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

  1. 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.
  2. Dixon, A. C. (1890). "On the doubly periodic functions arising out of the curve x3 + y3 - 3αxy = 1". Quarterly Journal of Pure and Applied Mathematics. XXIV: 167–233.
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