Lemniscate elliptic functions

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

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 [3] the lemniscate sine relates the arc length to the chord length of a lemniscate

The lemniscate functions have periods related to a number 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) 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 [4] and are a special case of two Jacobi elliptic functions on that lattice, .

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

The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function .

Lemniscate sine and cosine functions

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Definitions

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

 

or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners  [6]

 

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:

 

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

 

Relation to the lemniscate constant

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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   and   for a constant ϖ called the lemniscate constant,[7]

 

The lemniscate functions satisfy the basic relation   analogous to the relation  

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:

 

An analogous formula for ϖ is:[8]

 

The Machin formula for π is   and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula  . Analogous formulas can be developed for ϖ, including the following found by Gauss:  [9]

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

 

Argument identities

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Zeros, poles and symmetries

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  in the complex plane.[11] In the picture, it can be seen that the fundamental periods   and   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,

 

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

 

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

 

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

 

This makes them elliptic functions (doubly periodic meromorphic functions in the complex plane) with a diagonal square period lattice of fundamental periods   and  .[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:

 

The sl function has simple zeros at Gaussian integer multiples of ϖ, complex numbers of the form   for integers a and b. It has simple poles at Gaussian half-integer multiples of ϖ, complex numbers of the form  , with residues  . The cl function is reflected and offset from the sl function,  . It has zeros for arguments   and poles for arguments   with residues  

Also

 

for some   and

 

The last formula is a special case of complex multiplication. Analogous formulas can be given for   where   is any Gaussian integer – the function   has complex multiplication by  .[15]

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

 
 

Pythagorean-like identity

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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:

 

As a result, the parametric equation   parametrizes the quartic curve  

This identity can alternately be rewritten:[18]

 
 

Defining a tangent-sum operator as   gives:

 

The functions   and   satisfy another Pythagorean-like identity:

 

Derivatives and integrals

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The derivatives are as follows:

 
 

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

 
 

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

 

Argument sum and multiple identities

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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]

 

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   and tangent-difference operator   the argument sum and difference identities can be expressed as:[20]

 

These resemble their trigonometric analogs:

 

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

 

Gauss discovered that

 

where   such that both sides are well-defined.

Also

 

where   such that both sides are well-defined; this resembles the trigonometric analog

 

Bisection formulas:

 
 

Duplication formulas:[21]

 
 

Triplication formulas:[21]

 
 

Note the "reverse symmetry" of the coefficients of numerator and denominator of  . This phenomenon can be observed in multiplication formulas for   where   whenever   and   is odd.[15]

Lemnatomic polynomials

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Let   be the lattice

 

Furthermore, let  ,  ,  ,  ,   (where  ),   be odd,   be odd,   and  . Then

 

for some coprime polynomials   and some  [22] where

 

and

 

where   is any  -torsion generator (i.e.   and   generates   as an  -module). Examples of  -torsion generators include   and  . The polynomial   is called the  -th lemnatomic polynomial. It is monic and is irreducible over  . The lemnatomic polynomials are the "lemniscate analogs" of the cyclotomic polynomials,[23]

 

The  -th lemnatomic polynomial   is the minimal polynomial of   in  . For convenience, let   and  . So for example, the minimal polynomial of   (and also of  ) in   is

 

and[24]

 
 [25]

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

 

which is the minimal polynomial of   (and also of  ) in  

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

 

which can be compared to the cyclotomic analog

 

Specific values

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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   where k is a non-negative integer and each pi (if any) is a distinct Fermat prime.[28]

     
     
     
     
     
     
     
     
     

Relation to geometric shapes

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Arc length of Bernoulli's lemniscate

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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.

 , 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   and   which are unit distance apart, let   be the reflection of   about  . Then   is the closure of the locus of the points   such that   is a right angle.[29]

Focal characterization:   is the locus of points in the plane such that the product of their distances from the two focal points   and   is the constant  .

Explicit coordinate characterization:   is a quartic curve satisfying the polar equation   or the Cartesian equation  

The perimeter of   is  .[30]

The points on   at distance   from the origin are the intersections of the circle   and the hyperbola  . The intersection in the positive quadrant has Cartesian coordinates:

 

Using this parametrization with   for a quarter of  , the arc length from the origin to a point   is:[31]

 

Likewise, the arc length from   to   is:

 

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  , respectively.

