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), 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 (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 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 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
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 (that is, or ), 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 (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 :
Pythagorean-like identity
The lemniscate functions satisfy a Pythagorean-like identity: