thumb|upright=2.0|The lemniscate sine (red) and lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric sine  (pale dashed red).
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.
The lemniscate sine and lemniscate cosine functions, usually written with the symbols  and  (sometimes the symbols  and  or  and  are used instead)Gauss used the symbols  and  for the lemniscate sine and cosine, respectively.
Ayoub (1984) uses  and .
Whittaker and Watson (1920) use the symbols  and .
Some sources use the generic letters  and  .
Prasolov & Solovyev (1997) use the letter  for the lemniscate sine and  for its derivative.
are analogous to the trigonometric functions sine and cosine.
While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle x^2+y^2 = x , the lemniscate sine relates the arc length to the chord length of a lemniscate \bigl(x^2+y^2\bigr){}^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.
The  and  functions have a square period lattice (a multiple of the Gaussian integers) with fundamental periods \{(1 + i)\varpi, (1 - i)\varpi\},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.
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 functions  and  have a square period lattice with fundamental periods \bigl\{\sqrt2\varpi, \sqrt2\varpi i\bigr\}.
The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function \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:Robinson (2019a) starts from this definition and thence derives other properties of the lemniscate functions.
\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{cl} z = -\operatorname{sl} z\, \bigl(1 + \operatorname{cl}^2 z\bigr),\ \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{sl} z = \operatorname{cl} z\, \bigl(1 + \operatorname{sl}^2 z\bigr),\  \operatorname{cl}(0) = 1,\  \operatorname{sl}(0) = 0,
or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners \big\{\tfrac12\varpi, \tfrac12\varpi i, -\tfrac12\varpi, -\tfrac12\varpi i\big\}\colonThis map was the first ever picture of a Schwarz–Christoffel mapping, in Schwarz (1869) p. 113.
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}}.
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:
\frac{\mathrm{d}}{\mathrm{d}z} \cos z = -\sin z,\ \frac{\mathrm{d}}{\mathrm{d}z} \sin z = \cos z,\  \cos(0) = 1,\  \sin(0) = 0,
or as inverses of a map from the upper half-plane to a half-infinite strip with real part between -\tfrac12\pi, \tfrac12\pi and positive imaginary part:
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}}.
Arc length of Bernoulli's lemniscate
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.
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.
The lemniscate of Bernoulli with half-width  is the locus of points in the plane such that the product of their distances from the two focal points F_1 = \bigl({-\tfrac1\sqrt2},0\bigr) and F_2 = \bigl(\tfrac1\sqrt2,0\bigr) is the constant \tfrac12.
This is a quartic curve satisfying the polar equation r^2 = \cos 2\theta or the Cartesian equation \bigl(x^2+y^2\bigr){}^2=x^2-y^2.
The points on the lemniscate 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:
\big(x(r), y(r)\big) = \biggl(\!\sqrt{\tfrac12r^2\bigl(1 + r^2\bigr)}, \sqrt{\tfrac12r^2\bigl(1 - r^2\bigr)}\,\biggr).
Using this parametrization with r \in [0, 1] for a quarter of the lemniscate, the arc length from the origin to a point \big(x(r), y(r)\big) is:Euler (1761), Siegel (1969).
Prasolov & Solovyev (1997) use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same.
\begin{aligned} &\int_0^r \sqrt{x'(t)^2 + y'(t)^2} \mathop{\mathrm{d}t} \\ & \quad {}= \int_0^r \sqrt{\frac{(1+2t^2)^2}{2(1+t^2)} + \frac{(1-2t^2)^2}{2(1-t^2)}} \mathop{\mathrm{d}t} \\[6mu] & \quad {}= \int_0^r \frac{\mathrm{d}t}{\sqrt{1-t^4}} \\[6mu] & \quad {}= \operatorname{arcsl} r.
\end{aligned}
Likewise, the arc length from (1,0) to \big(x(r), y(r)\big) is:
\begin{aligned} &\int_r^1 \sqrt{x'(t)^2 + y'(t)^2} \mathop{\mathrm{d}t} \\ & \quad {}= \int_r^1 \frac{\mathrm{d}t}{\sqrt{1-t^4}} \\[6mu] & \quad {}= \operatorname{arccl} r = \tfrac12\varpi - \operatorname{arcsl} r.
\end{aligned}
Or in the inverse direction, the lemniscate sine and cosine functions give the distance from the origin as functions of arc length from the origin and the point (1,0), respectively.
Analogously, the circular sine and cosine functions relate the chord length to the arc length for the unit diameter circle with polar equation r = \cos \theta or Cartesian equation x^2 + y^2 = x, using the same argument above but with the parametrization:
\big(x(r), y(r)\big) = \biggl(r^2, \sqrt{r^2\bigl(1-r^2\bigr)}\,\biggr).