Analogously, the circular sine and cosine functions relate the chord length to the arc length for the unit diameter circle with polar equation   or Cartesian equation   using the same argument above but with the parametrization:

 

Alternatively, just as the unit circle   is parametrized in terms of the arc length   from the point   by

 

  is parametrized in terms of the arc length   from the point   by[32]

 

The notation   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]

 
 
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   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   is a power of two (where   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  . Then the n-division points for   are the points

 

where   is the floor function. See below for some specific values of  .

Arc length of rectangular elastica

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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:

 

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

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The lemniscate elliptic functions and an ellipse

Let   be a point on the ellipse   in the first quadrant and let   be the projection of   on the unit circle  . The distance   between the origin   and the point   is a function of   (the angle   where  ; equivalently the length of the circular arc  ). The parameter   is given by

 

If   is the projection of   on the x-axis and if   is the projection of   on the x-axis, then the lemniscate elliptic functions are given by

 
 

Series Identities

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Power series

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

 

where the coefficients   are determined as follows:

 
 

where   stands for all three-term compositions of  . For example, to evaluate  , it can be seen that there are only six compositions of   that give a nonzero contribution to the sum:   and  , so

 

The expansion can be equivalently written as[38]

 

where

 

The power series expansion of   at the origin is

 

where   if   is even and[39]

 

if   is odd.

The expansion can be equivalently written as[40]

 

where

 
 

For the lemniscate cosine,[41]

 
 

where

 
 

Ramanujan's cos/cosh identity

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Ramanujan's famous cos/cosh identity states that if

 

then[39]

 

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

 
 

and

 

Continued fractions

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For  :[43]

 
 

Methods of computation

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

  •      
  • for each   do
    •      
    • if   then
      •   break
  •  
  • for each n from N to 0 do
    •  
  • return  

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

Several methods of computing   involve first making the change of variables   and then computing  

A hyperbolic series method:[46][47]

 
 

Fourier series method:[48]

 
 
 

The lemniscate functions can be computed more rapidly by

 

where

 

are the Jacobi theta functions.[49]

Fourier series for the logarithm of the lemniscate sine:

 

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

 
 

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

 
 
 
 

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

 
 

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

The lemniscate functions as a ratio of entire functions

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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]

 

where

 

Here,   and   denote, respectively, the zeros and poles of sl which are in the quadrant  . 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

 

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

 

Therefore

 

It is known that

 

Then from

 

and

 

we get

 

Hence

 

Therefore

 

for some constant   for   but this result holds for all   by analytic continuation. Using

 

gives   which completes the proof.  

Proof by Liouville's theorem

Let

 

with patches at removable singularities. The shifting formulas

 

imply that   is an elliptic function with periods   and  , just as  . It follows that the function   defined by

 

when patched, is an elliptic function without poles. By Liouville's theorem, it is a constant. By using  ,   and  , this constant is  , which proves the theorem.  

Gauss conjectured that   (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   and   as infinite series (see below). He also discovered several identities involving the functions   and  , such as

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

and

 

Thanks to a certain theorem[58] on splitting limits, we are allowed to multiply out the infinite products and collect like powers of  . Doing so gives the following power series expansions that are convergent everywhere in the complex plane:[59][60][61][62][63]

 
 

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

We define   and   by

 

Then the lemniscate cosine can be written as

 

where[64]

 
 

Furthermore, the identities

 
 
 

and the Pythagorean-like identities

 
 

hold for all  .

The quasi-addition formulas

 
 

(where  ) imply further multiplication formulas for   and   by recursion.[65]

Gauss'   and   satisfy the following system of differential equations:

 
 

where  . Both   and   satisfy the differential equation[66]

 

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

 
 

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   and   is

 
 

where  .

Relation to other functions

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Relation to Weierstrass and Jacobi elliptic functions

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The lemniscate functions are closely related to the Weierstrass elliptic function   (the "lemniscatic case"), with invariants g2 = 1 and g3 = 0. This lattice has fundamental periods   and  . The associated constants of the Weierstrass function are  

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   is called the "pseudolemniscatic case".[67]

The square of the lemniscate sine can be represented as

 

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]

 

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

 
 

where the second arguments denote the elliptic modulus  .

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

 
 

Relation to the modular lambda function

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

 

For example:

 

Inverse functions

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

 

It can also be represented by the hypergeometric function:

 

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:

 

For x in the interval  ,   and  

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

 

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

 
 
 
 
 

Expression using elliptic integrals

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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]

 
 

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]

 

The lemniscate arccosine has this expression:[citation needed]

 

Use in integration

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

 
 
 
 
 
 
 
 
 
 
 

Hyperbolic lemniscate functions

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Fundamental information

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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 in the complex plane. Dark areas represent zeros and bright areas represent poles. The complex argument is represented by varying hue.

