Lemniscate elliptic functions

Short description: Mathematical functions

The lemniscate sine (red) and lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric sine y = sin(πx/ϖ) (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.[1]

The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead),[2] 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 x2+y2=x,[3] the lemniscate sine relates the arc length to the chord length of a lemniscate (x2+y2)2=x2−y2.

The lemniscate functions have periods related to a number ϖ= 2.622057... called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic) π= 3.141592..., ratio of perimeter to diameter of a circle.

As complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods {(1+i)ϖ,(1−i)ϖ},[4] and are a special case of two Jacobi elliptic functions on that lattice, sl⁡z=sn⁡(z;−1), cl⁡z=cd⁡(z;−1).

Similarly, the hyperbolic lemniscate sine slh and hyperbolic lemniscate cosine clh have a square period lattice with fundamental periods {2ϖ,2ϖi}.

The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function ℘(z;a,0).

## Contents

* 1 Lemniscate sine and cosine functions
* 1.1 Definitions
* 1.2 Relation to the lemniscate constant
* 2 Argument identities
* 2.1 Zeros, poles and symmetries
* 2.2 Pythagorean-like identity
* 2.3 Derivatives and integrals
* 2.4 Argument sum and multiple identities
* 2.5 Lemnatomic polynomials
* 2.6 Specific values
* 3 Relation to geometric shapes
* 3.1 Arc length of Bernoulli's lemniscate
* 3.2 Arc length of rectangular elastica
* 3.3 Elliptic characterization
* 4 Series Identities
* 4.1 Power series
* 4.2 Ramanujan's cos/cosh identity
* 4.3 Continued fractions
* 4.4 Methods of computation
* 4.5 The lemniscate functions as a ratio of entire functions
* 5 Relation to other functions
* 5.1 Relation to Weierstrass and Jacobi elliptic functions
* 5.2 Relation to the modular lambda function
* 6 Inverse functions
* 6.1 Expression using elliptic integrals
* 6.2 Use in integration
* 7 Hyperbolic lemniscate functions
* 7.1 Fundamental information
* 7.2 Relation to quartic Fermat curve
* 7.2.1 Hyperbolic Lemniscate Tangent and Cotangent
* 7.2.2 Derivation of the Hyperbolic Lemniscate functions
* 7.2.3 First proof: comparison with the derivative of the arctangent
* 7.2.4 Second proof: integral formation and area subtraction
* 7.3 Specific values
* 7.4 Combination and halving theorems
* 7.5 Coordinate Transformations
* 8 Number theory
* 8.1 Hurwitz numbers
* 8.1.1 Appearances in Laurent series
* 8.2 A quartic analog of the Legendre symbol
* 9 World map projections
* 10 See also
* 11 Notes
* 12 References
* 13 External links

## Lemniscate sine and cosine functions

### Definitions

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

ddzsl⁡z=(1+sl2z)cl⁡z, ddzcl⁡z=−(1+cl2z)sl⁡z, sl⁡0=0, 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 {12ϖ,12ϖi,−12ϖ,−12ϖi}:[6]

z=∫0sl⁡zdt1−t4=∫cl⁡z1dt1−t4.

Beyond that square, the functions can be extended to the complex plane via analytic continuation by successive reflections.

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

ddzsin⁡z=cos⁡z, ddzcos⁡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 −12π,12π and positive imaginary part:

z=∫0sin⁡zdt1−t2=∫cos⁡z1dt1−t2.

### Relation to the lemniscate constant

Main article: Lemniscate constant

The lemniscate sine function and hyperbolic lemniscate sine functions are defined as inverses of elliptic integrals. The complete integrals are related to the lemniscate constant ϖ.

The lemniscate functions have minimal real period 2ϖ, minimal imaginary period 2ϖi and fundamental complex periods (1+i)ϖ and (1−i)ϖ for a constant ϖ called the lemniscate constant,[7]

ϖ=2∫01dt1−t4=2.62205…

The lemniscate functions satisfy the basic relation cl⁡z=sl(12ϖ−z), analogous to the relation cos⁡z=sin(12π−z).

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

2π=12⋅12+1212⋅12+1212+1212⋯

An analogous formula for ϖ is:[8]

2ϖ=12⋅12+12/12⋅12+12/12+12/12⋯

The Machin formula for π is 14π=4arctan⁡15−arctan⁡1239, and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula 14π=arctan⁡12+arctan⁡13. Analogous formulas can be developed for ϖ, including the following found by Gauss: 12ϖ=2arcsl⁡12+arcsl⁡723.[9]

The lemniscate and circle constants were found by Gauss to be related to each-other by the arithmetic-geometric mean M:[10]

πϖ=M(1,2)

## Argument identities

### Zeros, poles and symmetries

sl in the complex plane.[11] In the picture, it can be seen that the fundamental periods (1+i)ϖ and (1−i)ϖ are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.

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

cl⁡(−z)=cl⁡zsl⁡(−z)=−sl⁡z

At translations of 12ϖ, cl and sl are exchanged, and at translations of 12iϖ they are additionally rotated and reciprocated:[12]

cl(z±12ϖ)=∓sl⁡z,cl(z±12iϖ)=∓isl⁡zsl(z±12ϖ)=±cl⁡z,sl(z±12iϖ)=±icl⁡z

Doubling these to translations by a unit-Gaussian-integer multiple of ϖ (that is, ±ϖ or ±iϖ), negates each function, an involution:

cl⁡(z+ϖ)=cl⁡(z+iϖ)=−cl⁡zsl⁡(z+ϖ)=sl⁡(z+iϖ)=−sl⁡z

As a result, both functions are invariant under translation by an even-Gaussian-integer multiple of ϖ.[13] That is, a displacement (a+bi)ϖ, with a+b=2k for integers a, b, and k.

cl(z+(1+i)ϖ)=cl(z+(1−i)ϖ)=cl⁡zsl(z+(1+i)ϖ)=sl(z+(1−i)ϖ)=sl⁡z

This makes them elliptic functions (doubly periodic meromorphic functions in the complex plane) with a diagonal square period lattice of fundamental periods (1+i)ϖ and (1−i)ϖ.[14] 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:

cl⁡z¯=cl⁡z‾sl⁡z¯=sl⁡z‾cl⁡iz=1cl⁡zsl⁡iz=isl⁡z

The sl function has simple zeros at Gaussian integer multiples of ϖ, complex numbers of the form aϖ+bϖi for integers a and b. It has simple poles at Gaussian half-integer multiples of ϖ, complex numbers of the form (a+12)ϖ+(b+12)ϖi, with residues (−1)a−b+1i. The cl function is reflected and offset from the sl function, cl⁡z=sl(12ϖ−z). It has zeros for arguments (a+12)ϖ+bϖi and poles for arguments aϖ+(b+12)ϖi, with residues (−1)a−bi.

Also

sl⁡z=sl⁡w↔z=(−1)m+nw+(m+ni)ϖ

for some m,n∈ℤ and

sl⁡((1±i)z)=(1±i)sl⁡zsl′⁡z.

The last formula is a special case of complex multiplication. Analogous formulas can be given for sl⁡((n+mi)z) where n+mi is any Gaussian integer – the function sl has complex multiplication by ℤ[i].[15]

There are also infinite series reflecting the distribution of the zeros and poles of sl:[16][17]

1sl⁡z=∑(n,k)∈ℤ2(−1)n+kz+nϖ+kϖi
sl⁡z=−i∑(n,k)∈ℤ2(−1)n+kz+(n+1/2)ϖ+(k+1/2)ϖi.

### Pythagorean-like identity

Curves x² ⊕ y² = a for various values of a. Negative a in green, positive a in blue, a = ±1 in red, a = ∞ in black.

The lemniscate functions satisfy a Pythagorean-like identity:

cl2⁡z+sl2⁡z+cl2⁡zsl2⁡z=1

As a result, the parametric equation (x,y)=(cl⁡t,sl⁡t) parametrizes the quartic curve x2+y2+x2y2=1.