The lemniscate integral and lemniscate functions satisfy an argument duplication identity discovered by Fagnano in 1718:Siegel (1969), Schappacher (1997)
\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}.
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 straightedge and compass whenever  is of the form n = 2^kp_1p_2\cdots p_m where  and  are non-negative integers and each  (if any) is a distinct Fermat prime.Such numbers are OEIS sequence A003401.
This was demonstrated by Niels Abel in 1827–1828.Abel (1827–1828), Rosen (1981), Prasolov & Solovyev (1997) Arc length of rectangular elastica
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  relative to the  coordinate of the rectangular elastica.Euler (1786), Sridharan (2004), Levien (2008) This curve has  coordinate and arc length:
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}}
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.
Lemniscate constant
thumb|upright=1.3|The lemniscate constant is twice the value of the complete lemniscate integral.
The lemniscate functions have minimal real period  and fundamental complex periods (1+i)\varpi and (1-i)\varpi for a constant  called the lemniscate constant,Schappacher (1997).
OEIS sequence A062539 lists the lemniscate constant's decimal digits.
\begin{aligned} \varpi &= 2\int_0^1\frac{\mathrm{d}t}{\sqrt{1-t^4}}  = \sqrt2\int_0^\infty\frac{\mathrm{d}t}{\sqrt{1+t^4}}  = \int_0^1\frac{\mathrm dt}{\sqrt{t-t^3}} \\[6mu] &= 4\int_0^\infty\Bigl(\sqrt[4]{1+t^{4}}-t\Bigr)\,\mathrm{d}t  = 2\sqrt2\int_0^1 \sqrt[4]{1-t^{4}}\mathop{\mathrm{d}t} =3\int_0^1 \sqrt{1-t^4}\,\mathrm dt\\[2mu] &= \tfrac{1}{2}\Beta\bigl( \tfrac14, \tfrac12\bigr)  = \frac{\Gamma\bigl(\tfrac14\bigr)^2}{2\sqrt{2\pi}}  = \frac{\pi}{{\operatorname{M}}\bigl(1, \sqrt2\bigr)}  = \sqrt{\pi}e^{\beta '(0)}\\[5mu] &= 2.62205\;75542\;92119\;81046\;48395\;89891\;11941\ldots, \end{aligned}
where  is the beta function,  is the gamma function,  is the arithmetic–geometric mean and  is the Dirichlet beta function.
Geometrically,  is the ratio of the perimeter of Bernoulli's lemniscate to its diameter.
The lemniscate constant was proven transcendental by Theodor Schneider in 1937.Schneider (1937) In 1975, Gregory Chudnovsky proved that  and  are algebraically independent over \mathbb{Q}.G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486G.
V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6 The related constant G = \varpi / \pi = 0.8346\ldots is Gauss's constant.
The lemniscate functions satisfy the basic relation \operatorname{cl}z = {\operatorname{sl}}\bigl(\tfrac12\varpi - z\bigr).
Furthermore,  is related to the area under the curve x^4 + y^4 = 1.
Defining \pi_n \mathrel{:=} \Beta\bigl(\tfrac1n, \tfrac1n \bigr), twice the area in the positive quadrant under the curve x^n + y^n = 1 is 2 \int_0^1 \sqrt[n]{1 - x^n}\mathop{\mathrm{d}x} = \tfrac1n \pi_n.
In the quartic case, \tfrac14 \pi_4 = \tfrac1\sqrt{2} \varpi.
Euler discovered in 1738 that for the rectangular elastica:Levien (2008).
Todd (1975) calls these two factors A = \varpi/2 and B = \pi/2\varpi the lemniscate constants, and discusses methods for computing them.
\textrm{arc}\ \textrm{length}\cdot\textrm{height} = \int_0^1 \frac{\mathrm{d}x}{\sqrt{1 - x^4}} \cdot \int_0^1 \frac{x^2 \mathop{\mathrm{d}x}}{\sqrt{1 - x^4}} = \frac\varpi2 \cdot \frac\pi{2\varpi} = \frac\pi4
Viète's formula for  can be written:
\frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots
An analogous formula for  is:Levin (2006)
\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 +  \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 +  \frac12 \Bigg/ \!\sqrt{\frac12 +  \frac12 \bigg/ \!\sqrt\frac12}} \cdots
The Wallis product for  is:
\frac{\pi}{2} = \prod_{n=1}^{\infty} \left(\frac{2n}{2n-1} \cdot \frac{2n}{2n+1}\right) = \biggl(\frac{2}{1} \cdot \frac{2}{3}\biggr) \biggl(\frac{4}{3} \cdot \frac{4}{5}\biggr) \biggl(\frac{6}{5} \cdot \frac{6}{7}\biggr) \cdots
An analogous formula for  is:Hyde (2014) proves the validity of a more general Wallis-like formula for clover curves; here the special case of the lemniscate is slightly transformed, for clarity.
