functions

The inverse function of the lemniscate sine is the lemniscate arcsine, defined as

\operatorname{arcsl} x = \int ^{x} _{0} (dt)/(\sqrt{ 1- t ^{4}) }.

It can also be represented by the hypergeometric function:

\operatorname{arcsl}x = x{} _{2F} ^{4 \right).} ^{1 \left(\tfrac 12 ,\tfrac 14 ;\tfrac 54 ;x}

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

\operatorname{arccl} x = \int ^{1} _{x} (dt)/(\sqrt{ 1- t ^{4}) } = \tfrac12\varpi - \operatornamex

For in the interval

-1 \leq x \leq 1 , \operatorname{sl}\operatorname{arcsl} x = x and \operatorname{cl}\operatorname{arccl} x = x

For the halving of the lemniscate arc length these formulas are valid:

\begin{aligned} {\operatorname{sl}} l (\tfrac 12 \operatorname{arcsl} x r ) & = {\sin} l (\tfrac 12 \arcsin x r ) {\operatorname{sech}} l (\tfrac 12 \operatorname{arsinh} x r ) \\ {\operatorname{sl}} l (\tfrac 12 \operatorname{arcsl} x r ) ^{2} & = {\tan} l (\tfrac 14 \arcsin x ^{2r) \end{aligned}}

Furthermore there are the so called Hyperbolic lemniscate area functions:

\operatorname{aslh}(x) = \int ^{x} _{0} (1)/(\sqrt{y ^{4} +1) } \mathrmy = \fracF\bigl[2\arctan(x);\frac{1}{2}\sqrt{2}\,\bigr] \operatorname{aclh}(x) = \int ^{inf ty} _{x} (1)/(\sqrt{y ^{4} +1) } \mathrmy = \fracF\bigl[2\arccot(x);\frac{1}{2}\sqrt{2}\,\bigr] \operatorname{aclh}(x) = (\varpi)/(\sqrt{ 2) } - \operatorname(x) \operatorname{aslh}(x) = \sqrt{ 2 }\operatorname{arcsl} l [x(\sqrt{x ^{4} + 1 } + 1 ) ^{-1/2} r ] \operatorname{arcsl}(x) = \sqrt{ 2 }\operatorname{aslh} l [x( 1 + \sqrt{ 1 - x ^{4})} ^{-1/2} r ]

Expression using elliptic integrals

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

These functions can be displayed directly by using the incomplete elliptic integral of the first kind:

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

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

The lemniscate arccosine has this expression:

\operatorname{arccl} x = (1)/(\sqrt 2) F\left(\arccos x; (1)/(\sqrt 2) \right)

Use in integration

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

\int (1)/(\sqrt{ 1- x ^{4}) }\,\mathrm dx=\operatorname x\int (1)/(\sqrt{(x ^{2+1)(2x} ^{2+1)}) }\,\mathrm dx=\int (1)/(\sqrt{x ^{4+6x} ^{2+1}) }\,\mathrm dx=\int (1)/(\sqrt{x ^{4+1}) }\,\mathrm dx=\int (1)/(\sqrt[ 4 ]{( 1- x ^{4)} ^{3}) }\,\mathrm dx=\int (1)/(\sqrt[ 4 ]{(x ^{4+1)} ^{3}) }\,\mathrm dx=\int (1)/(\sqrt[ 4 ]{( 1- x ^{2)} ^{3}) }\,\mathrm dx=\int (1)/(\sqrt[ 4 ]{(x ^{2+1)} ^{3}) }\,\mathrm dx=\int (1)/(\sqrt[ 4 ]{(ax ^{2+bx+c)} ^{3}) }\,\mathrm dx=\int\sqrt{\operatorname{sech} x} dx={2\operatorname{arcsl}}\tanh \tfrac 12 x \int\sqrt{\sec x} dx={2\operatorname{arcsl}}\tan \tfrac 12 x

Hyperbolic lemniscate functions

Fundamental information

For convenience, let

\sigma = \sqrt{ 2 }\varpi . \sigma is the "squircular" analog of \pi (see below). The decimal expansion of \sigma (i.e. 3 . 7081 \ldots ^[53]) appears in entry 34e of chapter 11 of Ramanujan's second notebook.^[54]

The hyperbolic lemniscate sine and cosine can be defined as inverses of elliptic integrals as follows:

z l{\overset{*}{=}} \int ^{\operatorname{slh} _{0} z} (d t)/(\sqrt{ 1+ t ^{4}) } = \int_^\infty \frac

where in

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

The complete integral has the value:

\int ^{inf ty} _{0} (d t)/(\sqrt{t ^{4} +1) } = \tfrac14 \Beta\bigl(\tfrac14, \tfrac14\bigr) = \frac = 1.85407\;46773\;01371\ldots

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

\operatorname{slh} z = {\operatorname{clh}}{ l ( (\sigma)/(2) - z r )}

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

\operatorname{slh}z\operatorname{clh}z = 1

The functions

\operatorname{slh} and \operatorname{clh} have a square period lattice with fundamental periods \{\sigma,\sigma i\} .

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

\operatorname{slh} l (\sqrt 2 z r ) = (( 1+ \operatorname{cl)/(^{2}) z)\operatorname{sl}z}{\sqrt 2 \operatorname{cl}z} \operatorname{clh} l (\sqrt 2 z r ) = (( 1+ \operatorname{sl)/(^{2}) z)\operatorname{cl}z}{\sqrt 2 \operatorname{sl}z}

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

\operatorname{slh}z = (\operatorname{sn)/((z; 1 /\sqrt 2 )}{\operatorname{cd}(z; 1 /\sqrt 2 )}) \operatorname{clh}z = (\operatorname{cd)/((z; 1 /\sqrt 2 )}{\operatorname{sn}(z; 1 /\sqrt 2 )})

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

\operatorname{slh}z = (1- i)/(\operatorname{sl}\left( \sqrt 2) (1+ i)/(\sqrt 2) z\right) = (\operatorname{sl)/(\left(\sqrt[ 4 ]{ -1 }z\right)) }{ \sqrt[ 4 ]{ -1 } }

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

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

Relation to quartic Fermat curve

Hyperbolic Lemniscate Tangent and Cotangent

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

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, the hyperbolic lemniscate sine of twice the enclosed area between this line and the x-axis is the y-coordinate of the intersection of with the line x = 1 . Just as \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 \sigma . Moreover,M( 1 , 1 /\sqrt{ 2 }) = (\pi)/(\sigma)

where

M is the arithmetic–geometric mean.