This identity can alternately be rewritten:[18]

(1+cl2⁡z)(1+sl2⁡z)=2

cl2⁡z=1−sl2⁡z1+sl2⁡z,sl2⁡z=1−cl2⁡z1+cl2⁡z

Defining a tangent-sum operator as a⊕b:=tan⁡(arctan⁡a+arctan⁡b)=a+b1−ab, gives:

cl2⁡z⊕sl2⁡z=1.

The functions cl~ and sl~ satisfy another Pythagorean-like identity:

(∫0xcl~tdt)2+(1−∫0xsl~tdt)2=1.

### Derivatives and integrals

The derivatives are as follows:

ddzcl⁡z=cl'⁡z=−(1+cl2⁡z)sl⁡z=−2sl⁡zsl2z+1cl'2⁡z=1−cl4⁡zddzsl⁡z=sl'⁡z=(1+sl2⁡z)cl⁡z=2cl⁡zcl2z+1sl'2⁡z=1−sl4⁡z

ddzcl~z=−2sl~zcl⁡z−sl~zcl⁡zddzsl~z=2cl~zcl⁡z−cl~zcl⁡z

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

d2dz2cl⁡z=−2cl3⁡z

d2dz2sl⁡z=−2sl3⁡z

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

∫cl⁡zdz=arctan⁡sl⁡z+C∫sl⁡zdz=−arctan⁡cl⁡z+C∫cl~zdz=sl~zcl⁡z+C∫sl~zdz=−cl~zcl⁡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:[19]

sl⁡(u+v)=sl⁡usl'⁡v+sl⁡vsl'⁡u1+sl2⁡usl2⁡v

The derivative and Pythagorean-like identities can be used to rework the identity used by Fagano in terms of sl and cl. Defining a tangent-sum operator a⊕b:=tan⁡(arctan⁡a+arctan⁡b) and tangent-difference operator a⊖b:=a⊕(−b), the argument sum and difference identities can be expressed as:[20]

cl⁡(u+v)=cl⁡ucl⁡v⊖sl⁡usl⁡v=cl⁡ucl⁡v−sl⁡usl⁡v1+sl⁡ucl⁡usl⁡vcl⁡vcl⁡(u−v)=cl⁡ucl⁡v⊕sl⁡usl⁡vsl⁡(u+v)=sl⁡ucl⁡v⊕cl⁡usl⁡v=sl⁡ucl⁡v+cl⁡usl⁡v1−sl⁡ucl⁡usl⁡vcl⁡vsl⁡(u−v)=sl⁡ucl⁡v⊖cl⁡usl⁡v

These resemble their trigonometric analogs:

cos⁡(u±v)=cos⁡ucos⁡v∓sin⁡usin⁡vsin⁡(u±v)=sin⁡ucos⁡v±cos⁡usin⁡v

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

cl⁡(x+iy)=cl⁡x−isl⁡xsl⁡ycl⁡ycl⁡y+isl⁡xcl⁡xsl⁡y=cl⁡xcl⁡y(1−sl2xsl2y)cl2y+sl2xcl2xsl2y−isl⁡xsl⁡y(cl2x+cl2y)cl2y+sl2xcl2xsl2ysl⁡(x+iy)=sl⁡x+icl⁡xsl⁡ycl⁡ycl⁡y−isl⁡xcl⁡xsl⁡y=sl⁡xcl⁡y(1−cl2xsl2y)cl2y+sl2xcl2xsl2y+icl⁡xsl⁡y(sl2x+cl2y)cl2y+sl2xcl2xsl2y

Gauss discovered that

sl⁡(u−v)sl⁡(u+v)=sl⁡((1+i)u)−sl⁡((1+i)v)sl⁡((1+i)u)+sl⁡((1+i)v)

where u,v∈ℂ such that both sides are well-defined.

Also

sl⁡(u+v)sl⁡(u−v)=sl2u−sl2v1+sl2usl2v

where u,v∈ℂ such that both sides are well-defined; this resembles the trigonometric analog

sin⁡(u+v)sin⁡(u−v)=sin2u−sin2v.

Bisection formulas:

cl212x=1+cl⁡x1+sl2x1+1+sl2x

sl212x=1−cl⁡x1+sl2x1+1+sl2x

Duplication formulas:[21]

cl⁡2x=−1+2cl2x+cl4x1+2cl2x−cl4x

sl⁡2x=2sl⁡xcl⁡x1+sl2x1+sl4x

Triplication formulas:[21]

cl⁡3x=−3cl⁡x+6cl5x+cl9x1+6cl4x−3cl8x

sl⁡3x=3sl⁡x−6sl5x−1sl9x1+6sl4x−3sl8x

Note the "reverse symmetry" of the coefficients of numerator and denominator of sl⁡3x. This phenomenon can be observed in multiplication formulas for sl⁡βx where β=m+ni whenever m,n∈ℤ and m+n is odd.[15]

### Lemnatomic polynomials

Let L be the lattice

L=ℤ(1+i)ϖ+ℤ(1−i)ϖ.

Furthermore, let K=ℚ(i), 𝒪=ℤ[i], z∈ℂ, β=m+in, γ=m′+in′ (where m,n,m′,n′∈ℤ), m+n be odd, m′+n′ be odd, γ≡1mod⁡2(1+i) and sl⁡βz=Mβ(sl⁡z). Then

Mβ(x)=iεxPβ(x4)Qβ(x4)

for some coprime polynomials Pβ(x),Qβ(x)∈𝒪[x] and some ε∈{0,1,2,3}[22] where

xPβ(x4)=∏γ|βΛγ(x)

and

Λβ(x)=∏[α]∈(𝒪/β𝒪)×(x−sl⁡αδβ)

where δβ is any β-torsion generator (i.e. δβ∈(1/β)L and [δβ]∈(1/β)L/L generates (1/β)L/L as an 𝒪-module). Examples of β-torsion generators include 2ϖ/β and (1+i)ϖ/β. The polynomial Λβ(x)∈𝒪[x] is called the β-th lemnatomic polynomial. It is monic and is irreducible over K. The lemnatomic polynomials are the "lemniscate analogs" of the cyclotomic polynomials,[23]

Φk(x)=∏[a]∈(ℤ/kℤ)×(x−ζka).

The β-th lemnatomic polynomial Λβ(x) is the minimal polynomial of sl⁡δβ in K[x]. For convenience, let ωβ=sl⁡(2ϖ/β) and ω~β=sl⁡((1+i)ϖ/β). So for example, the minimal polynomial of ω5 (and also of ω~5) in K[x] is

Λ5(x)=x16+52x12−26x8−12x4+1,

and[24]

ω5=−13+65+285−3854
ω~5=−13−65+285+3854[25]

(an equivalent expression is given in the table below). Another example is[23]

Λ−1+2i(x)=x4−1+2i

which is the minimal polynomial of ω−1+2i (and also of ω~−1+2i) in K[x].

If p is prime and β is positive and odd,[26] then[27]

deg⁡Λβ=β2∏p|β(1−1p)(1−(−1)(p−1)/2p)

which can be compared to the cyclotomic analog

deg⁡Φk=k∏p|k(1−1p).

### Specific values

Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into n parts of equal length, using only basic arithmetic and square roots, if and only if n is of the form n=2kp1p2⋯pm where k is a non-negative integer and each pi (if any) is a distinct Fermat prime.[28]

n | cl⁡nϖ | sl⁡nϖ

|
|

1 | −1 | 0
56 | −23−34 | 12(3+1−124)
34 | −2−1 | 2−1
23 | −12(3+1−124) | 23−34
12 | 0 | 1
13 | 12(3+1−124) | 23−34
14 | 2−1 | 2−1
16 | 23−34 | 12(3+1−124)

## Relation to geometric shapes

### Arc length of Bernoulli's lemniscate

The lemniscate sine and cosine relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin. 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.

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 is the closure of the locus of the points P such that |APB−APB′| is a right angle.[29]

Focal characterization: ℒ is the locus of points in the plane such that the product of their distances from the two focal points F1=(−12,0) and F2=(12,0) is the constant 12.

Explicit coordinate characterization: ℒ is a quartic curve satisfying the polar equation r2=cos⁡2θ or the Cartesian equation (x2+y2)2=x2−y2.