\frac{\varpi}{2} = \prod_{n=1}^{\infty} \left(\frac{4n-1}{4n-2} \cdot \frac{4n}{4n+1}\right) = \biggl(\frac{3}{2} \cdot \frac{4}{5}\biggr) \biggl(\frac{7}{6} \cdot \frac{8}{9}\biggr) \biggl(\frac{11}{10} \cdot \frac{12}{13}\biggr) \cdots
The ratio of these two is:
\frac{\varpi}{\pi} = \prod_{n=1}^{\infty} \left(\frac{4n-1}{4n} \cdot \frac{4n+2}{4n+1}\right) = \biggl(\frac{3}{4} \cdot \frac{6}{5}\biggr) \biggl(\frac{7}{8} \cdot \frac{10}{9}\biggr) \biggl(\frac{11}{12} \cdot \frac{14}{13}\biggr) \cdots
An infinite series for \varpi / \pi discovered by Gauss is:Bottazzini & Gray (2013), p. 60
\frac{\varpi}{\pi} = \sum_{n=0}^\infty (-1)^n \prod_{k=1}^n \frac{(2k-1)^2}{(2k)^2}   = 1 - \frac{1^2}{2^2} + \frac{1^2\cdot3^2}{2^2\cdot4^2} - \frac{1^2\cdot3^2\cdot5^2}{2^2\cdot4^2\cdot6^2} + \cdots
The Machin formula for  is \tfrac14\pi = 4 \arctan \tfrac15 - \arctan \tfrac1{239}, and several similar formulas for  can be developed using trigonometric angle sum identities, e.g. Euler's formula \tfrac14\pi = \arctan\tfrac12 + \arctan\tfrac13.
Analogous formulas can be developed for , including the following found by Gauss: \tfrac12\varpi = 2 \operatorname{arcsl} \tfrac12 + \operatorname{arcsl} \tfrac7{23}.Todd (1975)
In a spirit similar to that of the Basel problem,
\sum_{z\in\mathbf{Z}[i]\setminus\{0\}}\frac{1}{z^4}=G_4(i)=\frac{\varpi ^4}{15}
where \mathbf{Z}[i] are the Gaussian integers and G_4(\tau) is the Eisenstein series of weight 4.
p. 249, 250 Symmetries
At translations of \tfrac12\varpi the lemniscate functions  and  are exchanged, and at translations of \tfrac12i\varpi they are additionally rotated and reciprocated:
\begin{aligned} {\operatorname{cl}}\bigl(z \pm \tfrac12\varpi\bigr) &= \mp\operatorname{sl} z,& {\operatorname{cl}}\bigl(z \pm \tfrac12i\varpi\bigr) &= \frac{\mp i}{\operatorname{sl} z} \\[6mu] {\operatorname{sl}}\bigl(z \pm \tfrac12\varpi\bigr) &= \pm\operatorname{cl} z,&  {\operatorname{sl}}\bigl(z \pm \tfrac12i\varpi\bigr) &= \frac{\pm i}{\operatorname{cl} z} \end{aligned}
Doubling these to translations by a unit-Gaussian-integer multiple of \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 \\[4mu] \operatorname{sl} (z + \varpi) &= \operatorname{sl} (z + i\varpi) = -\operatorname{sl} z \end{aligned}
As a result, both functions are invariant under translation by an even-Gaussian-integer multiple of \varpi.The even Gaussian integers are the residue class of 0, modulo , the black squares on a checkerboard.
That is, a displacement (a + bi)\varpi, with a + b = 2k for integers , , and  .
\begin{aligned} {\operatorname{cl}}\bigl(z + (1 + i)\varpi\bigr) &= {\operatorname{cl}} \bigl(z + (1 - i)\varpi\bigr) = \operatorname{cl} z \\[4mu] {\operatorname{sl}}\bigl(z + (1 + i)\varpi\bigr) &= {\operatorname{sl}} \bigl(z + (1 - i)\varpi\bigr) = \operatorname{sl} z \end{aligned}
This makes them elliptic functions (doubly periodic meromorphic functions in the complex plane) with a diagonal square period lattice of fundamental periods (1 + i)\varpi and (1 - i)\varpi.Prasolov & Solovyev (1997), Robinson (2019a) Elliptic functions with a square period lattice are more symmetrical than arbitrary elliptic functions, following the symmetries of the square.