The hyperbolic lemniscate sine satisfies the argument addition identity:

\operatorname{slh}(a + b) = (\operatorname{slh)/(a\operatorname{slh}'b) + \operatorname{slh}b\operatorname{slh}'a}{ 1- \operatorname{slh} ^{2a\operatorname{slh}} ^{2b}}

When

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

(d)/(d u) \operatorname{slh}(u) = \sqrt{ 1 + \operatorname{slh}(u) ^{4}} (d)/(d u) \operatorname{clh}(u) = - \sqrt{ 1 + \operatorname{clh}(u) ^{4}} (d)/(d u) (1)/(2) \operatorname{arsinh} l [ \operatorname{slh}(u) ^{2} r ] = \operatorname{slh}(u) (d)/(- d u) (1)/(2) \operatorname{arsinh} l [ \operatorname{clh}(u) ^{2} r ] = \operatorname{clh}(u)

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:

tlh (\sqrt{ 2 }u) = \sin _{4} (\sqrt{ 2 }u) = \operatorname{sl}(u)\sqrt{ (\operatorname{cl)/(^{2}) u +1 }{\operatorname{sl} ^{2} u + \operatorname{cl} ^{2} u}} ctlh (\sqrt{ 2 }u) = \cos _{4} (\sqrt{ 2 }u) = \operatorname{cl}(u)\sqrt{ (\operatorname{sl)/(^{2}) u +1 }{\operatorname{sl} ^{2} u + \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:

tlh (u) = (sn (u;\tfrac{ 1)/(2) \sqrt{ 2 })}{\sqrt[ 4 ]{ cd (u;\tfrac{ 1 }{ 2 }\sqrt{ 2 }) ^{4} + sn (u;\tfrac{ 1 }{ 2 }\sqrt{ 2 }) ^{4}}} ctlh (u) = (cd (u;\tfrac{ 1)/(2) \sqrt{ 2 })}{\sqrt[ 4 ]{ cd (u;\tfrac{ 1 }{ 2 }\sqrt{ 2 }) ^{4} + sn (u;\tfrac{ 1 }{ 2 }\sqrt{ 2 }) ^{4}}}

When

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

(d)/(d u) \operatorname{tlh}(u) = \operatorname{ctlh}(u) ^{3} (d)/(d u) \operatorname{ctlh}(u) = - \operatorname{tlh}(u) ^{3} \int ^{\varpi/\sqrt{ 2} _{0} } \operatorname(u) \,\mathrmu = \frac \int ^{\varpi/\sqrt{ 2} _{0} } \operatorname(u) \,\mathrmu = \frac

Derivation of the Hyperbolic Lemniscate functions

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 (d)/(d w) x(w) = - y(w) ^{3} (d)/(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) = \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} 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}

The following therefore applies to the quotient:

(y(w))/(x(w)) = (\operatorname{sl)/((\tfrac{ 1 }{ 2 }\sqrt{ 2 }w)) [\operatorname{cl}(\tfrac{ 1 }{ 2 }\sqrt{ 2 }w) ^{2+1]} ^{1/2} } = \operatorname(w) The functions x(w) and y(w) are called cotangent hyperbolic lemniscatus and hyperbolic tangent.x(w) = ctlh (w) y(w) = 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.

First proof: comparison with the derivative of the arctangent

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) = \sqrt{ l ( (1)/(\sqrt[ 4 ]{s ^{4} +1) }\biggr)^2 + \biggl(\frac\biggr)^2} = \frac

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:

(d)/(d s) \arctan(s) = (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:

(d)/(d s) aslh (s) = l [ (d)/(d s) \arctan(s) r ] D(s) ^{2} = (1)/(s ^{2} +1) D(s) ^{2} = (1)/(l ( s ^{2} +1) (\sqrt{s ^{2} +1)/() }\biggr)^2 = \frac

Second proof: integral formation and area subtraction

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:

atlh (v) = 2 l (\int ^{v} _{0} \sqrt[ 4 ]{ 1 - w ^{4}} d w r ) - v\sqrt[ 4 ]{ 1 - v ^{4}} (d)/(d v) atlh (v) = 2 \sqrt[ 4 ]{ 1 - v ^{4}} - l ( (d)/(d v) v\sqrt[ 4 ]{ 1 - v ^{4}r)} = (1)/(( 1- v ^{4)} ^{3/4})

The following transformation applies:

aslh (x) =atlh l ( (x)/(\sqrt[ 4 ]{x ^{4} +1) }\biggr)

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

(d)/(d x) aslh (x) = (d)/(atlh l ( d x) (x)/(\sqrt[ 4 ]{x ^{4} +1) }\biggr) = \biggl(\frac \frac \biggr) \biggl[1 - \biggl(\frac{x}{\sqrt[4]}\biggr)^4\biggr]^ = = (1)/((x ^{4} +1 ) ^{5/4}) l [ 1 - l ( (x)/(\sqrt[ 4 ]{x ^{4} +1) }\biggr)^4\biggr]^ = \frac \biggl(\frac\biggr)^ = \frac

