In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.; ;
The lemniscate sine and lemniscate cosine functions, usually written with the symbols and (sometimes the symbols and or and are used instead), p. 199 used the symbols and for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. p. 316, p. 204, and p. 240. uses and . use the symbols and . Some sources use the generic letters and . use the letter for the lemniscate sine and for its derivative. are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle The circle is the unit-diameter circle centered at with polar equation the degree-2 clover under the definition from . This is not the unit-radius circle centered at the origin. Notice that the lemniscate is the degree-4 clover. 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 (quadratic) , ratio of perimeter to diameter of a circle.
As complex functions, and have a square period lattice (a multiple of the Gaussian integers) with fundamental periods 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. 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: starts from this definition and thence derives other properties of the lemniscate functions.
or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners This map was the first ever picture of a Schwarz–Christoffel mapping, in p. 113.
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,. OEIS sequence A062539 lists the lemniscate constant's decimal digits.
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:Combining the first and fourth identity gives . This identity is (incorrectly) given in p. 226, without the minus sign at the front of the right-hand side.
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 .The even Gaussian integers are the residue class of , modulo , the black squares on a checkerboard. That is, a displacement with for integers , , and .
This makes them elliptic functions (doubly periodic meromorphic functions in the complex plane) 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 : §22.12.6, §22.12.12Analogously,
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: generalizes the first of these forms.
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: §44 p. 79, §47 pp. 80–81
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: §46 p. 80
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 , , , , (where ), be odd, be odd, and . Then
for some coprime polynomials and some In fact, . 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,
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
The fourth root with the least positive principal argument is chosen. (an equivalent expression is given in the table below). Another example is
which is the minimal polynomial of (and also of ) in
If is prime and is positive and odd,The restriction to positive and odd can be dropped in . then p. 142, Example 7.29(c)
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 (if any) 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 (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. p. 200
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 .And the area enclosed by is , which stands in stark contrast to the unit circle (whose enclosed area is a non-constructible number).
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:; . use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same.
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 §22.18.E6
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 (if any) is a distinct Fermat prime.Such numbers are OEIS sequence A003401. 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 (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 -division points for are the points
where is the floor function. See below for some specific values of .
Arc length of rectangular elastica