Lemniscate elliptic functions

In mathematics, the lemniscate elliptic functions are elliptic functions [|related] to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.
The lemniscate sine and lemniscate cosine functions, usually written with the symbols and , 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 the lemniscate sine relates the arc length to the chord length of a lemniscate
The lemniscate functions have periods related to a number called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the , ratio of perimeter to diameter of a circle.
As complex functions, and have a square period lattice with fundamental periods and are a special case of two Jacobi elliptic functions on that lattice, .
Similarly, the hyperbolic lemniscate sine and hyperbolic lemniscate cosine 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 and can be defined as the solution to the initial value problem:
or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners
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:
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 functions have minimal real period, minimal imaginary period and fundamental complex periods and for a constant called the lemniscate constant,
The lemniscate functions satisfy the basic relation analogous to the relation
The lemniscate constant is a close analog of the circle constant, and many identities involving have analogues involving, as identities involving the trigonometric functions have analogues involving the lemniscate functions. For example, Viète's formula for can be written:
An analogous formula for is:
The Machin formula for is and several similar formulas for can be developed using trigonometric angle sum identities, e.g. Euler's formula. Analogous formulas can be developed for, including the following found by Gauss:
The lemniscate and circle constants were found by Gauss to be related to each-other by the arithmetic-geometric mean :

Argument identities

Zeros, poles and symmetries

The lemniscate functions and are even and odd functions, respectively,
At translations of and are exchanged, and at translations of they are additionally rotated and reciprocated:
Doubling these to translations by a unit-Gaussian-integer multiple of , negates each function, an involution:
As a result, both functions are invariant under translation by an even-Gaussian-integer multiple of. That is, a displacement with for integers,, and .
This makes them elliptic functions with a diagonal square period lattice of fundamental periods and. 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 function has simple zeros at Gaussian integer multiples of, complex numbers of the form for integers and. It has simple poles at Gaussian half-integer multiples of, complex numbers of the form, with residues. The function is reflected and offset from the 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.
There are also infinite series reflecting the distribution of the zeros and poles of :

Pythagorean-like identity

The lemniscate functions satisfy a Pythagorean-like identity:
As a result, the parametric equation parametrizes the quartic curve
This identity can alternately be rewritten:
Defining a tangent-sum operator as gives:

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:
The derivative and Pythagorean-like identities can be used to rework the identity used by Fagano in terms of and. Defining a tangent-sum operator and tangent-difference operator the argument sum and difference identities can be expressed as:
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:
Triplication formulas:
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.

Lemnatomic polynomials

Let be the lattice
Furthermore, let,,,, , be odd, be odd, and. Then
for some coprime polynomials
and some where
and
where is any -torsion generator. 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,
The -th lemnatomic polynomial is the minimal polynomial of in. For convenience, let and. So for example, the minimal polynomial of in is
and
. Another example is
which is the minimal polynomial of in
If is prime and is positive and odd, then
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 parts of equal length, using only basic arithmetic and square roots, if and only if is of the form where is a non-negative integer and each is a distinct Fermat prime.

Relation to geometric shapes

Arc length of Bernoulli's lemniscate

, the lemniscate of Bernoulli with unit distance from its center to its furthest point, 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.
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.
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:
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
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:
Later mathematicians generalized this result. Analogously to the constructible polygons in the circle, the lemniscate can be divided into sections of equal arc length using only straightedge and compass if and only if is of the form where is a non-negative integer and each is a distinct Fermat prime. 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. Equivalently, the lemniscate can be divided into sections of equal arc length using only straightedge and compass if and only if is a power of two. 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 or its two foci.
Let. Then the -division points for are the points
where is the floor function. See [|below] for some specific values of.