Specific values

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

\int ^{inf ty} _{0} (\operatorname{d)/(t}{\sqrt{t) ^{4} + 1 }} = \tfrac 14 \Beta l (\tfrac 14 , \tfrac 14 r ) = (\varpi)/(\sqrt 2) = (\sigma)/(2) = 1 . 85407 \ldots

whereas

\tfrac 12 \Beta l (\tfrac 12 , \tfrac 12 r ) = \tfrac{\pi} 2 , so the values below such as {\operatorname{slh}} l (\tfrac{\varpi}{ 2 \sqrt{ 2 }} r ) = {\operatorname{slh}} l (\tfrac{\sigma}{ 4 } r ) = 1 are analogous to the trigonometric {\sin} l (\tfrac{\pi} 2 r ) = 1 .\operatorname{slh}\left( (\varpi)/(2 \sqrt{ 2) }\right) = 1\operatorname{slh}\left( (\varpi)/(3 \sqrt{ 2) }\right) = \frac\sqrt[4]\operatorname{slh}\left( (2 \varpi)/(3 \sqrt{ 2) }\right) = \sqrt[4]\operatorname{slh}\left( (\varpi)/(4 \sqrt{ 2) }\right) = \frac(\sqrt-1)\operatorname{slh}\left( (3 \varpi)/(4 \sqrt{ 2) }\right) = \frac(\sqrt+1)\operatorname{slh}\left( (\varpi)/(5 \sqrt{ 2) }\right) = \frac\sqrt\sqrt = 2\sqrt[4]\sqrt\operatorname{slh}\left( (2 \varpi)/(5 \sqrt{ 2) }\right) = \frac(\sqrt+1)\sqrt = 2\sqrt[4]\sqrt\operatorname{slh}\left( (3 \varpi)/(5 \sqrt{ 2) }\right) = \frac\sqrt\sqrt = 2\sqrt[4]\sqrt\operatorname{slh}\left( (4 \varpi)/(5 \sqrt{ 2) }\right) = \frac(\sqrt+1)\sqrt = 2\sqrt[4]\sqrt\operatorname{slh}\left( (\varpi)/(6 \sqrt{ 2) }\right) = \frac(\sqrt+1)(1-\sqrt[4])\operatorname{slh}\left( (5 \varpi)/(6 \sqrt{ 2) }\right) = \frac(\sqrt+1)(1+\sqrt[4])

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

z	\operatorname{clh} z	\operatorname{slh} z	\operatorname{ctlh} z = \cos _{4} z	\operatorname{tlh} z = \sin _{4} z
0	inf ty	0	1	0
{\tfrac 14 }\sigma	1	1	1 / \sqrt[ 4 ]{ 2 }	1 / \sqrt[ 4 ]{ 2 }
{\tfrac 12 }\sigma	0	inf ty	0	1
{\tfrac 34 }\sigma	-1	-1	-1 / \sqrt[ 4 ]{ 2 }	1 / \sqrt[ 4 ]{ 2 }
\sigma	inf ty	0	-1	0

Combination and halving theorems

In combination with the Hyperbolic Lemniscate Areasine, the following identities can be established:

tlh l [ aslh (x) r ] = ctlh l [ aclh (x) r ] = (x)/(\sqrt[ 4 ]{x ^{4} +1) } ctlh l [ aslh (x) r ] = tlh l [ aclh (x) r ] = (1)/(\sqrt[ 4 ]{x ^{4} +1) }

The square of the Hyperbolic Lemniscate Tangent is the Pythagorean counterpart of the square of the Hyperbolic Lemniscate cotangent because the sum of the fourth powers of

\operatorname{tlh} and \operatorname{ctlh} is always equal to the value one.

The bisection theorem of the hyperbolic sinus lemniscatus reads as follows:

slh l [\tfrac{ 1 }{ 2 } aslh (x) r ] = (\sqrt{ 2)/(x}{\sqrt{x) ^{2} + 1 + \sqrt{x ^{4} + 1 }} + \sqrt{\sqrt{x ^{4} + 1 } - x ^{2} + 1 }}

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

aslh (x) = \sqrt{ 2 } arcsl l [x(\sqrt{x ^{4} + 1 } + 1 ) ^{-1/2} r ] arcsl (x) = \sqrt{ 2 }aslh l ( (\sqrt{ 2)/(x}{\sqrt{ 1) + x ^{2}} + \sqrt{ 1 - x ^{2}}r)}

In addition, the following formulas are valid for all real values

x \in \R :slh l [\tfrac{ 1 }{ 2 } aclh (x) r ] = \sqrt{\sqrt{x ^{4} + 1 } + x ^{2} - \sqrt{ 2 }x\sqrt{\sqrt{x ^{4} + 1 } + x ^{2}}} = l (\sqrt{x ^{4} + 1 } - x ^{2} + 1 r ) ^{-1/2} l (\sqrt{\sqrt{x ^{4} + 1 } + 1 } - x r ) clh l [\tfrac{ 1 }{ 2 } aclh (x) r ] = \sqrt{\sqrt{x ^{4} + 1 } + x ^{2} + \sqrt{ 2 }x\sqrt{\sqrt{x ^{4} + 1 } + x ^{2}}} = l (\sqrt{x ^{4} + 1 } - x ^{2} + 1 r ) ^{-1/2} l (\sqrt{\sqrt{x ^{4} + 1 } + 1 } + x r ) These identities follow from the last-mentioned formula:tlh [\tfrac{ 1 }{ 2 } aclh (x)] ^{2} = \tfrac{ 1 }{ 2 }\sqrt{ 2-2 \sqrt{ 2 }x\sqrt{\sqrt{x ^{4+1}-x} ^{2}}} = l ( 2 x ^{2} + 2 + 2 \sqrt{x ^{4} + 1 } r ) ^{-1} l (\sqrt{\sqrt{x ^{4} + 1 } + 1 } - x r ) ctlh [\tfrac{ 1 }{ 2 } aclh (x)] ^{2} = \tfrac{ 1 }{ 2 }\sqrt{ 2+2 \sqrt{ 2 }x\sqrt{\sqrt{x ^{4+1}-x} ^{2}}} = l ( 2 x ^{2} + 2 + 2 \sqrt{x ^{4} + 1 } r ) ^{-1} l (\sqrt{\sqrt{x ^{4} + 1 } + 1 } + x r ) The following formulas for the lemniscatic sine and lemniscatic cosine are closely related:sl [\tfrac{ 1 }{ 2 }\sqrt{ 2 } aclh (x)] = cl [\tfrac{ 1 }{ 2 }\sqrt{ 2 } aslh (x)] = \sqrt{\sqrt{x ^{4} + 1 } - x ^{2}} sl [\tfrac{ 1 }{ 2 }\sqrt{ 2 } aslh (x)] = cl [\tfrac{ 1 }{ 2 }\sqrt{ 2 } aclh (x)] = x l (\sqrt{x ^{4} + 1 } + 1 r ) ^{-1/2}