Reflections and quarter-turn rotations of lemniscate function arguments have simple expressions:
\begin{aligned} \operatorname{cl} \bar{z} &= \overline{\operatorname{cl} z} \\[6mu] \operatorname{sl} \bar{z} &= \overline{\operatorname{sl} z} \\[4mu] \operatorname{cl} iz &= \frac{1}{\operatorname{cl} z} \\[6mu] \operatorname{sl} iz &= i \operatorname{sl} z \end{aligned}
The  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 \bigl(a + \tfrac12\bigr)\varpi + \bigl(b + \tfrac12\bigr)\varpi i, with residues (-1)^{a-b+1}i.
The  function is reflected and offset from the  function, \operatorname{cl}z = {\operatorname{sl}}\bigl(\tfrac12\varpi - z\bigr).
It has zeros for arguments \bigl(a + \tfrac12\bigr)\varpi + b\varpi i and poles for arguments a\varpi + \bigl(b + \tfrac12\bigr)\varpi i, with residues (-1)^{a-b}i.
Pythagorean-like identity
thumb|upright=1.3|Curves  for various values of a.
Negative  in green, positive  in blue,  in red,  in black.
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:Lindqvist & Peetre (2001) generalizes the first of these forms.
\bigl(1 + \operatorname{cl^2} z\bigr) \bigl(1+\operatorname{sl^2} z\bigr) = 2
\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}
Defining a tangent-sum operator as a \oplus b \mathrel{:=} \tan(\arctan a + \arctan b), gives:
\operatorname{cl^2} z \oplus \operatorname{sl^2} z = 1
Derivatives and integrals
The derivatives are as follows:
\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cl} z = \operatorname{cl'}z &= -\operatorname{sl}z\, \bigl(1 + \operatorname{cl^2} z\bigr) \\ \operatorname{cl'^2} z &= 1 -  \operatorname{cl^4} z \\[5mu]
\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sl} z = \operatorname{sl'}z &= \operatorname{cl}z\, \bigl(1 + \operatorname{sl^2} z\bigr) \\ \operatorname{sl'^2} z &= 1 -  \operatorname{sl^4} z
\end{aligned}
The second derivatives of lemniscate sine and lemniscate cosine are their negative duplicated cubes:
\frac{\mathrm{d}^2}{\mathrm{d}z^2}\operatorname{cl}z = -2\operatorname{cl^3}z
\frac{\mathrm{d}^2}{\mathrm{d}z^2}\operatorname{sl}z = -2\operatorname{sl^3}z
The lemniscate functions can be integrated using the inverse tangent function:
\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
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:Ayoub (1984), Prasolov & Solovyev (1997)
\operatorname{sl}(u+v) = \frac{\operatorname{sl}u\,\operatorname{sl'}v + \operatorname{sl}v\,\operatorname{sl'}u} {1 + \operatorname{sl^2}u\, \operatorname{sl^2}v}
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 \oplus b \mathrel{:=} \tan(\arctan a + \arctan b) and tangent-difference operator a \ominus b \mathrel{:=} a \oplus (-b), the argument sum and difference identities can be expressed as:Euler (1761), §44 p. 79, §47 pp.