For convenience, let  .   is the "squircular" analog of   (see below). The decimal expansion of   (i.e.  [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:

 

where in  ,   is in the square with corners  . Beyond that square, the functions can be analytically continued to meromorphic functions in the whole complex plane.

The complete integral has the value:

 

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

 

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

 

The functions   and   have a square period lattice with fundamental periods  .

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

 
 

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

 
 

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

 

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

 

Relation to quartic Fermat curve

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Hyperbolic Lemniscate Tangent and Cotangent

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

 
Superellipse with the relation  

In a quartic Fermat curve   (sometimes called a squircle) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle   (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  .[74] Just as   is the area enclosed by the circle  , the area enclosed by the squircle   is  . Moreover,

 

where   is the arithmetic–geometric mean.

The hyperbolic lemniscate sine satisfies the argument addition identity:

 

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

 

 

 

 

There are also the Hyperbolic lemniscate tangent and the Hyperbolic lemniscate coangent als further functions:

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

 
 

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:

 
 

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

 

 

 

 

Derivation of the Hyperbolic Lemniscate functions

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With respect to the quartic Fermat curve  , the hyperbolic lemniscate sine is analogous to the trigonometric tangent function. Unlike   and  , the functions   and   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:

 
 
 
 
 

The solutions to this system of equations are as follows:

 
 

The following therefore applies to the quotient:

 

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

 
 

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

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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:

 

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:

 

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:

 

Second proof: integral formation and area subtraction

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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:

 
 

The following transformation applies:

 

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

 
 

Specific values

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

 

whereas   so the values below such as   are analogous to the trigonometric  .

 
 
 
 
 
 
 
 
 
 
 

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

         
         
         
         
         
         

Combination and halving theorems

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

 
 

Then the following identities can be established,

 
 

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

 
 

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

 

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

 
 

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

 
 

These identities follow from the last-mentioned formula:

 
 

Hence, their 4th powers again equal one,

 

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

 
 

Coordinate Transformations

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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   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:

 
 
 

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

 
 
 

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

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In algebraic number theory, every finite abelian extension of the Gaussian rationals   is a subfield of   for some positive integer  .[23][76] This is analogous to the Kronecker–Weber theorem for the rational numbers   which is based on division of the circle – in particular, every finite abelian extension of   is a subfield of   for some positive integer  . Both are special cases of Kronecker's Jugendtraum, which became Hilbert's twelfth problem.

The field   (for positive odd  ) is the extension of   generated by the  - and  -coordinates of the  -torsion points on the elliptic curve  .[76]

Hurwitz numbers

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The Bernoulli numbers   can be defined by

 

and appear in

 

where   is the Riemann zeta function.

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

 

where   is the Weierstrass zeta function with lattice invariants   and  . They appear in

 

where   are the Gaussian integers and   are the Eisenstein series of weight  , and in

 

The Hurwitz numbers can also be determined as follows:  ,

 

and   if   is not a multiple of  .[79] This yields[77]

 

Also[80]

 

where   such that   just as

 

where   (by the von Staudt–Clausen theorem).

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

 

(sequence A000146 in the OEIS) where   is any prime, and an analogous theorem holds for the Hurwitz numbers: suppose that   is odd,   is even,   is a prime such that  ,   (see Fermat's theorem on sums of two squares) and  . Then for any given  ,   is uniquely determined; equivalently   where   is the number of solutions of the congruence   in variables   that are non-negative integers.[81] The Hurwitz theorem then determines the fractional part of the Hurwitz numbers:[77]

 

The sequence of the integers   starts with  [77]

Let  . If   is a prime, then  . If   is not a prime, then  .[82]

Some authors instead define the Hurwitz numbers as  .

Appearances in Laurent series

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

 

Analogously, in terms of the Bernoulli numbers:

 

A quartic analog of the Legendre symbol

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Let   be a prime such that  . A quartic residue (mod  ) is any number congruent to the fourth power of an integer. Define   to be   if   is a quartic residue (mod  ) and define it to be   if   is not a quartic residue (mod  ).

If   and   are coprime, then there exist numbers   (see[84] for these numbers) such that[85]

 

This theorem is analogous to

 

where   is the Legendre symbol.