Coordinate Transformations

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) = \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:

l [\int ^{inf ty} _{0} \exp( - x ^{2)} d x r ] ^{2} = \int ^{inf ty} _{0} \int ^{inf ty} _{0} \exp( - y ^{2-z} ^{2)} d y d z = = \int ^{\pi/ 2} _{0} \int ^{inf ty} _{0} \det\begin{bmatrix} \partial/\partial rr\cos(\phi) & \partial/\partial \phir\cos(\phi) \ \partial/\partial rr\sin(\phi) & \partial/\partial \phi r\sin(\phi) \end{bmatrix} \exp l \{ - l [r\cos(\phi) r ] ^{2-l[r\sin(\phi)r]} ^{2r\}} d r d \phi = = \int ^{\pi/ 2} _{0} \int ^{inf ty} _{0} r\exp( - r ^{2)} d r d \phi = \int ^{\pi/ 2} _{0} (1)/(2) d \phi = (\pi)/(4) And this is the analogous coordinate transformation for the lemniscatory case:l [\int ^{inf ty} _{0} \exp( - x ^{4)} d x r ] ^{2} = \int ^{inf ty} _{0} \int ^{inf ty} _{0} \exp( - y ^{4-z} ^{4)} d y d z = = \int ^{\varpi/\sqrt{ 2} _{0} } \int_^ \det\begin \partial/\partial r\,\,r\,\text(\phi) & \partial/\partial \phi\,\,r\,\text(\phi) \\ \partial/\partial r\,\,r\, \text(\phi) & \partial/\partial \phi\,\,r\,\text(\phi) \end \exp\bigl\ \,\mathrmr \,\mathrm\phi = = \int ^{\varpi/\sqrt{ 2} _{0} } \int_^ r\exp(-r^4) \,\mathrmr \,\mathrm\phi = \int_^ \frac \,\mathrm\phi = \frac

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.

Number theory

In algebraic number theory, every finite abelian extension of the Gaussian rationals

Q (i) is a subfield of Q (i,\omega _{n)} for some positive integer n .^[55] This is analogous to the Kronecker–Weber theorem for the rational numbers Q which is based on division of the circle – in particular, every finite abelian extension of Q is a subfield of Q (\zeta _{n)} for some positive integer n . Both are special cases of Kronecker's Jugendtraum, which became Hilbert's twelfth problem.Q (i,\operatorname{sl}(\varpi /n)) (for positive odd n ) is the extension of 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} .^[55]

Hurwitz numbers

The Bernoulli numbers

B _{n} can be defined byB _{n =} \lim _{z\to} (d ^{n})/(dz ^{n}) (z)/(e ^{z-1}) , n\ge 0

and appear in

\sum _{k\inZ\setminus\{0\} }\frac= (-1)^\mathrm_\frac=2\zeta (2n),\quad n\ge 1

where

\zeta is the Riemann zeta function.

The Hurwitz numbers

H _{n,} named after Adolf Hurwitz, are the "lemniscate analogs" of the Bernoulli numbers. They can be defined by^{[56] [57]} H _{n =} - \lim _{z\to} (d ^{n})/(dz ^{n}) z\zeta (z; 1 / 4 , 0 ), n\ge 0

where

\zeta ( ⋅ ; 1 / 4 , 0 ) is the Weierstrass zeta function with lattice invariants 1 / 4 and 0 . They appear in\sum _{z\inZ[i]\setminus\{0\} }\frac= \mathrm_\frac= G_(i),\quad n\ge 1

where

Z [i] are the Gaussian integers and G _{4n} are the Eisenstein series of weight 4 n , and in\displaystyle \begin{array}{ll} \displaystyle\sum ^{inf ty\dfrac{n} _{n =1} ^{k}{e} ^{2\pi} -1 } = \begin{cases} \dfrac{ 1 }{ 24 } - \dfrac{ 1 }{ 8 \pi} & { if }\ k=1 \\\dfrac & \ k\equiv1\, (\mathrm\, 4)\ \ k\ge 5 \\ \dfrac+\dfrac\left(\dfrac\right)^ & \ k\equiv 3\,(\mathrm\,4)\ \ k\ge 3. \\\end\end

The Hurwitz numbers can also be determined as follows:

H _{4=1/10} ,H _{4n} = (3)/(( 2 n -3 )( 16 n ^{2-1)}) \sum ^{n -1} _{k =1} \binom{ 4 n}{ 4 k}( 4 k -1 )( 4 (n - k) -1 ) H _{4k} H _{4(n-k)} , n\ge 2

and

H _{n=0} if n is not a multiple of 4 .^[58] This yields^[56] H _{8= (3)/(10)} , H _{12} = (567)/(130) , H _{16} = (43659)/(170) ,\ldots

Also^[59]

\operatorname{denom} H _{4n} =2 \prod _{(p-1)|4n} p where p\in P such that p\equiv 1 ( mod 4 ), just as\operatorname{denom} B _{2n} = \prod _{(p-1)|2n} p where p\in P (by the von Staudt–Clausen theorem).

In fact, the von Staudt–Clausen theorem states that

B _{2n} + \sum _{(p-1)|2n} (1)/(p) \in Z , n\ge 1

where

p is any prime, and an analogous theorem holds for the Hurwitz numbers: suppose that a\in Z is odd, b\in Z is even, p is a prime such that p\equiv 1 ( mod 4 ) , p = a ^{2+b} ^{2} (see Fermat's theorem on sums of two squares) and a\equiv b +1 ( mod 4 ) . Then for any given p , a = a _{p} is uniquely determined and^[56] H _{4n} - (1)/(2) - \sum _{(p-1)|4n} (( 2 a ^{4 n/(p -1 )} _{p)})/(p) l{\overset{ def }} \mathrm_n\in\mathbb,\quad n\ge 1,\operatorname{sl}z = \sum ^{inf ty} _{n =0} k _{4n+1} (z ^{4n+1})/(( 4 n +1 )!) , \left|z\right| < (\varpi)/(\sqrt{ 2) }\implies k_p\equiv 2a_p\,(\text\, p).

The sequence of the integers

G _{n} starts with 0 , -1 , 5 , 253 ,\ldots . ^[56]

Let

n\ge 2 . If 4 n +1 is a prime, then G _{n\equiv} 1 ( mod 4 ) . If 4 n +1 is not a prime, then G _{n\equiv} 3 ( mod 4 ) .^[60]

Some authors instead define the Hurwitz numbers as

H _{n'=H} _{4n} .

