# Dixon's elliptic functions

In mathematics, **Dixon's elliptic functions**, are two doubly periodic meromorphic functions on the complex plane that have regular hexagons as repeating units: the plane can be tiled by regular hexagons in such a way that the restriction of the function to such a hexagon is simply a shift of its restriction to any of the other hexagons. This in no way contradicts the fact that a doubly periodic meromorphic function has a fundamental region that is a parallelogram: the vertices of such a parallelogram (indeed, in this case a rectangle) may be taken to be the centers of four suitably located hexagons.

These functions are named after Alfred Cardew Dixon,[1] who introduced them in 1890.[2]

Dixon's elliptic functions are denoted sm and cm, and they satisfy the following identities:

- where and is the Beta function
- where is Weierstrass's elliptic function

## See also

## Notes and references

- van Fossen Conrad, Eric; Flajolet, Philippe (July 2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion".
*Séminaire Lotharingien de Combinatoire*.**54**: Art. B54g, 44. arXiv:math/0507268. Bibcode:2005math......7268V. MR 2223029. - Dixon, A. C. (1890). "On the doubly periodic functions arising out of the curve
*x*^{3}+*y*^{3}- 3αxy = 1".*Quarterly Journal of Pure and Applied Mathematics*.**XXIV**: 167–233.