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 [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 sineslh and hyperbolic lemniscate cosineclh 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
Definitions
The lemniscate functions sl and cl can be defined as the solution to the initial value problem:[5]
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
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 meanM:[10]
Argument identities
Zeros, poles and symmetries
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.
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 squareperiod 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
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
The functions and satisfy another Pythagorean-like identity:
Derivatives and integrals
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
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]
In particular, to compute the complex-valued functions in real components,
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
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
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]
Further values
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.
, 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 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:[30]
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[31]
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:[32]
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.[33] 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.[34] 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; the theorem refers to constructing the division points only.
Let . Then the n-division points for are the points
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.[35] 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
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
Power series
The power series expansion of the lemniscate sine at the origin is[36]
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[37]
where
The power series expansion of at the origin is
where if is even and[38]
if is odd.
The expansion can be equivalently written as[39]
where
For the lemniscate cosine,[40]
where
Ramanujan's cos/cosh identity
Ramanujan's famous cos/cosh identity states that if
then[38]
There is a close relation between the lemniscate functions and . Indeed,[38][41]
and
Continued fractions
For :[42]
Methods of computation
A fast algorithm, returning approximations to (which get closer to with increasing ), is the following:[43]
for eachdo
ifthen
break
for eachn from N to 0do
return
This is effectively using the arithmetic-geometric mean and is based on Landen's transformations.[44]
Several methods of computing involve first making the change of variables and then computing
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:[38][41][51]
Two other fast computation methods use the following sum and product series:
The following identities come from product representations of the theta functions:[52]
A similar formula involving the function can be given.[53]
In the same pattern following sum formulas can be set up with the help of the tangent duplication theorem:
where
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]
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
for some constant for but this result holds for all by analytic continuation. Using
gives which completes the proof.
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]
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[63]
Furthermore, the identities
and the Pythagorean-like identities
hold for all .
An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see Lemniscate elliptic functions § Methods of computation; the theta functions and the above functions are not equivalent).
Relation to other functions
Relation to Weierstrass and Jacobi elliptic functions
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".[64]
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:[65]
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.[66][67]
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
The lemniscate sine can be used for the computation of values of the modular lambda function:
For example:
Inverse functions
The inverse function of the lemniscate sine is the lemniscate arcsine, defined as
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:
Furthermore there are the so called Hyperbolic lemniscate area functions:
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:
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):
The lemniscate arccosine has this expression:
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):
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).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. [68]) appears in entry 34e of chapter 11 of Ramanujan's second notebook.[69]
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
Hyperbolic Lemniscate Tangent and Cotangent
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 .[70] Just as is the area enclosed by the circle , the area enclosed by the squircle is . Moreover,
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
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.[71]
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
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:
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
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
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
In combination with the Hyperbolic Lemniscate Areasine, the following identities can be established:
The square of the Hyperbolic Lemniscate Tangent is the Pythagorean counterpart of the square of the Hyperbolic Lemniscate cotangent because the sum of the fourth powers of and is always equal to the value one.
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:
The following formulas for the lemniscatic sine and lemniscatic cosine are closely related:
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 integrated in relation to x. In the following, the proofs of both integrals are given in a parallel way of displaying.
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.
In fact, the von Staudt–Clausen theorem states that
(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 and[73]
The sequence of the integers starts with [73]
Let . If is a prime, then . If is not a prime, then .[77]
Some authors instead define the Hurwitz numbers as .
Appearances in Laurent series
The Hurwitz numbers appear in several Laurent series expansions related to the lemniscate functions:[78]
Analogously, in terms of the Bernoulli numbers:
A quartic analog of the Legendre symbol
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[79] for these numbers) such that[80]
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.[82] 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.[83] Because many partial differential equations can be effectively solved by conformal mapping, this map from sphere to cube is convenient for atmospheric modeling.[84]
^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.
^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.
^Robinson (2019a) starts from this definition and thence derives other properties of the lemniscate functions.
^This map was the first ever picture of a Schwarz–Christoffel mapping, in Schwarz (1869) p. 113.
^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 .
^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.
^The even Gaussian integers are the residue class of 0, modulo 1 + i, the black squares on a checkerboard.
^Cartan, H. (1961). Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes (in French). Hermann. pp. 160–164.
^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).
^More precisely, if for each , exists and there is a convergent series of nonnegative real numbers such that for all and , then
^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.
^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 .
^The function satisfies the differential equation (see Gauss (1866), p. 408). The function satisfies the differential equation
^The Bernoulli numbers can be determined by an analogous recurrence: where and .
^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)
^Hurwitz, Adolf (1963). Mathematische Werke: Band II (in German). Springer Basel AG. p. 370
^Eisenstein, G. (1846). "Beiträge zur Theorie der elliptischen Functionen". Journal für die reine und angewandte Mathematik (in German). 30. Eisenstein uses and .
^Ogawa, Takuma (2005). "Similarities between the trigonometric function and the lemniscate function from arithmetic view point". Tsukuba Journal of Mathematics. 29 (1).
Parker, Matt (2021). "What is the area of a Squircle?". Stand-up Maths. YouTube. Archived from the original on 2021-12-19.
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