Appearances in Laurent series

The Hurwitz numbers appear in several Laurent series expansions related to the lemniscate functions:^[61]

\begin{align} \operatorname{sl} ^{2z} & = \sum ^{inf ty} _{n =1} (2 ^{4n} ( 1- ( -1 ) ^{n} 2 ^{2n} ) H _{4n})/(4 n) (z ^{4n-2})/(( 4 n -2 )!) , \left|z\right|< (\varpi)/(\sqrt{ 2) } \\\frac &= \frac-\sum_^\infty \frac\frac,\quad \left|z\right|<\frac \\\frac &= \frac-\sum_^\infty \frac\frac,\quad \left|z\right|<\varpi \\\frac &= \frac+\sum_^\infty \frac\frac,\quad \left|z\right|<\varpi\end

Analogously, in terms of the Bernoulli numbers:

(1)/(\sinh ^{2} z) = (1)/(z ^{2}) - \sum ^{inf ty} _{n =1} (2 ^{2n} B _{2n})/(2 n) (z ^{2n-2})/(( 2 n -2 )!) , \left|z\right|<\pi.

A quartic analog of the Legendre symbol

Let

p be a prime such that p\equiv 1 ( mod 4 ) . A quartic residue (mod p ) is any number congruent to the fourth power of an integer. Define \left(\tfrac{a}{p}\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'\in Z [i] (see^[62] for these numbers) such that^[63] \left( (a)/(p) \right) _{4=\prod} _{p'} (\operatorname{sl)/(( 2 \varpi) ap'/p)}{\operatorname{sl}( 2 \varpi p'/p)}. This theorem is analogous to\left( (a)/(p) \right) = \prod ^{(p -1)/(2)} _{n =1} (\sin ( 2 \pi a n/p))/(\sin ( 2 \pi n/p)) where \left(\tfrac{ ⋅ }{ ⋅ }\right) is the Legendre symbol.

World map projections

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).^[64]

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. 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.^[65] Because many partial differential equations can be effectively solved by conformal mapping, this map from sphere to cube is convenient for atmospheric modeling.^[66]

See also

• Elliptic function
  • Abel elliptic functions
  • Dixon elliptic functions
  • Jacobi elliptic functions
  • Weierstrass elliptic function
• Elliptic Gauss sum
• Gauss's constant
• Peirce quincuncial projection
• Schwarz–Christoffel mapping

External links

• Parker . Matt . Matt Parker . What is the area of a Squircle?. https://ghostarchive.org/varchive/youtube/20211219/gjtTcyWL0NA . 2021-12-19 . live. Stand-up Maths . 2021 . YouTube .