The perimeter of ℒ is 2ϖ.[30]

The points on ℒ at distance r from the origin are the intersections of the circle x2+y2=r2 and the hyperbola x2−y2=r4. The intersection in the positive quadrant has Cartesian coordinates:

(x(r),y(r))=(12r2(1+r2),12r2(1−r2)).

Using this parametrization with r∈[0,1] for a quarter of ℒ, the arc length from the origin to a point (x(r),y(r)) is:[31]

∫0rx′(t)2+y′(t)2dt=∫0r(1+2t2)22(1+t2)+(1−2t2)22(1−t2)dt=∫0rdt1−t4=arcsl⁡r.

Likewise, the arc length from (1,0) to (x(r),y(r)) is:

∫r1x′(t)2+y′(t)2dt=∫r1dt1−t4=arccl⁡r=12ϖ−arcsl⁡r.

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⁡θ or Cartesian equation x2+y2=x, using the same argument above but with the parametrization:

(x(r),y(r))=(r2,r2(1−r2)).

Alternatively, just as the unit circle x2+y2=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 is parametrized in terms of the arc length s from the point (1,0) by[32]

(x(s),y(s))=(cl⁡s1+sl2s,sl⁡scl⁡s1+sl2s)=(cl~s,sl~s).

The notation cl~,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:[33]

∫0zdt1−t4=2∫0udt1−t4,if z=2u1−u41+u4 and 0≤u≤2−1.
A lemniscate divided into 15 sections of equal arclength (red curves). Because the prime factors of 15 (3 and 5) are both Fermat primes, this polygon (in black) is constructible using a straightedge and compass.

Later mathematicians generalized this result. Analogously to the constructible polygons in the circle, the lemniscate can be divided into n sections of equal arc length using only straightedge and compass if and only if n is of the form n=2kp1p2⋯pm where k is a non-negative integer and each pi (if any) is a distinct Fermat prime.[34] 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.[35] Equivalently, the lemniscate can be divided into n sections of equal arc length using only straightedge and compass if and only if φ(n) is a power of two (where φ is Euler's totient function). The lemniscate is not assumed to be already drawn, as that would go against the rules of straightedge and compass constructions; instead, it is assumed that we are given only two points by which the lemniscate is defined, such as its center and radial point (one of the two points on the lemniscate such that their distance from the center is maximal) or its two foci.

Let rj=sl⁡2jϖn. Then the n-division points for ℒ are the points

(rj12(1+rj2), (−1)⌊4j/n⌋12rj2(1−rj2)),j∈{1,2,…,n}

where ⌊⋅⌋ is the floor function. See below for some specific values of sl⁡2ϖn.

### Arc length of rectangular elastica

The lemniscate sine relates the arc length to the x coordinate in the rectangular elastica.

The inverse lemniscate sine also describes the arc length s relative to the x coordinate of the rectangular elastica.[36] This curve has y coordinate and arc length:

y=∫x1t2dt1−t4,s=arcsl⁡x=∫0xdt1−t4

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

The lemniscate elliptic functions and an ellipse

Let C be a point on the ellipse x2+2y2=1 in the first quadrant and let D be the projection of C on the unit circle x2+y2=1. The distance r between the origin A and the point C is a function of φ (the angle BAC where B=(1,0); equivalently the length of the circular arc BD). The parameter u is given by

u=∫0φr(θ)dθ=∫0φdθ1+sin2θ.

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

cl⁡u=AF‾,sl⁡u=DE‾,
cl~u=AF‾AC‾,sl~u=AF‾FC‾.

## Series Identities

### Power series

The power series expansion of the lemniscate sine at the origin is[37]

sl⁡z=∑n=0∞anzn=z−12z55!+3024z99!−4390848z1313!+⋯,|z|<ϖ2

where the coefficients an are determined as follows:

n≢1(mod4)⟹an=0,
a1=1,∀n∈ℕ0:an+2=−2(n+1)(n+2)∑i+j+k=naiajak

where i+j+k=n stands for all three-term compositions of n. For example, to evaluate a13, 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

a13=−212⋅13(a9a1a1+a1a9a1+a1a1a9+a5a5a1+a5a1a5+a1a5a5)=−1115600.

The expansion can be equivalently written as[38]

sl⁡z=∑n=0∞p2nz4n+1(4n+1)!,|z|<ϖ2

where

pn+2=−12∑j=0n(2n+22j+2)pn−j∑k=0j(2j+12k+1)pkpj−k,p0=1,p1=0.

The power series expansion of sl~ at the origin is

sl~z=∑n=0∞αnzn=z−9z33!+153z55!−4977z77!+⋯,|z|<ϖ2

where αn=0 if n is even and[39]

αn=2πϖ(−1)(n−1)/2n!∑k=1∞(2kπ/ϖ)n+1cosh⁡kπ,|αn|∼2n+5/2n+1ϖn+2

if n is odd.

The expansion can be equivalently written as[40]

sl~z=∑n=0∞(−1)n2n+1(∑l=0n2l(2n+22l+1)sltn−l)z2n+1(2n+1)!,|z|<ϖ2

where

sn+2=3sn+1+24∑j=0n(2n+22j+2)sn−j∑k=0j(2j+12k+1)sksj−k,s0=1,s1=3,
tn+2=3tn+1+3∑j=0n(2n+22j+2)tn−j∑k=0j(2j+12k+1)tktj−k,t0=1,t1=3.

For the lemniscate cosine,[41]

cl⁡z=1−∑n=0∞(−1)n(∑l=0n2l(2n+22l+1)qlrn−l)z2n+2(2n+2)!=1−2z22!+12z44!−216z66!+⋯,|z|<ϖ2,
cl~z=∑n=0∞(−1)n2nqnz2n(2n)!=1−3z22!+33z44!−819z66!+⋯,|z|<ϖ2

where

rn+2=3∑j=0n(2n+22j+2)rn−j∑k=0j(2j+12k+1)rkrj−k,r0=1,r1=0,
qn+2=32qn+1+6∑j=0n(2n+22j+2)qn−j∑k=0j(2j+12k+1)qkqj−k,q0=1,q1=32.

### Ramanujan's cos/cosh identity

Ramanujan's famous cos/cosh identity states that if

R(s)=πϖ2∑n∈ℤcos⁡(2nπs/ϖ)cosh⁡nπ,

then[39]

R(s)−2+R(is)−2=2,|Re⁡s|<ϖ2,|Im⁡s|<ϖ2.

There is a close relation between the lemniscate functions and R(s). Indeed,[39][42]

sl~s=−ddsR(s)|Im⁡s|<ϖ2
cl~s=dds1−R(s)2,|Re⁡s−ϖ2|<ϖ2,|Im⁡s|<ϖ2

and

R(s)=11+sl2s,|Im⁡s|<ϖ2.

### Continued fractions

For z∈ℂ∖{0}:[43]

∫0∞e−tz2cl⁡tdt=1/2z+a1z+a2z+a3z+⋱,an=n24((−1)n+1+3)
∫0∞e−tz2sl⁡tcl⁡tdt=1/2z2+b1−a1z2+b2−a2z2+b3−⋱,an=n2(4n2−1),bn=3(2n−1)2

### Methods of computation

A fast algorithm, returning approximations to sl⁡x (which get closer to sl⁡x with increasing N), is the following:[44]

* a0<-1; b0<-12; c0<-12
* for each n≥1 do

* an<-12(an−1+bn−1); bn<-an−1bn−1; cn<-12(an−1−bn−1)
* if cn<tolerance then

* N<-n; break

* ϕN<-2NaN2x
* for each n from N to 0 do

* ϕn−1<-12(ϕn+arcsin(cnansin⁡ϕn))

* return sin⁡ϕ02−sin2ϕ0

This is effectively using the arithmetic-geometric mean and is based on Landen's transformations.[45]

Several methods of computing sl⁡x involve first making the change of variables πx=ϖx~ and then computing sl⁡(ϖx~/π).