80–81
\begin{aligned} \operatorname{cl}(u+v) &= \operatorname{cl}u\,\operatorname{cl}v \ominus \operatorname{sl}u\, \operatorname{sl}v = \frac{\operatorname{cl}u\, \operatorname{cl}v - \operatorname{sl}u\, \operatorname{sl}v} {1 + \operatorname{sl}u\, \operatorname{cl}u\, \operatorname{sl}v\, \operatorname{cl}v} \\[2mu] \operatorname{cl}(u-v) &= \operatorname{cl}u\,\operatorname{cl}v \oplus \operatorname{sl}u\, \operatorname{sl}v \\[2mu] \operatorname{sl}(u+v) &= \operatorname{sl}u\,\operatorname{cl}v \oplus \operatorname{cl}u\,\operatorname{sl}v = \frac{\operatorname{sl}u\, \operatorname{cl}v + \operatorname{cl}u\, \operatorname{sl}v} {1 - \operatorname{sl}u\, \operatorname{cl}u\, \operatorname{sl}v\, \operatorname{cl}v} \\[2mu] \operatorname{sl}(u-v) &= \operatorname{sl}u\,\operatorname{cl}v \ominus \operatorname{cl}u\,\operatorname{sl}v
\end{aligned}
These resemble their trigonometric analogs:
\begin{aligned} \cos(u \pm v) &= \cos u\,\cos v \mp \sin u\,\sin v \\[6mu] \sin(u \pm v) &= \sin u\,\cos v \pm \cos u\,\sin v
\end{aligned}
Bisection formulas:
\operatorname{cl}^2 \tfrac12x = \frac{1+\operatorname{cl}x \sqrt{1+\operatorname{sl}^2x}}{\sqrt{1+\operatorname{sl}^2x}+1}
\operatorname{sl}^2 \tfrac12x = \frac{1-\operatorname{cl}x\sqrt{1+\operatorname{sl}^2x}}{\sqrt{1+\operatorname{sl}^2x}+1}
Duplication formulas:Euler (1761) §46 p. 80
\operatorname{cl} 2x = \frac{-1+2\,\operatorname{cl}^2x + \operatorname{cl}^4x}{1+2\,\operatorname{cl}^2x - \operatorname{cl}^4x}
\operatorname{sl} 2x = 2\,\operatorname{sl}x\,\operatorname{cl}x\frac{1+\operatorname{sl}^2x}{1+\operatorname{sl}^4x}
Triplication formulas:
\operatorname{cl} 3x = \frac{-3\,\operatorname{cl}x + 6\,\operatorname{cl}^5x + \operatorname{cl}^9x}{1+6\,\operatorname{cl}^4x - 3\,\operatorname{cl}^8x}
\operatorname{sl} 3x = \frac{3\,\operatorname{sl}x - 6\,\operatorname{sl}^5x - \operatorname{sl}^9x}{1 + 6\,\operatorname{sl}^4x - 3\,\operatorname{sl}^8x}
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, whenever  is of the form n = 2^kp_1p_2\cdots p_m where  and  are non-negative integers and each  (if any) is a distinct Fermat prime.Rosen (1981) However, the expressions become unwieldy as  grows.
Below are the expressions for dividing the lemniscate (x^2+y^2)^2=x^2-y^2 into  parts of equal length for some .
The -division points are obtained by intersecting the lemniscate (x^2+y^2)^2=x^2-y^2 with the circle x^2+y^2=\operatorname{sl}^2\tfrac{2j\varpi}{n} where j\in\{1,2,\ldots ,n\}.
Power series
The power series expansion of the lemniscate sine at the origin is
\operatorname{sl}z=\sum_{n=0}^\infty a_n z^n=z-\tfrac{1}{10}z^5+\tfrac{1}{120}z^9-\tfrac{11}{15600}z^{13}+\cdots,\quad z\in\mathbb{C},\, |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+2)\in\mathbb{N}_0:\,a_{n+2}=-\frac{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, so
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}.
Relation to Weierstrass and Jacobi elliptic functions
The lemniscate functions are closely related to the Weierstrass elliptic function \wp(z; 1, 0) (the "lemniscatic case"), with invariants  and .
This lattice has fundamental half periods \omega_1 = \varpi/\sqrt2, 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".Robinson (2019a) The square of the lemniscate sine can be represented as
\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)\varpi/\sqrt2;1,0\right)}
where the second and third argument of \wp denote the lattice invariants.
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.The identity \operatorname{cl} z = {\operatorname{cn}}\bigl(\sqrt2z;\tfrac{1}{\sqrt2}\bigr) can be found in Greenhill (1892), p. 33.
\operatorname{sl} z = \operatorname{sn}(z;i)={\tfrac1{\sqrt2}\operatorname{sd}}\bigl(\sqrt2z;\tfrac{1}{\sqrt2}\bigr)
\operatorname{cl} z = \operatorname{cd}(z;i)= {\operatorname{cn}}\bigl(\sqrt2z;\tfrac{1}{\sqrt2}\bigr)
where the second arguments denote the elliptic modulus.
