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 hexgaons.
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 . MR 2223029.
- ^ Dixon, A. C. (1890). "On the doubly periodic functions arising out of the curve x^{3} + y^{3} = 1". Quarterly Journal of Pure and Applied Mathematics. XXIV: 167–245.