References

• • Book: Adams . Oscar Sherman . 1925 . [ftp://ftp.library.noaa.gov/docs.lib/htdocs/rescue/cgs_specpubs/QB275U35no1121925.pdf Elliptic functions applied to conformal world maps ]. US Government Printing Office . 297 .
• Ayoub . Raymond . Raymond Ayoub . 1984 . The Lemniscate and Fagnano's Contributions to Elliptic Integrals . Archive for History of Exact Sciences . 29 . 2 . 131–149 . 10.1007/BF00348244 .
• Book: Berndt . Bruce C. . Ramanujan's Notebooks Part IV . Springer . 1994 . First . 978-1-4612-6932-8 .
• Book: Borwein . Jonatham M. . Jonathan Borwein . Borwein . Peter B. . Peter Borwein . 1987 . 2.7 The Landen Transformation . 60 . Pi and the AGM . Wiley-Interscience .
• Book: Bottazzini . Umberto . Umberto Bottazzini . Gray . Jeremy . Jeremy Gray . 2013 . Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory . Springer . 10.1007/978-1-4614-5725-1 .
• Book: Carlson, Billie C. . 2010 . Bille C. Carlson . 1 . Olver . Frank . Frank W. J. Olver . Lozier . Daniel . Boisvert . Ronald . Clark . Charles . NIST Handbook of Mathematical Functions . Cambridge . 19. Elliptic Integrals . https://dlmf.nist.gov/19 . Digital Library of Mathematical Functions.
• Cox . David Archibald . David A. Cox . January 1984 . The Arithmetic-Geometric Mean of Gauss. L'Enseignement Mathématique. 30. 2. 275–330.
• Cox . David Archibald . Shurman . Jerry . 2005 . Geometry and number theory on clovers . The American Mathematical Monthly . 112 . 8 . 682–704 . 10.1080/00029890.2005.11920241 .
• Book: Cox . David Archibald . 2012 . The Lemniscate . 463–514 . Galois Theory . Wiley . 10.1002/9781118218457.ch15 .
• Book: Cox . David Archibald . Primes of the Form . Wiley . 2013 . Second .
• Cox . David Archibald . Trevor . Hyde . 2014 . The Galois theory of the lemniscate . Journal of Number Theory . 135 . 43–59 . 10.1016/j.jnt.2013.08.006 . 1208.2653 .
• Book: Enneper . Alfred . Alfred Enneper . 1890 . 1st ed. 1876 . Note III: Historische Notizen über geometrische Anwendungen elliptischer Integrale. . Historical notes on geometric applications of elliptic integrals . 524–547 . Elliptische Functionen, Theorie und Geschichte . de . Nebert . https://archive.org/details/elliptischefunct00enneuoft/page/524/ .
• Leonhard Euler . Euler . Leonhard . 1761 . Observationes de comparatione arcuum curvarum irrectificibilium . Observations on the comparison of arcs of irrectifiable curves . Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae . 6 . 58–84 . la. E252. (Figures)
• Euler . Leonhard . 1786 . De miris proprietatibus curvae elasticae sub aequatione $y\; =\; \backslash int\; xx\backslash mathop\; \backslash big/\; \backslash sqrt$ contentae . On the amazing properties of elastic curves contained in equation $y\; =\; \backslash int\; xx\backslash mathop\; \backslash big/\; \backslash sqrt$ . Acta Academiae Scientiarum Imperialis Petropolitanae . 1782 . 2 . 34–61 . la. E 605.
• Book: Eymard . Pierre . Lafon. Jean-Pierre . The Number Pi . Wilson. Stephen. American Mathematical Society . 2004 . 0-8218-3246-8.
  • Book: Gauss . Carl Friedrich . Carl Friedrich Gauss . Werke (Band X, Abteilung I) . Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen . Latin, German. 1917.
• Gómez-Molleda . M. A. . Lario . Joan-C. . 2019 . Ruler and Compass Constructions of the Equilateral Triangle and Pentagon in the Lemniscate Curve . 10.1007/s00283-019-09892-w . The Mathematical Intelligencer . 41 . 4 . 17–21 .
• Book: Greenhill . Alfred George . Alfred George Greenhill. 1892 . The Applications of Elliptic Functions. MacMillan .
• Guyou . Émile . 1887 . Nouveau système de projection de la sphère: Généralisation de la projection de Mercator . New system of projection of the sphere . Annales Hydrographiques . Série 2 . 9 . 16–35 . fr .
• Book: Houzel . Christian . Christian Houzel . 1978 . Fonctions elliptiques et intégrales abéliennes . Elliptic functions and Abelian integrals . Dieudonné . Jean . Jean Dieudonné . Abrégé d'histoire des mathématiques, 1700–1900. II . Hermann . 1–113 . fr.
• Hyde . Trevor . 2014 . A Wallis product on clovers . The American Mathematical Monthly . 121 . 3 . 237–243 . 10.4169/amer.math.monthly.121.03.237 .
• Kubota . Tomio . Tomio Kubota . 1964 . Some arithmetical applications of an elliptic function . Crelle's Journal . 214/215 . 141–145 . 10.1515/crll.1964.214-215.141 .
• Langer . Joel C. . David A. . Singer . 2010 . Reflections on the Lemniscate of Bernoulli: The Forty-Eight Faces of a Mathematical Gem . Milan Journal of Mathematics . 78 . 2 . 643–682 . 10.1007/s00032-010-0124-5 .
• Langer . Joel C. . David A. . Singer . 2011 . The lemniscatic chessboard . Forum Geometricorum . 11 . 183–199 .
• Book: Lawden . Derek Frank . Derek Frank Lawden . Elliptic Functions and Applications . Applied Mathematical Sciences . 1989 . 80 . Springer-Verlag . 10.1007/978-1-4757-3980-0 .
• Book: Lee, L. P. . Laurence Patrick Lee. 1976. Conformal Projections Based on Elliptic Functions. Toronto . B. V. Gutsell, York University. Cartographica Monographs . 16. limited. 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.
• Book: Lemmermeyer . Franz . Reciprocity Laws: From Euler to Eisenstein . Springer . 2000 . 3-540-66957-4.
• Levien . Raph . 2008 . Raph Levien . The elastica: a mathematical history . UCB/EECS-2008-103 . University of California at Berkeley .
• Levin . Aaron . A Geometric Interpretation of an Infinite Product for the Lemniscate Constant . The American Mathematical Monthly . 2006 . 113 . 6 . 510–520 . 10.2307/27641976 .
• Lindqvist . Peter . Peetre . Jaak . Jaak Peetre . Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals . The American Mathematical Monthly . 2001 . 108 . 5 . 403–410 . 10.1080/00029890.2001.11919766 .
• Book: Markushevich . Aleksei Ivanovich . Aleksei Ivanovich Markushevich . The Remarkable Sine Functions . 1966 . Elsevier .
• Book: Markushevich . Aleksei Ivanovich . Introduction to the Classical Theory of Abelian Functions . Translations of Mathematical Monographs . 1992 . 96 . American Mathematical Society . 10.1090/mmono/096 .
• McGregor . John L. . 2005 . C-CAM: Geometric Aspects and Dynamical Formulation . . 70 .
• Book: McKean . Henry . Henry McKean . Moll . Victor . Victor Hugo Moll . 1999 . Elliptic Curves: Function Theory, Geometry, Arithmetic . Cambridge . 9780521582285.
• Book: Milne-Thomson . Louis Melville . L. M. Milne-Thomson . 1964 . 16. Jacobian Elliptic Functions and Theta Functions . 567–585 . https://archive.org/details/handbookofmathem00abra/page/567/ . Milton . Abramowitz . Milton Abramowitz . Irene Ann . Stegun . Irene Stegun . Handbook of Mathematical Functions . Abramowitz and Stegun . National Bureau of Standards .
• Neuman . Edward . Edward Neuman . 2007 . On Gauss lemniscate functions and lemniscatic mean . Mathematica Pannonica . 18 . 1 . 77–94 .
• Nishimura . Ryo . New properties of the lemniscate function and its transformation . Journal of Mathematical Analysis and Applications . 2015 . 427 . 1 . 460–468 . 10.1016/j.jmaa.2015.02.066 . free .
• Ogawa . Takuma . Similarities between the trigonometric function and the lemniscate function from arithmetic view point . Tsukuba Journal of Mathematics . 2005 . 29 . 1 .
• Peirce . Charles Sanders . Charles Sanders Peirce . 1879 . A Quincuncial Projection of the Sphere . American Journal of Mathematics . 2 . 4 . 394–397 . 10.2307/2369491 .
• Book: Popescu-Pampu, Patrick . 2016 . What is the Genus? . Lecture Notes in Mathematics . 2162 . Springer . 10.1007/978-3-319-42312-8 .
• Book: Prasolov . Viktor . Solovyev . Yuri . 1997 . 4. Abel's Theorem on Division of Lemniscate . Elliptic functions and elliptic integrals . Translations of Mathematical Monographs . 170 . American Mathematical Society. . 10.1090/mmono/170 .
• Rančić . Miodrag . Purser . R. James . Mesinger . Fedor . 1996 . A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates . Quarterly Journal of the Royal Meteorological Society . 122 . 532 . 959–982 . 10.1002/qj.49712253209 .
• Book: Reinhardt . William P. . Walker . Peter L. . 2010a . 1 . Olver . Frank . Lozier . Daniel . Boisvert . Ronald . Clark . Charles . NIST Handbook of Mathematical Functions . Cambridge . 22. Jacobian Elliptic Functions . https://dlmf.nist.gov/22 . Digital Library of Mathematical Functions .
• Book: Reinhardt . William P. . Walker . Peter L. . 2010b . 1 . Olver . Frank . Lozier . Daniel . Boisvert . Ronald . Clark . Charles . NIST Handbook of Mathematical Functions . Cambridge . 23. Weierstrass Elliptic and Modular Functions . https://dlmf.nist.gov/23 . Digital Library of Mathematical Functions .
• Robinson . Paul L. . The Lemniscatic Functions . 2019a . 1902.08614 .
• Robinson . Paul L. . The Elliptic Functions in a First-Order System . 2019b . 1903.07147 .
• Rosen . Michael . Michael Rosen (mathematician) . Abel's Theorem on the Lemniscate . The American Mathematical Monthly . 1981 . 88 . 6 . 387–395 . 10.2307/2321821 .
• Book: Roy . Ranjan . Elliptic and Modular Functions from Gauss to Dedekind to Hecke . Cambridge University Press . 28 . 2017 . 978-1-107-15938-9.
• Book: Some milestones of lemniscatomy . Schappacher . Norbert . Norbert Schappacher . 1997 . Sertöz . S. . Algebraic Geometry . Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey . Marcel Dekker . 257–290 . http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf.
• Schneider . Theodor . Theodor Schneider . 1937 . Arithmetische Untersuchungen elliptischer Integrale . Arithmetic investigations of elliptic integrals . de . Mathematische Annalen . 113 . 1 . 1–13. 10.1007/BF01571618 .
• Schwarz . Hermann Amandus . Hermann Schwarz. 1869 . Ueber einige Abbildungsaufgaben . About some mapping problems . de . Crelle's Journal . 70 . 105–120 . 10.1515/crll.1869.70.105 .
• Book: Siegel . Carl Ludwig . Carl Ludwig Siegel . Topics in Complex Function Theory, Vol. I . 1. Elliptic Functions . 1–89 . Wiley-Interscience . 1969 . 0-471-60844-0.
• Snape . Jamie . 2004 . Bernoulli's Lemniscate . 50–56 . Applications of Elliptic Functions in Classical and Algebraic Geometry . https://www.jamiesnape.io/publications/mmath/ . University of Durham.
• Book: Southard . Thomas H. . 1964 . 18. Weierstrass Elliptic and Related Functions . 627–683 . https://archive.org/details/handbookofmathem00abra/page/627/ . Milton . Abramowitz . Milton Abramowitz . Irene Ann . Stegun . Irene Stegun . Handbook of Mathematical Functions . Abramowitz and Stegun . National Bureau of Standards .
  • Todd . John . John (Jack) Todd . The lemniscate constants . Communications of the ACM . 1975 . 18 . 1 . 14–19 . 10.1145/360569.360580 . free .
• Vigren . Erik . Dieckmann . Andreas . Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges . Symmetry . 21 June 2020 . 12 . 6 . 1040 . 10.3390/sym12061040 . free .
• Book: Whittaker . Edmund Taylor . Edmund T. Whittaker . Watson . George Neville . George N. Watson . 1920 . 1st ed. 1902 . 22.8 The lemniscate functions . 524–528 . https://archive.org/details/courseofmodernan00whit/page/524/ . 3rd . A Course of Modern Analysis . A Course of Modern Analysis . Cambridge .
• Book: Whittaker . Edmund Taylor . Edmund T. Whittaker . Watson . George Neville . George N. Watson . 1927 . 4th ed. 1927 . 21 The theta functions . 469–470 . 4th . A Course of Modern Analysis . A Course of Modern Analysis . Cambridge .