Relation to the modular lambda function
The lemniscate sine can be used for the computation of values of the modular lambda function:
\prod_{k=1}^n \;{\operatorname{sl}}{\left(\frac{2k-1}{2n+1}\frac{\varpi}{2}\right)} =\sqrt[8]{\frac{\lambda ((2n+1)i)}{1-\lambda ((2n+1)i)}}
For example:
\begin{aligned} &{\operatorname{sl}}\bigl(\tfrac1{14}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac3{14}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac5{14}\varpi\bigr) \\[7mu] &\quad {}= \sqrt[8]{\frac{\lambda (7i)}{1-\lambda (7i)}}  = {\tan}\Bigl({\tfrac{1}{2}\arccsc}\Bigl(\tfrac{1}{2}\sqrt{8\sqrt{7}+21}+\tfrac{1}{2}\sqrt{7}+1\Bigr)\Bigr) \\[18mu] & {\operatorname{sl}}\bigl(\tfrac1{18}\varpi\bigr)\, {\operatorname{sl}}\bigl(\tfrac3{18}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac5{18}\varpi\bigr)\,{\operatorname{sl}}\bigl(\tfrac7{18}\varpi\bigr) \\[-3mu] &\quad {}= \sqrt[8]{\frac{\lambda (9i)}{1-\lambda (9i)}} = {\tan}\Biggl( \frac\pi4 - {\arctan}\Biggl(\frac{2\sqrt[3]{2\sqrt{3}-2}-2\sqrt[3]{2-\sqrt{3}}+\sqrt{3}-1}{\sqrt[4]{12}}\Biggr)\Biggr) \end{aligned}
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:Vigren & Dieckmann (2020), p. 7
\operatorname{sl}\Bigl(\frac\varpi\pi x\Bigr) = \frac\varpi\pi \Bigg/ \sum_{n=-\infty}^\infty \frac{(-1)^n}{{\sinh} {\left(x-n\pi\right)}}
Fourier series method:Reinhardt & Walker (2010), 22.11
\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 |\operatorname{Im}x|<\frac{\pi}{2}
The lemniscate sine can be computed more rapidly by
\operatorname{sl}\Bigl(\frac\varpi\pi x\Bigr) = \frac{{\theta_1}{\left(x, e^{-\pi}\right)}}{{\theta_3}{\left(x, e^{-\pi}\right)}}
where
\begin{aligned} \theta_1(x,q)&=\sum_{n\in\mathbb{Z}}q^{(n+1/2)^2}\exp\left((2n+1)ix+\bigl(n-\tfrac12\bigr)i\pi \right) \\ &=2\sum_{n=0}^\infty (-1)^n q^{(n+1/2)^2}\sin ((2n+1)x),\end{aligned}
\theta_3(x,q)=\sum_{n\in\mathbb{Z}}q^{n^2}e^{2nix}=1+2\sum_{n=1}^\infty q^{n^2}\cos (2nx)
are the Jacobi theta functions.Reinhardt & Walker (2010), 22.2.E7
A fast algorithm, returning approximations to \operatorname{sl} x (which get closer to \operatorname{sl}x with increasing N), is the following:Reinhardt & Walker (2010), §22.20(ii)
This is effectively using the arithmetic-geometric mean and is based on Landen's transformations.Carlson (2010), §19.8
Two other fast computation methods use the following sum and product series:
\text{sl}\Bigl(\frac\varpi\pi x\Bigr) = f\biggl(\frac{4\pi}\varpi\sin(x)\sum_{n = 1}^{\infty} \frac{\cosh[(2n-1)\pi]}{\cosh[(2n-1)\pi]^2 - \cos(x)^2}\biggr)
\text{cl}\Bigl(\frac\varpi\pi x\Bigr) = f\biggl(\frac{4\pi}\varpi\cos(x)\sum_{n = 1}^{\infty} \frac{\cosh[(2n-1)\pi]}{\cosh[(2n-1)\pi]^2 - \sin(x)^2}\biggr)
\mathrm{sl}\Bigl(\frac\varpi\pi x\Bigr) = 2\exp\bigl({-\tfrac{1}{4}\pi}\bigr)\sin(x)\prod_{n = 1}^{\infty} \frac{1-2\cos(2x)\exp(-2n\pi)+\exp(-4n\pi)}{1+2\cos(2x)\exp[-(2n-1)\pi]+\exp[-(4n-2)\pi]}
\mathrm{cl}\Bigl(\frac\varpi\pi x\Bigr) = 2\exp\bigl({-\tfrac{1}{4}\pi}\bigr)\cos(x)\prod_{n = 1}^{\infty} \frac{1+2\cos(2x)\exp(-2n\pi)+\exp(-4n\pi)}{1-2\cos(2x)\exp[-(2n-1)\pi]+\exp[-(4n-2)\pi]}
where f(x) = \tan(2 \arctan x) = 2x / (1 - x^2).
The following series identities were discovered by Ramanujan: p. 247, 248, 253
\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}
\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)}
Inverse functions
The inverse function of the lemniscate sine is the lemniscate arcsine, defined as:
\operatorname{arcsl} x = \int_0^x \frac{1}{\sqrt{1-t^4}} \mathrm{d}t
The inverse function of the lemniscate cosine is the lemniscate arccosine.