A hyperbolic series method:[46][47]

sl⁡(ϖπx)=πϖ∑n∈ℤ(−1)ncosh⁡(x−(n+1/2)π),x∈ℂ

1sl⁡(ϖx/π)=πϖ∑n∈ℤ(−1)nsinh(x−nπ)=πϖ∑n∈ℤ(−1)nsin⁡(x−nπi),x∈ℂ

Fourier series method:[48]

sl⁡(ϖπx)=2πϖ∑n=0∞(−1)nsin⁡((2n+1)x)cosh⁡((n+1/2)π),|Im⁡x|<π2
cl⁡(ϖπx)=2πϖ∑n=0∞cos⁡((2n+1)x)cosh⁡((n+1/2)π),|Im⁡x|<π2
1sl⁡(ϖx/π)=πϖ(1sin⁡x−4∑n=0∞sin⁡((2n+1)x)e(2n+1)π+1),|Im⁡x|<π

The lemniscate functions can be computed more rapidly by

sl⁡(ϖπx)=θ1(x,e−π)θ3(x,e−π),x∈ℂcl⁡(ϖπx)=θ2(x,e−π)θ4(x,e−π),x∈ℂ

where

θ1(x,e−π)=∑n∈ℤ(−1)n+1e−π(n+1/2+x/π)2=∑n∈ℤ(−1)ne−π(n+1/2)2sin⁡((2n+1)x),θ2(x,e−π)=∑n∈ℤ(−1)ne−π(n+x/π)2=∑n∈ℤe−π(n+1/2)2cos⁡((2n+1)x),θ3(x,e−π)=∑n∈ℤe−π(n+x/π)2=∑n∈ℤe−πn2cos⁡2nx,θ4(x,e−π)=∑n∈ℤe−π(n+1/2+x/π)2=∑n∈ℤ(−1)ne−πn2cos⁡2nx

are the Jacobi theta functions.[49]

Fourier series for the logarithm of the lemniscate sine:

ln⁡sl⁡(ϖπx)=ln⁡2−π4+ln⁡sin⁡x+2∑n=1∞(−1)ncos⁡2nxn(enπ+(−1)n),|Im⁡x|<π2

The following series identities were discovered by Ramanujan:[50]

ϖ2π2sl2(ϖx/π)=1sin2x−1π−8∑n=1∞ncos⁡2nxe2nπ−1,|Im⁡x|<π
arctan⁡sl⁡(ϖπx)=2∑n=0∞sin⁡((2n+1)x)(2n+1)cosh⁡((n+1/2)π),|Im⁡x|<π2

The functions sl~ and cl~ analogous to sin and cos on the unit circle have the following Fourier and hyperbolic series expansions:[39][42][51]

sl~s=22π2ϖ2∑n=1∞nsin⁡(2nπs/ϖ)cosh⁡nπ,|Im⁡s|<ϖ2
cl~s=2π2ϖ2∑n=0∞(2n+1)cos⁡((2n+1)πs/ϖ)sinh⁡((n+1/2)π),|Im⁡s|<ϖ2
sl~s=π2ϖ22∑n∈ℤsinh⁡(π(n+s/ϖ))cosh2(π(n+s/ϖ)),s∈ℂ
cl~s=π2ϖ22∑n∈ℤ(−1)ncosh2(π(n+s/ϖ)),s∈ℂ

The following identities come from product representations of the theta functions:[52]

sl(ϖπx)=2e−π/4sin⁡x∏n=1∞1−2e−2nπcos⁡2x+e−4nπ1+2e−(2n−1)πcos⁡2x+e−(4n−2)π,x∈ℂ
cl(ϖπx)=2e−π/4cos⁡x∏n=1∞1+2e−2nπcos⁡2x+e−4nπ1−2e−(2n−1)πcos⁡2x+e−(4n−2)π,x∈ℂ

A similar formula involving the sn function can be given.[53]

### 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 sl has the following product expansion, reflecting the distribution of its zeros and poles:[54]

sl⁡z=M(z)N(z)

where

M(z)=z∏α(1−z4α4),N(z)=∏β(1−z4β4).

Here, α and β denote, respectively, the zeros and poles of sl which are in the quadrant Re⁡z>0,Im⁡z≥0. A proof can be found in.[54][55] 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.[56]

Proof of the infinite product for the lemniscate sine

Proof by logarithmic differentiation

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

M′(z)M(z)=−∑n=0∞24nH4nz4n−1(4n)!,|z|<ϖ

(where Hn are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers) and

N′(z)N(z)=(1+i)M′((1+i)z)M((1+i)z)−M′(z)M(z).

Therefore

N′(z)N(z)=∑n=0∞24n(1−(−1)n22n)H4nz4n−1(4n)!,|z|<ϖ2.

It is known that

1sl2z=∑n=0∞24n(4n−1)H4nz4n−2(4n)!,|z|<ϖ.

Then from

ddzsl′⁡zsl⁡z=−1sl2z−sl2z

and

sl2z=1sl2z−(1+i)2sl2((1+i)z)

we get

sl′⁡zsl⁡z=−∑n=0∞24n(2−(−1)n22n)H4nz4n−1(4n)!,|z|<ϖ2.

Hence

sl′⁡zsl⁡z=M′(z)M(z)−N′(z)N(z),|z|<ϖ2.

Therefore

sl⁡z=CM(z)N(z)

for some constant C for |z|<ϖ/2 but this result holds for all z∈ℂ by analytic continuation. Using

limz->0sl⁡zz=1

gives C=1 which completes the proof. ◼

Proof by Liouville's theorem

Let

f(z)=M(z)N(z)=(1+i)M(z)2M((1+i)z),

with patches at removable singularities. The shifting formulas

M(z+2ϖ)=e2πϖ(z+ϖ)M(z),M(z+2ϖi)=e−2πϖ(iz−ϖ)M(z)

imply that f is an elliptic function with periods 2ϖ and 2ϖi, just as sl. It follows that the function g defined by

g(z)=sl⁡zf(z),

when patched, is an elliptic function without poles. By Liouville's theorem, it is a constant. By using sl⁡z=z+O⁡(z5), M(z)=z+O⁡(z5) and N(z)=1+O⁡(z4), this constant is 1, which proves the theorem. ◼

Gauss conjectured that ln⁡N(ϖ)=π/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”.[57] Gauss expanded the products for M and N as infinite series (see below). He also discovered several identities involving the functions M and N, such as

The M function in the complex plane. The complex argument is represented by varying hue. The N function in the complex plane. The complex argument is represented by varying hue.

N(z)=M((1+i)z)(1+i)M(z),z∉ϖℤ[i]

and

N(2z)=M(z)4+N(z)4.

Thanks to a certain theorem[58] 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:[59][60][61][62][63]

M(z)=z−2z55!−36z99!+552z1313!+⋯,z∈ℂ
N(z)=1+2z44!−4z88!+408z1212!+⋯,z∈ℂ.

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

We define S and T by

S(z)=N(z1+i)2−iM(z1+i)2,T(z)=S(iz).

Then the lemniscate cosine can be written as

cl⁡z=S(z)T(z)

where[64]

S(z)=1−z22!−z44!−3z66!+17z88!−9z1010!+111z1212!+⋯,z∈ℂ
T(z)=1+z22!−z44!+3z66!+17z88!+9z1010!+111z1212!+⋯,z∈ℂ.

Furthermore, the identities

M(2z)=2M(z)N(z)S(z)T(z),
S(2z)=S(z)4−2M(z)4,
T(2z)=T(z)4−2M(z)4

and the Pythagorean-like identities

M(z)2+S(z)2=N(z)2,
M(z)2+N(z)2=T(z)2

hold for all z∈ℂ.

The quasi-addition formulas

M(z+w)M(z−w)=M(z)2N(w)2−N(z)2M(w)2,
N(z+w)N(z−w)=N(z)2N(w)2+M(z)2M(w)2

(where z,w∈ℂ) imply further multiplication formulas for M and N by recursion.[65]

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

M(z)M″(z)=M′(z)2−N(z)2,
N(z)N″(z)=N′(z)2+M(z)2

where z∈ℂ. Both M and N satisfy the differential equation[66]

X(z)X⁗(z)=4X′(z)X‴(z)−3X″(z)2+2X(z)2,z∈ℂ.