Notes and References

1. ;

2. p. 199 used the symbols and for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. p. 316, p. 204, and p. 240. uses and . use the symbols and . Some sources use the generic letters and . use the letter for the lemniscate sine and for its derivative.
3. The circle x ^{2+y} ^{2} = x is the unit-diameter circle centered at $\backslash bigl(\backslash tfrac12,\; 0\backslash bigr)$ with polar equation r = \cos \theta, the degree-2 clover under the definition from . This is not the unit-radius circle x ^{2+y} ^{2=1} centered at the origin. Notice that the lemniscate l (x ^{2+y} ^{2r){}} ^{2=x} ^{2-y} ^{2} is the degree-4 clover.
4. The fundamental periods ( 1+ i)\varpi and ( 1- i)\varpi are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.
5. starts from this definition and thence derives other properties of the lemniscate functions.
6. This map was the first ever picture of a Schwarz–Christoffel mapping, in p. 113.
7. . OEIS sequence A062539 lists the lemniscate constant's decimal digits.
8. Combining the first and fourth identity gives \operatorname{sl}z =- i/\operatorname{sl}(z - ( 1+ i)\varpi/ 2 ) . This identity is (incorrectly) given in p. 226, without the minus sign at the front of the right-hand side.
9. The even Gaussian integers are the residue class of 0, modulo, the black squares on a checkerboard.
10. §22.12.6, §22.12.12
11. Analogously, (1)/(\sin z) = \sum _{n\inZ} (( -1 ) ^{n})/(z + n\pi) .
12. generalizes the first of these forms.
13. §44 p. 79, §47 pp. 80–81
14. §46 p. 80
15. In fact, i ^{\varepsilon=\operatorname{sl}\tfrac{\beta\varpi}{2}} .

16. Lemnatomic polynomials

    Let

    L be the latticeL = Z ( 1+ i)\varpi + Z ( 1- i)\varpi. Furthermore, let K = Q (i) , l{O}= Z [i] , z\in C , \beta = m + in , \gamma = m' + in' (where m,n,m',n'\in Z ), m + n be odd, m' + n' be odd, \gamma\equiv 1 \operatorname{mod} 2 ( 1+ i) and \operatorname{sl} \beta z = M _{\beta} (\operatorname{sl}z) . ThenM _{\beta} (x) = i ^{\varepsilon} x (P _{\beta} (x ^{4)})/(Q _{\beta} (x ^{4)}) for some coprime polynomials P _{\beta} (x), Q _{\beta} (x)\in l{O}[x] and some \varepsilon\in \{ 0 , 1 , 2 , 3 \} ^[15]
17. The fourth root with the least positive principal argument is chosen.
18. The restriction to positive and odd \beta can be dropped in \operatorname{deg} Λ ^{x \right|} _{\beta = \left|( l{O}/\beta l{O})} .
19. p. 142, Example 7.29(c)
20. p. 200

21. . use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same.