World map projections

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"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

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Notes

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  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   is the unit-diameter circle centered at   with polar equation   the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle   centered at the origin. Notice that the lemniscate   is the degree-4 clover.
  4. ^ The fundamental periods   and   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   changes from   (excluding  ) to  , the colors go through cyan, blue  , magneta, red  , orange, yellow  , green, and back to cyan  .
  12. ^ Combining the first and fourth identity gives  . 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. ^ a b Cox (2012)
  16. ^ Reinhardt & Walker (2010a) §22.12.6, §22.12.12
  17. ^ Analogously,  
  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. ^ a b Euler (1761) §46 p. 80
  22. ^ In fact,  .
  23. ^ a b c 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   can be dropped in  .
  27. ^ Cox (2013) p. 142, Example 7.29(c)
  28. ^ Rosen (1981)
  29. ^ Eymard & Lafon (2004) p. 200
  30. ^ And the area enclosed by   is  , 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". The On-Line Encyclopedia of Integer Sequences.
  38. ^ Lomont, J.S.; Brillhart, John (2001). Elliptic Polynomials. CRC Press. pp. 12, 44. ISBN 1-58488-210-7.
  39. ^ a b c d "A193543 - Oeis".
  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. ^ a b "A289695 - Oeis".
  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,   and   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. ^ a b Eymard & Lafon (2004) p. 227.
  55. ^ Cartan, H. (1961). Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes (in French). Hermann. pp. 160–164.
  56. ^ More precisely, suppose   is a sequence of bounded complex functions on a set  , such that   converges uniformly on  . If   is any permutation of  , then   for all  . The theorem in question then follows from the fact that there exists a bijection between the natural numbers and  's (resp.  's).
  57. ^ Bottazzini & Gray (2013) p. 58
  58. ^ More precisely, if for each  ,   exists and there is a convergent series   of nonnegative real numbers such that   for all   and  , then
     
  59. ^ Alternatively, it can be inferred that these expansions exist just from the analyticity of   and  . 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   in the   series, and infinitely many others.
  60. ^ Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 405; there's an error on the page: the coefficient of   should be  , not  .
  61. ^ If  , then the coefficients   are given by the recurrence   with   where   are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers.
  62. ^ The power series expansions of   and   are useful for finding a  -division polynomial for the  -division of the lemniscate   (where   where   such that   is odd). For example, suppose we want to find a  -division polynomial. Given that
     
    for some constants  , from
     
    where
     
    we have
     
    Therefore, a  -division polynomial is
     
    (meaning one of its roots is  ). The equations arrived at by this process are the lemniscate analogs of
     
    (so that   is one of the solutions) which comes up when dividing the unit circle into   arcs of equal length. In the following note, the first few coefficients of the monic normalization of such  -division polynomials are described symbolically in terms of  .
  63. ^ By utilizing the power series expansion of the   function, it can be proved that a polynomial having   as one of its roots (with   from the previous note) is
     
    where
     
    and so on.
  64. ^ Zhuravskiy, A. M. (1941). Spravochnik po ellipticheskim funktsiyam (in Russian). 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
     
     
    so
     
    On dividing the numerator and the denominator by  , we obtain the triplication formula for  :
     
  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   can be found in Greenhill (1892) p. 33.
  71. ^ Siegel (1969)
  72. ^ http://oeis.org/A175576 [bare URL]
  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. ^ a b Cox (2012) p. 508, 509
  77. ^ a b c d Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203—206
  78. ^ Equivalently,   where   and   is the Jacobi epsilon function with modulus  .
  79. ^ The Bernoulli numbers can be determined by an analogous recurrence:   where   and  .
  80. ^ Katz, Nicholas M. (1975). "The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers". Mathematische Annalen. 216 (1): 1–4. See eq. (9)
  81. ^ For more on the   function, see Lemniscate constant.
  82. ^ Hurwitz, Adolf (1963). Mathematische Werke: Band II (in German). Springer Basel AG. p. 370
  83. ^ Arakawa et al. (2014) define   by the expansion of  
  84. ^ Eisenstein, G. (1846). "Beiträge zur Theorie der elliptischen Functionen". Journal für die reine und angewandte Mathematik (in German). 30. Eisenstein uses   and  .
  85. ^ Ogawa, Takuma (2005). "Similarities between the trigonometric function and the lemniscate function from arithmetic view point". Tsukuba Journal of Mathematics. 29 (1).
  86. ^ 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.
  87. ^ Adams (1925)
  88. ^ Adams (1925); Lee (1976).
  89. ^ Rančić, Purser & Mesinger (1996); McGregor (2005).
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References

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