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

M(z)=zexp⁡(−∫0z∫0w(1sl2v−1v2)dvdw),
N(z)=exp⁡(∫0z∫0wsl2vdvdw)

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

An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see Lemniscate elliptic functions § Methods of computation); the relation between M,N and θ1,θ3 is

M(z)=2−1/4eπz2/(2ϖ2)πϖθ1(πzϖ,e−π),
N(z)=2−1/4eπz2/(2ϖ2)πϖθ3(πzϖ,e−π)

where z∈ℂ.

## Relation to other functions

### Relation to Weierstrass and Jacobi elliptic functions

The lemniscate functions are closely related to the Weierstrass elliptic function ℘(z;1,0) (the "lemniscatic case"), with invariants g2=1 and g3=0. This lattice has fundamental periods ω1=2ϖ, and ω2=iω1. The associated constants of the Weierstrass function are e1=12, e2=0, e3=−12.

The related case of a Weierstrass elliptic function with g2=a, g3=0 may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: a>0 and a<0. The period parallelogram is either a square or a rhombus. The Weierstrass elliptic function ℘(z;−1,0) is called the "pseudolemniscatic case".[67]

The square of the lemniscate sine can be represented as

sl2z=1℘(z;4,0)=i2℘((1−i)z;−1,0)=−2℘(2z+(i−1)ϖ2;1,0)

where the second and third argument of ℘ denote the lattice invariants g2 and g3. The lemniscate sine is a rational function in the Weierstrass elliptic function and its derivative:[68]

sl⁡z=−2℘(z;−1,0)℘′(z;−1,0).

The lemniscate functions can also be written in terms of Jacobi elliptic functions. The Jacobi elliptic functions sn and cd with positive real elliptic modulus have an "upright" rectangular lattice aligned with real and imaginary axes. Alternately, the functions sn and cd with modulus i (and sd and cn with modulus 1/2) have a square period lattice rotated 1/8 turn.[69][70]

sl⁡z=sn⁡(z;i)=sc⁡(z;2)=12sd(2z;12)

cl⁡z=cd⁡(z;i)=dn⁡(z;2)=cn(2z;12)

where the second arguments denote the elliptic modulus k.

The functions sl~ and cl~ can also be expressed in terms of Jacobi elliptic functions:

sl~z=cd⁡(z;i)sd⁡(z;i)=dn⁡(z;2)sn⁡(z;2)=12cn⁡(2z;12)sn⁡(2z;12),
cl~z=cd⁡(z;i)nd⁡(z;i)=dn⁡(z;2)cn⁡(z;2)=cn⁡(2z;12)dn⁡(2z;12).

### Relation to the modular lambda function

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

∏k=1nsl(2k−12n+1ϖ2)=λ((2n+1)i)1−λ((2n+1)i)8

For example:

sl(114ϖ)sl(314ϖ)sl(514ϖ)=λ(7i)1−λ(7i)8=tan(12arccsc(1287+21+127+1))=22+7+21+87+214+67+455+1727sl(118ϖ)sl(318ϖ)sl(518ϖ)sl(718ϖ)=λ(9i)1−λ(9i)8=tan⁡(||π4−arctan⁡(||223−23−22−33+3−1124||))

## Inverse functions

The inverse function of the lemniscate sine is the lemniscate arcsine, defined as[71]

arcsl⁡x=∫0xdt1−t4.

It can also be represented by the hypergeometric function:

arcsl⁡x=x2F1(12,14;54;x4)

which can be easily seen by using the binomial series.

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

arccl⁡x=∫x1dt1−t4=12ϖ−arcsl⁡x

For x in the interval −1≤x≤1, sl⁡arcsl⁡x=x and cl⁡arccl⁡x=x

For the halving of the lemniscate arc length these formulas are valid:[citation needed]

sl(12arcsl⁡x)=sin(12arcsin⁡x)sech(12arsinh⁡x)sl(12arcsl⁡x)2=tan(14arcsin⁡x2)

Furthermore there are the so called Hyperbolic lemniscate area functions:[citation needed]

aslh⁡(x)=∫0x1y4+1dy=12F(2arctan⁡x;12)

aclh⁡(x)=∫x∞1y4+1dy=12F(2arccot⁡x;12)

aclh⁡(x)=ϖ2−aslh⁡(x)

aslh⁡(x)=2arcsl⁡(x/1+x4+1)

arcsl⁡(x)=2aslh⁡(x/1+1−x4)

### 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:[citation needed]

arcsl⁡x=12F(arcsin2x1+x2;12)

arcsl⁡x=2(2−1)F(arcsin(2+1)x1+x2+1;(2−1)2)

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):[citation needed]

arcsl⁡x=2+22E(arcsin(2+1)x1+x2+1;(2−1)2) −E(arcsin2x1+x2;12)+x1−x22(1+x2+1+x2)

The lemniscate arccosine has this expression:[citation needed]

arccl⁡x=12F(arccos⁡x;12)

### 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):

∫11−x4dx=arcsl⁡x

∫1(x2+1)(2x2+1)dx=arcslxx2+1

∫1x4+6x2+1dx=arcsl2xx4+6x2+1+x2+1

∫1x4+1dx=2arcslxx4+1+1

∫1(1−x4)34dx=2arcslx1+1−x4

∫1(x4+1)34dx=arcslxx4+14

∫1(1−x2)34dx=2arcslx1+1−x2

∫1(x2+1)34dx=2arcslxx2+1+1

∫1(ax2+bx+c)34dx=224a2c−ab24arcsl2ax+b4a(ax2+bx+c)+4ac−b2

∫sech⁡xdx=2arcsltanh⁡12x

∫sec⁡xdx=2arcsltan⁡12x

## Hyperbolic lemniscate functions

### Fundamental information

The hyperbolic lemniscate sine (red) and hyperbolic lemniscate cosine (purple) applied to a real argument, in comparison with the trigonometric tangent (pale dashed red). File:Slh in the complex plane.pngThe hyperbolic lemniscate sine in the complex plane. Dark areas represent zeros and bright areas represent poles. The complex argument is represented by varying hue.

For convenience, let σ=2ϖ. σ is the "squircular" analog of π (see below). The decimal expansion of σ (i.e. 3.7081…[72]) appears in entry 34e of chapter 11 of Ramanujan's second notebook.[73]

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

z=*∫0slh⁡zdt1+t4=∫clh⁡z∞dt1+t4

where in (*), z is in the square with corners {σ/2,σi/2,−σ/2,−σi/2}. Beyond that square, the functions can be analytically continued to meromorphic functions in the whole complex plane.

The complete integral has the value:

∫0∞dtt4+1=14B(14,14)=σ2=1.854074677301371…

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

slh⁡z=clh(σ2−z)

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

slh⁡zclh⁡z=1

The functions slh and clh have a square period lattice with fundamental periods {σ,σi}.

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

slh⁡(2z)=(1+cl2z)sl⁡z2cl⁡z

clh⁡(2z)=(1+sl2z)cl⁡z2sl⁡z

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

slh⁡z=sn⁡(z;1/2)cd⁡(z;1/2)

clh⁡z=cd⁡(z;1/2)sn⁡(z;1/2)

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

slh⁡z=1−i2sl⁡(1+i2z)=sl⁡(−14z)−14

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

sinh⁡z=−isin⁡(iz)=sin⁡(−12z)−12

### Relation to quartic Fermat curve

#### Hyperbolic Lemniscate Tangent and Cotangent

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

Superellipse with the relation x4+y4=1

In a quartic Fermat curve x4+y4=1 (sometimes called a squircle) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle x2+y2=1 (the quadratic Fermat curve). If the origin and a point on the curve are connected to each other by a line L, the hyperbolic lemniscate sine of twice the enclosed area between this line and the x-axis is the y-coordinate of the intersection of L with the line x=1.[74] Just as π is the area enclosed by the circle x2+y2=1, the area enclosed by the squircle x4+y4=1 is σ. Moreover,

M(1,1/2)=πσ

where M is the arithmetic–geometric mean.