This function is defined by following expression:
\operatorname{arccl} x = \int_{x}^{1} \frac{1}{\sqrt{1-t^4}} \mathrm{d}t = \tfrac12\varpi - \operatorname{arcsl}x
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}}\bigl(\tfrac12\operatorname{arcsl} x\bigr) &= {\sin}\bigl(\tfrac12\arcsin x\bigr) \,{\operatorname{sech}}\bigl(\tfrac12\operatorname{arsinh} x\bigr) \\ {\operatorname{sl}}\bigl(\tfrac12\operatorname{arcsl} x\bigr)^2 &= {\tan}\bigl(\tfrac14\arcsin x^2\bigr) \end{aligned}
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 = \frac{1}{\sqrt2}F\left[{\arcsin}{\left(\frac{\sqrt2x}{\sqrt{1+x^2}}\right)};\frac{1}{\sqrt2}\right]
\operatorname{arcsl} x = 2(\sqrt2-1)F\left[{\arcsin}{\left[\frac{(\sqrt2+1)x}{\sqrt{1+x^2}+1}\right]};(\sqrt2-1)^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 = {}&\frac{2+\sqrt2}{2}E\left[{\arcsin}{\left[\frac{(\sqrt2+1)x}{\sqrt{1+x^2}+1}\right]};(\sqrt2-1)^2\right] \\[5mu] &\ \ - E\left[{\arcsin}{\left(\frac{\sqrt2x}{\sqrt{1+x^2}}\right)};\frac{1}{\sqrt2}\right] + \frac{x\sqrt{1-x^2}}{\sqrt2(1+x^2+\sqrt{1+x^2})} \end{aligned}
The lemniscate arccosine has this expression:
\operatorname{arccl} x = \frac{1}{\sqrt2}F\left[\arccos x;\frac{1}{\sqrt2}\right]
Use in integration
The lemniscate can be used to integrate many functions.
Here is a list of important integrals (the constants of integration are omitted):
\int\frac{1}{\sqrt{1-x^4}}\,\mathrm dx=\operatorname{arcsl} x
\int\frac{1}{\sqrt{(x^2+1)(2x^2+1)}}\,\mathrm dx={\operatorname{arcsl}}{\left(\frac{x}{\sqrt{x^2+1}}\right)}
\int\frac{1}{\sqrt{x^4+6x^2+1}}\,\mathrm dx={\operatorname{arcsl}}{\left(\frac{\sqrt2x}{\sqrt{\sqrt{x^4+6x^2+1}+x^2+1}}\right)}
\int\frac{1}{\sqrt{x^4+1}}\,\mathrm dx={\sqrt2\operatorname{arcsl}}{\left(\frac{x}{\sqrt{\sqrt{x^4+1}+1}}\right)}
\int\frac{1}{\sqrt[4]{(1-x^4)^3}}\,\mathrm dx={\sqrt2\operatorname{arcsl}}{\left(\frac{x}{\sqrt{1+\sqrt{1-x^4}}}\right)}
\int\frac{1}{\sqrt[4]{(x^4+1)^3}}\,\mathrm dx={\operatorname{arcsl}}{\left(\frac{x}{\sqrt[4]{x^4+1}}\right)}
\int\frac{1}{\sqrt[4]{(1-x^2)^3}}\,\mathrm dx={2\operatorname{arcsl}}{\left(\frac{x}{1+\sqrt{1-x^2}}\right)}
\int\frac{1}{\sqrt[4]{(x^2+1)^3}}\,\mathrm dx={2\operatorname{arcsl}}{\left(\frac{x}{\sqrt{x^2+1}+1}\right)}
\int\frac{1}{\sqrt[4]{(ax^2+bx+c)^3}}\,\mathrm dx={\frac{2\sqrt2}{\sqrt[4]{4a^2c-ab^2}}\operatorname{arcsl}}{\left(\frac{2ax+b}{\sqrt{4a(ax^2+bx+c)}+\sqrt{4ac-b^2}}\right)}
\int\sqrt{\operatorname{sech} x}\,\mathrm dx={2\operatorname{arcsl}}\bigl(\tanh \tfrac12x\bigr)
\int\sqrt{\sec x}\,\mathrm dx={2\operatorname{arcsl}}\bigl(\tan \tfrac12x\bigr)
Hyperbolic lemniscate functions
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).