22. §22.18.E6
23. Such numbers are OEIS sequence A003401.

24. ;

25. ;

26. Web site: A104203. The On-Line Encyclopedia of Integer Sequences.
27. Book: Lomont . J.S.. Brillhart. John. Elliptic Polynomials. CRC Press . 2001 . 1-58488-210-7. 12, 44.
28. Book: Lomont . J.S.. Brillhart. John. Elliptic Polynomials. CRC Press . 2001 . 1-58488-210-7. p. 79, eq. 5.36
29. Book: Lomont . J.S.. Brillhart. John. Elliptic Polynomials. CRC Press . 2001 . 1-58488-210-7. p. 79, eq. 5. 36 and p. 78, eq. 5.33
30. Web site: A193543 - Oeis .
31. Web site: A289695 - Oeis .
32. Book: Wall . H. S. . Analytic Theory of Continued Fractions . Chelsea Publishing Company . 1948 . 374–375.
33. Web site: Dieckmann . Andreas . Collection of Infinite Products and Series .
34. §22.12.12; p. 7
35. In general, \sinh(x - n\pi) and \sin (x - n\pi i) =- i\sinh (ix + n\pi) are not equivalent, but the resulting infinite sum is the same.
36. §22.11
37. §22.2.E7
38. p. 247, 248, 253
39. §22.11.E1
40. p. 227.
41. Book: Cartan, H. . Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes. Hermann . 1961. French . 160–164.
42. More precisely, suppose \{a _{n\}} 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 \{n _{1,n} _{2,n} _{3,\ldots\}} is any permutation of \{ 1 , 2 , 3 ,\ldots\} , then $\backslash prod\_^\backslash infty\; (1+a\_n(z))=\backslash prod\_^\backslash infty\; (1+a\_(z))$ for all z\in S . The theorem in question then follows from the fact that there exists a bijection between the natural numbers and \alpha 's (resp. \beta 's).
43. p. 58
44. More precisely, if for each k , $\backslash lim\_\; a\_k(n)$ exists and there is a convergent series $\backslash sum\_^\backslash infty\; M\_k$ of nonnegative real numbers such that \left|a _{k(n)\right|\le} M _{k} for all n\in N and 1 \le k\le n , then\lim _{n\toinfty} \sum ^{n} _{k =1} a _{k(n)=\sum} ^{inf ty} ^{k =1} \lim _{n\toinfty} a _{k(n).}
45. 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 $\backslash sum\_\backslash frac=-\backslash ,\backslash text\backslash ,z^5$ in the M series, and infinitely many others.
46. Book: Gauss . C. F. . Werke (Band III) . Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen . 1866 . Latin, German. p. 405; there's an error on the page: the coefficient of \varphi ^{17} should be \tfrac{ 107 }{ 7410154752000 } , not \tfrac{ 107 }{ 207484333056000 } .
47. The M function satisfies the differential equation M(z)M''''(z) -4 M'(z)M'''(z) + +3 M''(z) ^{2-2M(z)} ^{2=0} (see Gauss (1866), p. 408). The N function satisfies the differential equation (N''(z) N(z) - N'(z) ^{2)} ^{2-M(z)} ^{4=0.}
48. If $M(z)=\backslash sum\_^\backslash infty\; a\_nz^$, then the coefficients a _{n} are given by the recurrence $a\_=-\backslash frac\backslash sum\_^n\; 2^\; a\_k\; \backslash frac$ with a _{0=1} where H _{n} are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers.
49. Book: Zhuravskiy, A. M. . Spravochnik po ellipticheskim funktsiyam . Izd. Akad. Nauk. U.S.S.R. . 1941 . Russian.
50. p. 234
51. Book: Armitage . J. V. . Elliptic Functions . Eberlein . W. F. . Cambridge University Press . 2006 . 978-0-521-78563-1 . 49.
52. The identity \operatorname{cl} z = {\operatorname{cn}}\left(\sqrt 2 z;\tfrac{ 1 }{\sqrt 2 }\right) can be found in p. 33.
53. http://oeis.org/A175576
54. Book: Berndt . Bruce C. . Ramanujan's Notebooks Part II . Springer . 1989 . 978-1-4612-4530-8. p. 96
55. p. 508, 509
56. Book: Arakawa . Tsuneo . Ibukiyama. Tomoyoshi . Kaneko. Masanobu. Bernoulli Numbers and Zeta Functions . Springer . 2014 . 978-4-431-54918-5. p. 203—206
57. Equivalently, H _{n=-\lim} _{z\to} (d ^{n})/(\left( dz ^{n}) (( 1+ i)z/ 2)/((( 1+ i)z/ 2 )} + \operatorname{sl) (z)/(2) l{E}\left( (z)/(2) ;i\right)\right) where n\ge 4 and l{E}(⋅ ;i) is the Jacobi epsilon function with modulus i .
58. The Bernoulli numbers can be determined by an analogous recurrence: B _{2n} =- (1)/(2 n +1) \sum ^{n -1} _{k =1} \binom{ 2 n}{ 2 k} B _{2k} B _{2(n-k)} where n\ge 2 and B _{2=1/6} .
59. Katz . Nicholas M. . 1975 . The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers . Mathematische Annalen . 216 . 1 . 1–4. See eq. (9)
60. Book: Hurwitz . Adolf . Mathematische Werke: Band II . German. Springer Basel AG . 1963. p. 370
61. Arakawa et al. (2014) define H _{4n} by the expansion of 1 /\operatorname{sl} ^{2.}
62. Eisenstein . G.. Beiträge zur Theorie der elliptischen Functionen . German. Journal für die reine und angewandte Mathematik. 1846 . 30. Eisenstein uses \varphi = \operatorname{sl} and \omega =2 \varpi .
63. Ogawa . Takuma . Similarities between the trigonometric function and the lemniscate function from arithmetic view point . Tsukuba Journal of Mathematics . 2005 . 29 . 1 .
64. . and introduced transverse and oblique aspects of the same projection, respectively. Also see . These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice.

65. .

66. .