The hyperbolic lemniscate sine satisfies the argument addition identity:

slh⁡(a+b)=slh⁡aslh′⁡b+slh⁡bslh′⁡a1−slh2aslh2b

When u is real, the derivative and the original antiderivative of slh and clh can be expressed in this way:

dduslh⁡(u)=1+slh⁡(u)4

dduclh⁡(u)=−1+clh⁡(u)4

ddu12arsinh⁡[slh⁡(u)2]=slh⁡(u)

ddu−12arsinh⁡[clh⁡(u)2]=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(2u)=sin4(2u)=sl⁡(u)cl2u+1sl2u+cl2u
ctlh(2u)=cos4(2u)=cl⁡(u)sl2u+1sl2u+cl2u

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;122)cd(u;122)4+sn(u;122)44
ctlh(u)=cd(u;122)cd(u;122)4+sn(u;122)44

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

ddutlh⁡(u)=ctlh⁡(u)3

dductlh⁡(u)=−tlh⁡(u)3

∫0ϖ/2tlh⁡(u)du=ϖ2

∫0ϖ/2ctlh⁡(u)du=ϖ2

#### Derivation of the Hyperbolic Lemniscate functions

With respect to the quartic Fermat curve x4+y4=1, the hyperbolic lemniscate sine is analogous to the trigonometric tangent function. Unlike slh and clh, the functions sin4 and cos4 cannot be analytically extended to meromorphic functions in the whole complex plane.[75]

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
ddwx(w)=−y(w)3
ddwy(w)=x(w)3
x(w=0)=1
y(w=0)=0

The solutions to this system of equations are as follows:

x(w)=cl⁡(122w)[sl⁡(122w)2+1]1/2[sl⁡(122w)2+cl⁡(122w)2]−1/2
y(w)=sl⁡(122w)[cl⁡(122w)2+1]1/2[sl⁡(122w)2+cl⁡(122w)2]−1/2

The following therefore applies to the quotient:

y(w)x(w)=sl⁡(122w)[cl⁡(122w)2+1]1/2cl⁡(122w)[sl⁡(122w)2+1]1/2=slh⁡(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)=(1s4+14)2+(ss4+14)2=s2+1s4+14

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:

ddsarctan⁡(s)=1s2+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:

ddsaslh(s)=[ddsarctan⁡(s)]D(s)2=1s2+1D(s)2=1s2+1(s2+1s4+14)2=1s4+1

#### 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(∫0v1−w44dw)−v1−v44
ddvatlh(v)=21−v44−(ddvv1−v44)=1(1−v4)3/4

The following transformation applies:

aslh(x)=atlh(xx4+14)

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

ddxaslh(x)=ddxatlh(xx4+14)=(ddxxx4+14)[1−(xx4+14)4]−3/4=
=1(x4+1)5/4[1−(xx4+14)4]−3/4=1(x4+1)5/4(1x4+1)−3/4=1x4+1

### Specific values

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

∫0∞d⁡tt4+1=14B(14,14)=ϖ2=σ2=1.85407…

whereas 12B(12,12)=π2, so the values below such as slh(ϖ22)=slh(σ4)=1 are analogous to the trigonometric sin(π2)=1.

slh⁡(ϖ22)=1
slh⁡(ϖ32)=13423−34
slh⁡(2ϖ32)=23+34
slh⁡(ϖ42)=124(2+1−1)
slh⁡(3ϖ42)=124(2+1+1)
slh⁡(ϖ52)=1845−1204−5+1=25−24sin⁡(120π)sin⁡(320π)
slh⁡(2ϖ52)=1224(5+1)204−5+1=25+24sin⁡(120π)sin⁡(320π)
slh⁡(3ϖ52)=1845−1204+5+1=25−24cos⁡(120π)cos⁡(320π)
slh⁡(4ϖ52)=1224(5+1)204+5+1=25+24cos⁡(120π)cos⁡(320π)
slh⁡(ϖ62)=12(23+3+1)(1−23−34)
slh⁡(5ϖ62)=12(23+3+1)(1+23−34)

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

z | clh⁡z | slh⁡z | ctlh⁡z=cos4z | tlh⁡z=sin4z

|
|
|
|

0 | ∞ | 0 | 1 | 0
14σ | 1 | 1 | 1/24 | 1/24
12σ | 0 | ∞ | 0 | 1
34σ | −1 | −1 | −1/24 | 1/24
σ | ∞ | 0 | −1 | 0

### Combination and halving theorems

Given the hyperbolic lemniscate tangent (tlh) and hyperbolic lemniscate cotangent (ctlh). Recall the hyperbolic lemniscate area functions from the section on inverse functions,

aslh⁡(x)=∫0x1y4+1dy
aclh⁡(x)=∫x∞1y4+1dy

Then the following identities can be established,

tlh[aslh(x)]=ctlh[aclh(x)]=xx4+14
ctlh[aslh(x)]=tlh[aclh(x)]=1x4+14

hence the 4th power of tlh and ctlh for these arguments is equal to one,

tlh[aslh(x)]4+ctlh[aslh(x)]4=1
tlh[aclh(x)]4+ctlh[aclh(x)]4=1

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

slh[12aslh(x)]=2xx2+1+x4+1+x4+1−x2+1

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

aslh(x)=2arcsl[x(x4+1+1)−1/2]
arcsl(x)=2aslh(2x1+x2+1−x2)

In addition, the following formulas are valid for all real values x∈ℝ:

slh[12aclh(x)]=x4+1+x2−2xx4+1+x2=(x4+1−x2+1)−1/2(x4+1+1−x)
clh[12aclh(x)]=x4+1+x2+2xx4+1+x2=(x4+1−x2+1)−1/2(x4+1+1+x)

These identities follow from the last-mentioned formula:

tlh[12aclh(x)]2=122−22xx4+1−x2=(2x2+2+2x4+1)−1/2(x4+1+1−x)
ctlh[12aclh(x)]2=122+22xx4+1−x2=(2x2+2+2x4+1)−1/2(x4+1+1+x)

Hence, their 4th powers again equal one,

tlh[12aclh(x)]4+ctlh[12aclh(x)]4=1

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

sl[122aclh(x)]=cl[122aslh(x)]=x4+1−x2
sl[122aslh(x)]=cl[122aclh(x)]=x(x4+1+1)−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⁡(−x4) 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:

[∫0∞exp⁡(−x2)dx]2=∫0∞∫0∞exp⁡(−y2−z2)dydz=
=∫0π/2∫0∞det⁡[∂/∂rrcos⁡(ϕ)∂/∂ϕrcos⁡(ϕ)∂/∂rrsin⁡(ϕ)∂/∂ϕrsin⁡(ϕ)]exp⁡{−[rcos⁡(ϕ)]2−[rsin⁡(ϕ)]2}drdϕ=
=∫0π/2∫0∞rexp⁡(−r2)drdϕ=∫0π/212dϕ=π4

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

[∫0∞exp⁡(−x4)dx]2=∫0∞∫0∞exp⁡(−y4−z4)dydz=
=∫0ϖ/2∫0∞det⁡[∂/∂rrctlh(ϕ)∂/∂ϕrctlh(ϕ)∂/∂rrtlh(ϕ)∂/∂ϕrtlh(ϕ)]exp⁡{−[rctlh(ϕ)]4−[rtlh(ϕ)]4}drdϕ=
=∫0ϖ/2∫0∞rexp⁡(−r4)drdϕ=∫0ϖ/2π4dϕ=ϖπ42

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,ωn) for some positive integer n.[23][76] 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(ζn) for some positive integer n. Both are special cases of Kronecker's Jugendtraum, which became Hilbert's twelfth problem.

The field ℚ(i,sl⁡(ϖ/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 y2=4x3+x.[76]

### Hurwitz numbers

The Bernoulli numbers Bn can be defined by

Bn=limz->0dndznzez−1,n≥0

and appear in

∑k∈ℤ∖{0}1k2n=(−1)n−1B2n(2π)2n(2n)!=2ζ(2n),n≥1

where ζ is the Riemann zeta function.