The hyperbolic lemniscate sine () and cosine () can be defined by their inverse functions as follows:
z = \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}}
The complete integral has the value:
\int_0^\infty \frac{\mathrm{d}t}{\sqrt{t^4 + 1}} = \tfrac14 \Beta\bigl(\tfrac14, \tfrac14\bigr) = \tfrac1{\sqrt2}\varpi = 1.85407\;46773\;01371\ldots
Therefore, the two defined functions have following relation to each other:
\operatorname{slh} z = {\operatorname{clh}}{\Bigl(\tfrac{1}{\sqrt2}\varpi - z \Bigr)}
The product of hyperbolic lemniscate sine and hyperbolic lemniscate cosine is equal to one:
\operatorname{slh}z\,\operatorname{clh}z = 1
The hyperbolic lemniscate functions can be expressed in terms of lemniscate sine and lemniscate cosine:
\operatorname{slh}\bigl(\sqrt2 x\bigr) = \frac{(1+\operatorname{cl}^2 x)\operatorname{sl}x}{\sqrt2\operatorname{cl}x}
\operatorname{clh}\bigl(\sqrt2 x\bigr) = \frac{(1 + \operatorname{sl}^2 x)\operatorname{cl}x}{\sqrt2\operatorname{sl}x}
But there is also a relation to the Jacobi elliptic functions with the elliptic modulus one by square root of two:
\operatorname{slh}x = \frac{\operatorname{sn}(x;1/\sqrt2)}{\operatorname{cd}(x;1/\sqrt2)}
\operatorname{clh}x = \frac{\operatorname{cd}(x;1/\sqrt2)}{\operatorname{sn}(x;1/\sqrt2)}
The hyperbolic lemniscate sine has following imaginary relation to the lemniscate sine:
\operatorname{slh}z = \frac{1-i}{\sqrt2} \operatorname{sl}\left(\frac{1+i}{\sqrt2}z\right) = \frac{\operatorname{sl}\left(\sqrt[4]{-1}z\right) }{ \sqrt[4]{-1} }
This is analogous to the relationship between hyperbolic and trigonometric sine:
\sinh z = -i \sin (iz) = \frac{\sin\left(\sqrt[2]{-1}z\right) }{ \sqrt[2]{-1}}
thumb|upright=1.3|With respect to the quartic Fermat curve x^4 + y^4 = 1, the hyperbolic lemniscate sine is analogous to the trigonometric tangent function.
In a quartic Fermat curve x^4 + y^4 = 1 (sometimes called a squircle) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle x^2 + y^2 = 1 (the quadratic Fermat curve).
If the origin and a point on the curve are connected to each other by a line , 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.Levin (2006), Robinson (2019b)
The hyperbolic lemniscate sine satisfies the argument addition identity:
\operatorname{slh}(a+b) = \frac{\operatorname{slh}a\sqrt{1+\operatorname{slh}^4b} + \operatorname{slh}b\sqrt{1 + \operatorname{slh}^4a}}{1-\operatorname{slh}^2a\,\operatorname{slh}^2b}
The derivative can be expressed in this way:
\frac{\mathrm{d}}{\mathrm{d}x}\operatorname{slh}x = \sqrt{1 + \operatorname{slh}^4 x}
Furthermore the function \sin_{4}(t) is called Hyperbolic Lemniscate Tangent and the function \cot_{4}(t) is called Hyperbolic Lemniscate Cotangent.
\text{tlh}(t) = \sin_{4}(t)
\text{ctlh}(t) = \cos_{4}(t)
Number theory
In algebraic number theory, every abelian extension of the Gaussian rationals \mathbf{Q}(i) is obtained by adjoining \operatorname{sl}z, where z is a solution of the equation \operatorname{sl}mz=0 over the complexes and m is a Gaussian integer.Ogawa (2005) This is analogous to the Kronecker–Weber theorem for the rational numbers \mathbf{Q} which is based on division of the circle.
Both are special cases of Kronecker's Jugendtraum, which became Hilbert's twelfth problem.
The field \mathbf{Q}(i,\operatorname{sl}(\varpi /n)) (for positive odd n) is the extension of \mathbf{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.
p. 508, 509 World map projections
thumb|upright=1.3|"The World on a Quincuncial Projection", from Peirce (1879).
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).Peirce (1879).
Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively.
Also see Lee (1976).
These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice.
When lines of constant real or imaginary part are projected onto the complex plane via the hyperbolic lemniscate sine, and thence stereographically projected onto the sphere (see Riemann sphere), the resulting curves are spherical conics, the spherical analog of planar ellipses and hyperbolas.Adams (1925) Thus the lemniscate functions (and more generally, the Jacobi elliptic functions) provide a parametrization for spherical conics.
A conformal map projection from the globe onto the 6 square faces of a cube can also be defined using the lemniscate functions.Adams (1925), Lee (1976).
Because many partial differential equations can be effectively solved by conformal mapping, this map from sphere to cube is convenient for atmospheric modeling.
Rančić, Purser, & Mesinger (1996); McGregor (2005).
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
Notes
References
E252. (Figures)
E605.
Reprinted in  (Figures)
Notes
