Dixon elliptic functions
In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these functions satisfy the identity, as real functions they parametrize the cubic Fermat curve, just as the trigonometric functions sine and cosine parametrize the unit circle.
They were named sm and cm by Alfred Dixon in 1890, by analogy to the trigonometric functions sine and cosine and the Jacobi elliptic functions sn and cn; Göran Dillner described them earlier in 1873.
Definition
The functions sm and cm can be defined as the solutions to the initial value problem:Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:
which can also be expressed using the hypergeometric function:
Parametrization of the cubic Fermat curve
Both sm and cm have a period along the real axis of with the beta function and the gamma function:They satisfy the identity. The parametric function parametrizes the cubic Fermat curve with representing the signed area lying between the segment from the origin to, the segment from the origin to, and the Fermat curve, analogous to the relationship between the argument of the trigonometric functions and the area of a sector of the unit circle. To see why, apply Green's theorem:
Notice that the area between the and can be broken into three pieces, each of area :
Symmetries
The function has zeros at the complex-valued points for any integers and, where is a cube root of unity, . The function has zeros at the complex-valued points. Both functions have poles at the complex-valued points.On the real line,, which is analogous to.
Fundamental reflections, rotations, and translations
Both and commute with complex conjugation,Analogous to the parity of trigonometric functions, the Dixon function is invariant under turn rotations of the complex plane, and turn rotations of the domain of cause turn rotations of the codomain:
Each Dixon elliptic function is invariant under translations by the Eisenstein integers scaled by
Negation of each of and is equivalent to a translation of the other,
For translations by give
Specific values
More specific values
Sum and difference identities
The Dixon elliptic functions satisfy the argument sum and difference identities:These formulas can be used to compute the complex-valued functions in real components:
Multiple-argument identities
Argument duplication and triplication identities can be derived from the sum identity:Specific value identities
The function satisfies the identitieswhere is lemniscate cosine and is Lemniscate constant.
Power series
The and functions can be approximated for by the Taylor serieswhose coefficients satisfy the recurrence
These recurrences result in:
Relation to other elliptic functions
Weierstrass elliptic function
The equianharmonic Weierstrass elliptic function with lattice a scaling of the Eisenstein integers, can be defined as:The function solves the differential equation:
We can also write it as the inverse of the integral:
In terms of, the Dixon elliptic functions can be written:
Likewise, the Weierstrass elliptic function can be written in terms of Dixon elliptic functions:
Jacobi elliptic functions
The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley. Let,,,, and. Then, letFinally, the Dixon elliptic functions are as so:
Generalized trigonometry
Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an case, and the functions sm and cm as an case.For example, defining and the inverses of an integral:
The area in the positive quadrant under the curve is
The quartic case results in a square lattice in the complex plane, related to the lemniscate elliptic functions.