The Hurwitz numbers Hn, named after Adolf Hurwitz, are the "lemniscate analogs" of the Bernoulli numbers. They can be defined by[77][78]

Hn=−limz->0dndznzζ(z;1/4,0),n≥0

where ζ(⋅;1/4,0) is the Weierstrass zeta function with lattice invariants 1/4 and 0. They appear in

∑z∈ℤ[i]∖{0}1z4n=H4n(2ϖ)4n(4n)!=G4n(i),n≥1

where ℤ[i] are the Gaussian integers and G4n are the Eisenstein series of weight 4n, and in

∑n=1∞nke2πn−1={124−18πif k=1Bk+12k+2if k≡1(mod4) and k≥5Bk+12k+2+Hk+12k+2(ϖπ)k+1if k≡3(mod4) and k≥3.

The Hurwitz numbers can also be determined as follows: H4=1/10,

H4n=3(2n−3)(16n2−1)∑k=1n−1(4n4k)(4k−1)(4(n−k)−1)H4kH4(n−k),n≥2

and Hn=0 if n is not a multiple of 4.[79] This yields[77]

H8=310,H12=567130,H16=43659170,…

Also[80]

denom⁡H4n=∏(p−1)|4np

where p∈ℙ such that p≢3(mod4), just as

denom⁡B2n=∏(p−1)|2np

where p∈ℙ (by the von Staudt–Clausen theorem).

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

B2n+∑(p−1)|2n1p∈ℤ,n≥1

(sequence A000146 in the OEIS) where p is any prime, and an analogous theorem holds for the Hurwitz numbers: suppose that a∈ℤ is odd, b∈ℤ is even, p is a prime such that p≡1(mod4), p=a2+b2 (see Fermat's theorem on sums of two squares) and a≡b+1(mod4). Then for any given p, 2a=ν(p) is uniquely determined; equivalently ν(p)=p−𝒩p where 𝒩p is the number of solutions of the congruence X3−X≡Y2(mod⁡p) in variables X,Y that are non-negative integers.[81] The Hurwitz theorem then determines the fractional part of the Hurwitz numbers:[77]

H4n−12−∑(p−1)|4nν(p)4n/(p−1)p=defGn∈ℤ,n≥1.

The sequence of the integers Gn starts with 0,−1,5,253,….[77]

Let n≥2. If 4n+1 is a prime, then Gn≡1(mod4). If 4n+1 is not a prime, then Gn≡3(mod4).[82]

Some authors instead define the Hurwitz numbers as Hn′=H4n.

#### Appearances in Laurent series

The Hurwitz numbers appear in several Laurent series expansions related to the lemniscate functions:[83]

sl2z=∑n=1∞24n(1−(−1)n22n)H4n4nz4n−2(4n−2)!,|z|<ϖ2sl′⁡zsl⁡z=1z−∑n=1∞24n(2−(−1)n22n)H4n4nz4n−1(4n−1)!,|z|<ϖ21sl⁡z=1z−∑n=1∞22n((−1)n2−22n)H4n4nz4n−1(4n−1)!,|z|<ϖ1sl2z=1z2+∑n=1∞24nH4n4nz4n−2(4n−2)!,|z|<ϖ

Analogously, in terms of the Bernoulli numbers:

1sinh2z=1z2−∑n=1∞22nB2n2nz2n−2(2n−2)!,|z|<π.

### A quartic analog of the Legendre symbol

Let p be a prime such that p≡1(mod4). A quartic residue (mod p) is any number congruent to the fourth power of an integer. Define (ap)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′∈ℤ[i] (see[84] for these numbers) such that[85]

(ap)4=∏p′sl⁡(2ϖap′/p)sl⁡(2ϖp′/p).

This theorem is analogous to

(ap)=∏n=1p−12sin⁡(2πan/p)sin⁡(2πn/p)

where (⋅⋅) is the Legendre symbol.

## World map projections

"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).[86]

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.[87] 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.[88] Because many partial differential equations can be effectively solved by conformal mapping, this map from sphere to cube is convenient for atmospheric modeling.[89]

## See also

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

## Notes

1. ↑ Fagnano (1718–1723); Euler (1761); Gauss (1917)
2. ↑ Gauss (1917) p. 199 used the symbols sl and cl for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. Cox (1984) p. 316, Eymard & Lafon (2004) p. 204, and Lemmermeyer (2000) p. 240. Ayoub (1984) uses sinlem and coslem. Whittaker & Watson (1920) use the symbols sin lemn and cos lemn. Some sources use the generic letters s and c. Prasolov & Solovyev (1997) use the letter φ for the lemniscate sine and φ′ for its derivative.
3. ↑ The circle x2+y2=x is the unit-diameter circle centered at (12,0) with polar equation r=cos⁡θ, the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle x2+y2=1 centered at the origin. Notice that the lemniscate (x2+y2)2=x2−y2 is the degree-4 clover.
4. ↑ The fundamental periods (1+i)ϖ and (1−i)ϖ are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.
5. ↑ Robinson (2019a) 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 Schwarz (1869) p. 113.
7. ↑ Schappacher (1997). OEIS sequence A062539 lists the lemniscate constant's decimal digits.
8. ↑ Levin (2006)
9. ↑ Todd (1975)
10. ↑ Cox (1984)
11. ↑ Dark areas represent zeros, and bright areas represent poles. As the argument of sl⁡z changes from −π (excluding −π) to π, the colors go through cyan, blue (Arg⁡≈−π/2), magneta, red (Arg⁡≈0), orange, yellow (Arg⁡≈π/2), green, and back to cyan (Arg⁡≈π).
12. ↑ Combining the first and fourth identity gives sl⁡z=−i/sl⁡(z−(1+i)ϖ/2). This identity is (incorrectly) given in Eymard & Lafon (2004) p. 226, without the minus sign at the front of the right-hand side.
13. ↑ The even Gaussian integers are the residue class of 0, modulo 1+i, the black squares on a checkerboard.
14. ↑ Prasolov & Solovyev (1997); Robinson (2019a)
15. ↑ 15.0 15.1 Cox (2012)
16. ↑ Reinhardt & Walker (2010a) §22.12.6, §22.12.12
17. ↑ Analogously, 1sin⁡z=∑n∈ℤ(−1)nz+nπ.
18. ↑ Lindqvist & Peetre (2001) generalizes the first of these forms.
19. ↑ Ayoub (1984); Prasolov & Solovyev (1997)
20. ↑ Euler (1761) §44 p. 79, §47 pp. 80–81
21. ↑ 21.0 21.1 Euler (1761) §46 p. 80
22. ↑ In fact, iε=sl⁡βϖ2.
23. ↑ 23.0 23.1 23.2 Cox & Hyde (2014)
24. ↑ Gómez-Molleda & Lario (2019)
25. ↑ The fourth root with the least positive principal argument is chosen.
26. ↑ The restriction to positive and odd β can be dropped in deg⁡Λβ=|(𝒪/β𝒪)×|.
27. ↑ Cox (2013) p. 142, Example 7.29(c)
28. ↑ Rosen (1981)
29. ↑ Eymard & Lafon (2004) p. 200
30. ↑ And the area enclosed by ℒ is 1, which stands in stark contrast to the unit circle (whose enclosed area is a non-constructible number).
31. ↑ 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.
32. ↑ Reinhardt & Walker (2010a) §22.18.E6
33. ↑ Siegel (1969); Schappacher (1997)
34. ↑ Such numbers are OEIS sequence A003401.
35. ↑ Abel (1827–1828); Rosen (1981); Prasolov & Solovyev (1997)
36. ↑ Euler (1786); Sridharan (2004); Levien (2008)
37. ↑ "A104203". https://oeis.org/A104203.
38. ↑ Lomont, J.S.; Brillhart, John (2001). Elliptic Polynomials. CRC Press. pp. 12, 44. ISBN 1-58488-210-7.
39. ↑ 39.0 39.1 39.2 39.3 "A193543 - Oeis". https://oeis.org/A193543.
40. ↑ Lomont, J.S.; Brillhart, John (2001). Elliptic Polynomials. CRC Press. ISBN 1-58488-210-7. p. 79, eq. 5.36
41. ↑ Lomont, J.S.; Brillhart, John (2001). Elliptic Polynomials. CRC Press. ISBN 1-58488-210-7. p. 79, eq. 5. 36 and p. 78, eq. 5.33
42. ↑ 42.0 42.1 "A289695 - Oeis". https://oeis.org/A289695.
43. ↑ Wall, H. S. (1948). Analytic Theory of Continued Fractions. Chelsea Publishing Company. pp. 374–375.
44. ↑ Reinhardt & Walker (2010a) §22.20(ii)
45. ↑ Carlson (2010) §19.8
46. ↑ Reinhardt & Walker (2010a) §22.12.12
47. ↑ In general, sinh⁡(x−nπ) and sin⁡(x−nπi)=−isinh⁡(ix+nπ) are not equivalent, but the resulting infinite sum is the same.
48. ↑ Reinhardt & Walker (2010a) §22.11
49. ↑ Reinhardt & Walker (2010a) §22.2.E7
50. ↑ Berndt (1994) p. 247, 248, 253
51. ↑ Reinhardt & Walker (2010a) §22.11.E1
52. ↑ Whittaker & Watson (1927)
53. ↑ Borwein & Borwein (1987)
54. ↑ 54.0 54.1 Eymard & Lafon (2004) p. 227.
55. ↑ Cartan, H. (1961) (in French). Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes. Hermann. pp. 160–164.
56. ↑ More precisely, suppose {an} is a sequence of bounded complex functions on a set S, such that ∑|an(z)| converges uniformly on S. If {n1,n2,n3,…} is any permutation of {1,2,3,…}, then ∏n=1∞(1+an(z))=∏k=1∞(1+ank(z)) for all z∈S. The theorem in question then follows from the fact that there exists a bijection between the natural numbers and α's (resp. β's).
57. ↑ Bottazzini & Gray (2013) p. 58
58. ↑ More precisely, if for each k, limn->∞ak(n) exists and there is a convergent series ∑k=1∞Mk of nonnegative real numbers such that |ak(n)|≤Mk for all n∈ℕ and 1≤k≤n, then

limn->∞∑k=1nak(n)=∑k=1∞limn->∞ak(n).
59. ↑ 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 ∑α1α4=−the coefficient ofz5 in the M series, and infinitely many others.
60. ↑ Gauss, C. F. (1866) (in Latin, German). Werke (Band III). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. https://gdz.sub.uni-goettingen.de/id/PPN235999628. p. 405; there's an error on the page: the coefficient of φ17 should be 1077410154752000, not 107207484333056000.
61. ↑ If M(z)=∑n=0∞anzn+1, then the coefficients an are given by the recurrence an+1=−1n+1∑k=0n2n−k+1akHn−k+1(n−k+1)! with a0=1 where Hn are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers.
62. ↑ The power series expansions of M and N are useful for finding a β-division polynomial for the β-division of the lemniscate ℒ (where β=m+ni where m,n∈ℤ such that m+n is odd). For example, suppose we want to find a 3-division polynomial. Given that

M(3z)=d9M(z)9+d5M(z)5N(z)4+d1M(z)N(z)8
for some constants d1,d5,d9, from

3z−2(3z)55!−36(3z)99!+O⁡(z13)=d9x9+d5x5y4+d1xy8,
where

x=z−2z55!−36z99!+O⁡(z13),y=1+2z44!−4z88!+O⁡(z12),
we have

{d1,d5,d9}={3,−6,−1}.
Therefore, a 3-division polynomial is

−X9−6X5+3X
(meaning one of its roots is sl⁡(2ϖ/3)). The equations arrived at by this process are the lemniscate analogs of

Xn=1
(so that e2πi/n is one of the solutions) which comes up when dividing the unit circle into n arcs of equal length. In the following note, the first few coefficients of the monic normalization of such β-division polynomials are described symbolically in terms of β.
63. ↑ By utilizing the power series expansion of the N function, it can be proved that a polynomial having sl⁡(2ϖ/β) as one of its roots (with β from the previous note) is

∑n=0(ββ‾−1)/4a4n+1(β)Xββ‾−4n
where

a1(β)=1,a5(β)=β4−ββ‾12,a9(β)=−β8−70β5β‾+336β4+35β2β‾2−300ββ‾10080
and so on.
64. ↑ Zhuravskiy, A. M. (1941) (in Russian). Spravochnik po ellipticheskim funktsiyam. Izd. Akad. Nauk. U.S.S.R..
65. ↑ For example, by the quasi-addition formulas, the duplication formulas and the Pythagorean-like identities, we have

M(3z)=−M(z)9−6M(z)5N(z)4+3M(z)N(z)8,
N(3z)=N(z)9+6M(z)4N(z)5−3M(z)8N(z),

sl⁡3z=−M(z)9−6M(z)5N(z)4+3M(z)N(z)8N(z)9+6M(z)4N(z)5−3M(z)8N(z).
On dividing the numerator and the denominator by N(z)9, we obtain the triplication formula for sl:

sl⁡3z=−sl9z−6sl5z+3sl⁡z1+6sl4z−3sl8z.
66. ↑ Gauss (1866), p. 408
67. ↑ Robinson (2019a)
68. ↑ Eymard & Lafon (2004) p. 234
69. ↑ Armitage, J. V.; Eberlein, W. F. (2006). Elliptic Functions. Cambridge University Press. p. 49. ISBN 978-0-521-78563-1.
70. ↑ The identity cl⁡z=cn(2z;12) can be found in Greenhill (1892) p. 33.
71. ↑ Siegel (1969)
72. ↑ Sloane, N. J. A., ed. "Sequence A175576". OEIS Foundation. https://oeis.org/A175576.
73. ↑ Berndt, Bruce C. (1989). Ramanujan's Notebooks Part II. Springer. ISBN 978-1-4612-4530-8. p. 96
74. ↑ Levin (2006); Robinson (2019b)
75. ↑ Levin (2006) p. 515
76. ↑ 76.0 76.1 Cox (2012) p. 508, 509
77. ↑ 77.0 77.1 77.2 77.3 Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203—206
78. ↑ Equivalently, Hn=−limz->0dndzn((1+i)z/2sl⁡((1+i)z/2)+z2ℰ(z2;i)) where n≥4 and ℰ(⋅;i) is the Jacobi epsilon function with modulus i.
79. ↑ The Bernoulli numbers can be determined by an analogous recurrence: B2n=−12n+1∑k=1n−1(2n2k)B2kB2(n−k) where n≥2 and B2=1/6.
80. ↑ Katz, Nicholas M. (1975). "The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers". Mathematische Annalen 216 (1): 1–4. doi:10.1007/BF02547966. See eq. (9)
81. ↑ For more on the ν function, see Lemniscate constant.
82. ↑ Hurwitz, Adolf (1963) (in German). Mathematische Werke: Band II. Springer Basel AG. p. 370
83. ↑ Arakawa et al. (2014) define H4n by the expansion of 1/sl2.
84. ↑ Eisenstein, G. (1846). "Beiträge zur Theorie der elliptischen Functionen" (in German). Journal für die reine und angewandte Mathematik 30. https://gdz.sub.uni-goettingen.de/id/PPN243919689_0030?tify={"pages":[202],"view":"scan"}. Eisenstein uses φ=sl and ω=2ϖ.
85. ↑ Ogawa (2005)
86. ↑ Peirce (1879). Guyou (1887) and Adams (1925) harv error: no target: CITEREFAdams1925 (help) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976) harv error: no target: CITEREFLee1976 (help). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice.
87. ↑ Adams (1925) harv error: no target: CITEREFAdams1925 (help)
88. ↑ Adams (1925) harv error: no target: CITEREFAdams1925 (help); Lee (1976) harv error: no target: CITEREFLee1976 (help).
89. ↑ Rančić, Purser & Mesinger (1996); McGregor (2005).

## References

* Abel, Niels Henrik (1827–1828) "Recherches sur les fonctions elliptiques" [Research on elliptic functions] (in French). Crelle's Journal.
Part 1. 1827. 2 (2): 101–181. doi:10.1515/crll.1827.2.101.
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## External links

* Parker, Matt (2021). "What is the area of a Squircle?". Stand-up Maths. YouTube.
* Ramalingam, Muthu Veerappan (2023). "Bernoulli Lemniscate and the Squircle || A remarkable Geometric fun fact!!?". Act of Learning. YouTube. Relation shown in the video amounts to cl⁡(2t)=cos42(t)−sin42(t)cos42(t)+sin42(t)

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